Research on preserver problems on Banach algebras and related topics

Citation preview

RIMS Kôkyûroku Bessatsu B93

Research on preserver problems on Banach algebras and related topics October 25㹼27, 2021 edited by Shiho Oi

July, 2023 Research Institute for Mathematical Sciences Kyoto University, Kyoto, Japan The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed. ‫ף‬2023 by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. All rights reserved. Printed in Japan.

i

Preface This volume is the proceedings of the conference “Research on preserver problems on Banach algebras and related topics”, held from October 25th to 27th, 2021. This conference was supported by the joint research program of the Research Institute for Mathematical Sciences (RIMS) of Kyoto University. Originally planned as an in-person event at RIMS, the conference had to be held online via Zoom due to the pandemic. Nevertheless, 23 speakers talked about their recent research concerning preserver problems and related topics. This volume comprises research papers, announcements, and survey articles. All papers in this volume have been reviewed and are in their final forms, thanks to the valuable and constructive feedback from the anonymous referees who devoted considerable time and effort to evaluating the submissions. The organizer deeply appreciates their efforts, which have greatly contributed to the quality of the proceedings. The organizer also acknowledges the patience and understanding of the authors during the review process, which may have taken longer than expected. The conference had 50 participants, 29 of whom are affiliated with institutions outside Japan. The organizer would like to expresses sincere gratitude to all the participants, invited speakers, authors, anonymous referees, and RIMS secretaries for their kind cooperation in making this conference a success. The organizer believes this volume will be a valuable resource for researchers in this fields and looks forward to further developments in the future. June, 2023 Shiho Oi

BB

S`Q;`K _2b2`+? QM T`2b2`p2` T`Q#H2Kb QM "M+? H;2#`b M/ `2Hi2/ iQTB+b k8 @ kd P+iQ#2`- kykR _AJa UPMHBM2V

P+iQ#2` k8- kykR PT2MBM; Re,k8@Re,jy

P+iQ#2` k8 Re,k8@Re,jy

wQQK

P+iQ#2` k8 Re,jy@R3,9y

wQQK

PT2MBM;

R 1p2MBM; a2bbBQM

*?B`KM, GDQb JQHM€` URe,jy@Rd,jyV _vQi`Q hMF URd,9y@R3,9yV Re,jy @ Rd,yy

PM qB;M2`Ƕb i?2Q`2K S2i2` ȑ2K`H U lMBp2`bBiv Q7 GDm#HDM V

Rd,yy @ Rd,jy

qB;M2`Ĝl?H?Q`M ivT2 i?2Q`2Kb 7Q` #BD2+iBp2 i`Mb7Q`KiBQMb #2ir22M b2ib Q7 KBMBKH i`BTQi2Mib MiQMBQ JX S2`Hi U AMbiBimiQ /2 Ji2K€iB+b UAJ:VX lMBp2`bB// /2 :`M/ V

Rd,jy @ Rd,9y Rd,9y @ R3,Ry

ě "`2F iBK2 ě LQMHBM2` "B`F?Qz@CK2b Q`i?Q;QMHBiv T`2b2`p2`b M/ i?2 qB;M2` 2[miBQM .BDM AHBȒ2pBÉ U lMBp2`bBiv Q7 w;`2# V

R3,Ry @ R3,9y

:`T? Q7 i?2 `2HiBQM BM/m+2/ #v "B`F?Qz@CK2b Q`i?Q;QMHBivX "QDM EmxK U lMBp2`bBiv Q7 S`BKQ`bF V

BBB

k LB;?i a2bbBQM

P+iQ#2` k8 ky,yy@kk,Ry

wQQK

*?B`KM, MiQMBQ JX S2`Hi Uky,yy@kR,yyV G2B GB UkR,Ry@kk,RyV ky,yy @ ky,jy

GBM2` M;H2 M/ T`HH2HBbK T`2b2`p2`b Q7 `2H Q` +QKTH2t >BH#2`i * @KQ/mH2b L;B@*?BM; qQM; U LiBQMH amM ui@b2M lMBp2`bBiv V

ky,jy @ kR,yy

h?2 Jxm`@lHK T`QT2`iv 7Q` mMB7Q`K H;2#`b PbKm >iQ`B U LBB;i lMBp2`biBv V

kR,yy @ kR,Ry kR,Ry @ kR,9y

ě "`2F iBK2 ě hBM;H2vǶb T`Q#H2K 7Q` mMB7Q`KHv +HQb2/ 7mM+iBQM H;2#`b hF2b?B JBm` U LBB;i lMBp2`bBiv V

kR,9y @ kk,Ry

 GB7iBM; S`QT2`iv 7Q` "M+? aT+2b 62`MM/ "Qi2H?Q U lMBp2`bBiv Q7 J2KT?Bb V

Bp

P+iQ#2` ke- kykR j 7i2`MQQM a2bbBQM

P+iQ#2` ke Rj,yy@R8,Ry

wQQK

*?B`KM, E2BB+?B qiM#2 URj,yy@R9,yyV hF2b?B JBm` UR9,Ry@R8,RyV Rj,yy @ Rj,jy

S`QT2`iB2b Q7 C@MQ`KH QT2`iQ`b CB 1mM G22 U a2DQM; lMBp2`bBiv V

Rj,jy @ R9,yy

HKQbi /BbDQBMiM2bb T`2b2`pBM; 7mM+iBQMHb QM "M+? HiiB+2b Q7 /Bz2`2MiB#H2 7mM+iBQMb G2B GB U LMFB lMBp2`bBiv V

R9,yy @ R9,Ry R9,Ry @ R9,9y

ě "`2F iBK2 ě am`D2+iBp2 BbQK2i`B2b QM  GBTb+?Bix bT+2 QM MHviB+ 7mM+iBQMb QM i?2 QT2M mMBi /Bb+ LQ`BQ LBr U LB?QM lMBp2`bBiv V

R9,9y @ R8,Ry

am`D2+iBp2 HBM2` BbQK2i`B2b QM mMBiH C ∗ @H;2#` pHm2/ GBTb+?Bix H;2#`b a?B?Q PB U LBB;i lMBp2`bBiv V

9 1p2MBM; a2bbBQM

P+iQ#2` ke Re,jy@R3,yy

wQQK

*?B`KM, a?B?Q PB Re,jy @ Rd,yy

k@BbQ@`2~2tBpBiv Q7 TQBMi2/ GBTb+?Bix bT+2b JQBbûb oBHH2;b@oHH2+BHHQb U lMBp2`bB// /2 *€/Bx V

Rd,yy @ Rd,jy

hQTQHQ;B+H `2~2tBpBiv Q7 BbQK2i`B2b QM H;2#`b Q7 Ki`Bt@pHm2/ GBTb+?Bix KTb J`ő /2 :€/Q` *#`2`@S/BHH U lMBp2`bBiv Q7 HK2`ő UaTBMV V CBKûM2x@o`;b MiQMBQ U lMBp2`bBiv Q7 HK2`ő UaTBMV V

Rd,jy @ R3,yy

am`D2+iBp2 BbQK2i`B2b QM i?2 "M+? H;2#` Q7 +QMiBMmQmbHv /Bz2`2MiB#H2 KTb rBi? pHm2b BM GBTb+?Bix H;2#` .BbmF2 >B`Qi U :`/mi2 a+?QQH Q7 a+B2M+2 M/ h2+?MQHQ;v- LBB;i lMBp2`bBiv V

p

8 LB;?i a2bbBQM

P+iQ#2` ke ky,yy@kk,Ry

wQQK

*?B`KM, S2i2` ȑ2K`H Uky,yy@kR,yyV "QDM EmxK UkR,Ry@kk,RyV ky,yy @ ky,jy

PM i?2 Q`/2` /2i2`KBMBM; T`QT2`iv Q7 i?2 MQ`K Q7  Em#Q@M/Q K2M BM QT2`iQ` H;2#`b GDQb JQHM€` U lMBp2`bBiv Q7 ax2;2/ M/ "m/T2bi lMBp2`bBiv Q7 h2+?MQH@ Q;v M/ 1+QMQKB+b V

ky,jy @ kR,yy

GiiB+2b Q7 T`QD2+iBQMb M/ i?2B` BbQKQ`T?BbKb JB+?Bv JQ`B U _AE1L V

kR,yy @ kR,Ry kR,Ry @ kR,9y

ě "`2F iBK2 ě  M2r Q`/2` `2HiBQM QM b2H7@/DQBMi QT2`iQ`b #b2/ QM i?2B` p`BM+2 :vƺ`;v S€H :2?û` U _Bp2`HM2- *K#`B/;2- lE V

kR,9y @ kk,Ry

GBM2` KTb T`2b2`pBM; BMM2` 7mM+iBQMb Cp/ Jb?`2;?B U GpH lMBp2`bBiv V

pB

P+iQ#2` kd- kykR e 1p2MBM; a2bbBQM

P+iQ#2` kd Re,yy@R3,98

wQQK

*?B`KM, PbKm >iQ`B URe,yy@Rd,yyV .BDM AHBȒ2pBÉ URd,98@R3,98V Re,yy @ Re,jy

Jƺ#Bmb ;v`Qp2+iQ` bT+2b M/ 7mM+iBQMH MHvbBb E2BB+?B qiM#2 U LBB;i lMBp2`bBiv V

Re,jy @ Rd,yy

 ;v`Q;`QmT QT2`iBQM QM BMp2`iB#H2 Ki`B+2b hQb?BFxm #2 U A#`FB lMBp2`bBiv V

Rd,yy @ Rd,R8 Rd,98 @ R3,R8

ě "`2F iBK2 ě AbQK2i`B2b M/ mMBi`B2b BM i?2 CQ`/M b2iiBM; J`ő *m2iQ@p2HHM2/ U lMBp2`bBiv Q7 E2Mi V

R3,R8 @ R3,98

AbQK2i`B2b Q7 qbb2`bi2BM bT+2b .€MB2H oB`Qbxi2F U H7`û/ _ûMvB AMbiBimi2 Q7 Ji?2KiB+b V

vii

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Articles Maps preserving triple transition pseudo-probabilities Antonio M. Peralta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The Mazur-Ulam property for a Banach space which satisfies a separation condition Osamu Hatori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Surjective isometries on an algebra of analytic functions with C n -boundary values Yuta Enami and Takeshi Miura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Applications of the Quotient Lifting Property Fernanda Botelho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Properties of C-normal operators Eungil Ko, Ji Eun Lee and Mee-Jung Lee . . . . . . . . . . . . . . . . . . . . . . . . . 117 Almost disjointness preserving functionals on Banach lattices of differentiable functions Guimei An, Jinxi Chen, Lei Li and Tong Liu . . . . . . . . . . . . . . . . . . . . . 125 Isometries, Jordan ∗-isomorphisms and order isomorphisms on spaces of a unital C ∗ -algebra-valued continuous maps Shiho Oi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Topological reflexivity of isometries on algebras of matrix-valued Lipschitz maps M. G. Cabrera-Padilla and A. Jim´enez-Vargas . . . . . . . . . . . . . . . . . . . . 143 Tingley’s problem for a Banach space of Lipschitz functions on the closed unit interval Daisuke Hirota and Takeshi Miura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

viii

On the norm of normal matrices Ludovick Bouthat, Javad Mashreghi and Fr´ed´eric Morneau-Gu´erin 183 M¨obius gyrovector spaces and functional analysis Keiichi Watanabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 On isometries of Wasserstein spaces Gy¨orgy P´al Geh´er, Tam´as Titkos and D´aniel Virosztek . . . . . . . . . . . 239

RIMS Kôkyûroku Bessatsu 㹀㸵㸶 Mathematical structures of integrable systems and their applications 㹀㸵㸷 Stochastic Analysis on Large Scale Interacting Systems 㹀㸶㸮 Regularity, singularity and long time behavior for partial differential equations with conservation law 㹀㸶㸯 Study of the History of Mathematics 2019 㹀㸶㸰 Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations 㹀㸶㸱 Algebraic Number Theory and Related Topics 2017 㹀㸶㸲 Inter-universal Teichmüller Theory Summit 2016 㹀㸶㸳 Study of the History of Mathematics 2020 㹀㸶㸴 Algebraic Number Theory and Related Topics 2018 㹀㸶㸵 Mathematical structures of integrable systems, its deepening and expansion 㹀㸶㸶 Harmonic Analysis and Nonlinear Partial Differential Equations 㹀㸶㸷 Study of the History of Mathematics 2021 㹀㸷㸮 Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties 㹀㸷㸯 Various aspects of integrable systems 㹀㸷㸰 Study of the History of Mathematics 2022

RIMS Kôkyûroku Bessatsu Vol. B93 䠎䠌䠎䠏ᖺ㻣᭶Ⓨ⾜ 䚷Ⓨ⾜ᡤ䚷䚷ி㒔኱Ꮫᩘ⌮ゎᯒ◊✲ᡤ 䚷༳ๅᡤ䚷㻌㻌᫂ᩥ⯋༳ๅᰴᘧ఍♫

RIMS Kˆ okyˆ uroku Bessatsu B93 (2023), 1–28

Maps preserving triple transition pseudo-probabilities By

Antonio M. Peralta∗

Abstract Let e and v be minimal tripotents in a JBW∗ -triple M . We introduce the notion of triple transition pseudo-probability from e to v as the complex number T T P (e, v) = φv (e), where φv is the unique extreme point of the closed unit ball of M∗ at which v attains its norm. In the case of two minimal projections in a von Neumann algebra, this correspond to the usual transition probability. We prove that every bijective transformation Φ preserving triple transition pseudo-probabilities between the lattices of tripotents of two atomic JBW∗ -triples M and N admits an extension to a bijective (complex) linear mapping between the socles of these JBW∗ -triples. If we additionally assume that Φ preserves orthogonality, then Φ can be extended to a surjective (complex-)linear (isometric) triple isomorphism from M onto N . In case that M and N are two spin factors or two type 1 Cartan factors we show, via techniques and results on preservers, that every bijection preserving triple transition pseudo-probabilities between the lattices of tripotents of M and N automatically preserves orthogonality, and hence admits an extension to a triple isomorphism from M onto N .

§ 1.

Introduction

The available mathematical models for quantum mechanic make use of complex Hilbert spaces to define the states of a quantum system. Given a complex Hilbert space H, the normal state space of S(H) is identified, via trace duality, with those Received April 5, 2022. Revised May 10, 2022. 2020 Mathematics Subject Classification(s): Primary 47B49, 46L60, 47N50 Secondary 81R15, 17C65 Key Words: Wigner theorem, minimal partial isometries, minimal tripotents, socle, triple transition pseudo-probability, preservers, Cartan factors, spin factors, triple isomorphism. Supported by MCIN/AEI/FEDER “Una manera de hacer Europa” project no. PGC2018-093332B-I00, Junta de Andaluc´ıa grants FQM375, A-FQM-242-UGR18 and PY20 00255, and by the IMAG–Mar´ıa de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033. ∗ Instituto de Matem´ aticas de la Universidad de Granada (IMAG). Departamento de An´ alisis Matem´ atico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

2

A.M. Peralta

positive norm-one elements (states) in the predual of the von Neumann algebra, B(H), of all bounded linear operators on H. Each observable is associated with a self-adjoint operator A ∈ B(H), and its expected value on the the system in state p is A(p) = tr(Ap), where tr(.) stands for the usual trace on B(H). The elements in S(H) are called the normal states of a quantum system associated to the Hilbert space H. The extreme points of S(H), as a convex set inside the closed unit ball of B(H)∗ , are called pure states, and they can be also identified with rank-one projections on H. The set of all rank-one projections on H will be denoted by P1 (H), while P(H) or P(B(H)) will stand for the set of all (orthogonal) projections on H. If two pure states are represented by the minimal projections p = ξ⊗ξ and q = η⊗η, with ξ and η in the unit sphere of H, according to Born’s rule, the transition probability from p to q is defined as T P (p, q) = tr(pq) = tr(pq ∗ ) = tr(qp∗ ) = |hξ, ηi|2 . Here, and along this note, for ξ in another complex Hilbert space K and η ∈ H, the symbol ξ ⊗ η will stand for the operator from H to K defined by ξ ⊗ η(ζ) := hζ, ηiξ. A bijective map Φ : P1 (H) → P1 (H) is called a symmetry transformation or a Wigner symmetry if it preserves the transition probability between minimal projections, that is, T P (Φ(p), Φ(q)) = tr(Φ(p)Φ(q)) = tr(pq) = T P (p, q), for all (p, q ∈ P1 (H)). A linear (respectively, conjugate-linear) mapping u : H → H is called a unitary (respectively, an anti-unitary) if uu∗ = u∗ u = 1. The celebrated Wigner’s theorem admits the following statement: Theorem 1.1. (Wigner theorem, [41], [36, page 12]) Let H be a complex Hilbert space. A bijective mapping Φ : P1 (H) → P1 (H) is a symmetry transformation if and only if there is an either unitary or anti-unitary operator u on H, unique up to multiplication by a unitary scalar, such that Φ(p) = upu∗ for all p ∈ P1 (H). Furthermore, the real linear (actually complex-linear or conjugate-linear) mapping T : B(H) → B(H), T (x) = uxu∗ is a ∗ -automorphism whose restriction to P1 (H) coincides with Φ. It is known (see, for example, [10, §4 and 6]) that for a complex Hilbert space H with dim(H) ≥ 3, the following mathematical models employed in the Hilbert space formulation of quantum mechanics are equivalent: (M.1) The set P of pure states on H (which algebraically corresponds to the set P1 (H)) whose automorphisms are the bijections preserving transition probabilities.

3

Maps preserving triple transition pseudo-probabilities

(M.2) The orthomodular lattice L of closed subspaces of H, or equivalently, the lattice of all projections in B(H), where the automorphisms are the bijections preserving orthogonality and order. The equivalence of these two models implies that if dim(H) ≥ 3, every bijection Φ : P(B(H)) → P(B(H)) preserving the partial ordering and orthogonality in both directions is given by a real linear ∗ -automorphism on B(H) determined either by a unitary or by an anti-unitary operator on H (cf. [10, §2.3 and Proposition 4.9]). The lattice of projections in B(H) is a subset of the strictly bigger lattice of partial isometries in B(H). We recall that an element e in B(H) is a partial isometry if ee∗ (equivalently, e∗ e) is a projection. Partial isometries are also called tripotents since an element e is a partial isometry if and only if ee∗ e = e. Let the symbol PI(H) = U(B(H)) stand for the set of all partial isometries on H. We shall write PI 1 (H) = Umin (B(H)) for the set of all rank–1 or minimal partial isometries on H. We say that e, v ∈ U(B(H)) are orthogonal if and only if {ee∗ , vv ∗ } and {e∗ e, v ∗ v} are two sets of orthogonal projections. The standard partial ordering on U(B(H)) is defined in the following terms: e ≤ u if u − e is a partial isometry orthogonal to e. L. Moln´ar seems to be the first author in considering a Wigner type theorem for bijections on the lattice of partial isometries of B(H) preserving the partial order and orthogonality in both directions. Theorem 1.2. [37, Theorem 1] Let H be a complex Hilbert space with dim(H) ≥ 3. Suppose that Φ : U(B(H)) → U (B(H)) is a bijective transformation which preserves the partial ordering and the orthogonality between partial isometries in both directions. If Φ is continuous (in the operator norm) at a single element of U(B(H)) different from 0, then Φ extends to a real-linear triple isomorphism. During the mini-symposium “Research on preserver problems on Banach algebras and related topics” held at RIMS (Research Institute for Mathematical Sciences), Kyoto University on October 25–27, 2021, the author of this note presented the following generalization of the previous theorem to the case of atomic JBW∗ -triples (i.e. JB∗ triples which are ℓ∞ -sums of Cartan factors). Theorem 1.3.

[18, Theorem 6.1] Let M =

ℓ∞ M i∈I

Ci and N =

ℓ∞ M j∈J

C˜j be atomic

JBW∗ -triples, where Ci and Cj are Cartan factors with rank ≥ 2. Suppose that Φ : U(M ) → U (N ) is a bijective transformation which preserves the partial ordering in both directions and orthogonality between tripotents. We shall additionally assume that Φ is continuous at a tripotent u = (ui )i in M with ui 6= 0 for all i (or we shall simply assume that Φ|Tu is continuous at a tripotent (ui )i in M with ui 6= 0 for all i). Then

4

A.M. Peralta

there exists a real linear triple isomorphism T : M → N such that T (w) = Φ(w) for all w ∈ U (M ). It should be remarked that the hypothesis concerning the ranks of the Cartan factors in the previous theorem cannot be relaxed (cf. [18, Remark 3.6]). Anyway, the validity of the result for rank-2 Cartan factors is undoubtedly an advantage. Back to the essence of Wigner theorem expressed in Theorem 1.1, we find the following contribution by L. Moln´ar. Theorem 1.4. [37, Theorem 2] Let Φ : Umin (B(H)) → Umin (B(H)) be a bijective mapping satisfying (1.1)

∗

tr(Φ(e) Φ(v)) = tr(e∗ v), for all e, v ∈ Umin (B(H)).

Then Φ extends to a surjective complex-linear isometry. Moreover, one of the following statements holds: (a) there exist unitaries u, w on H such that Φ(e) = uew (e ∈ Umin (B(H))); (b) there exist anti-unitaries u, w on H such that Φ(e) = ue∗ w (e ∈ Umin (B(H))). Let us observe that for each minimal partial isometry e in B(H), the functional φe (x) = tr(e∗ x) is the unique extreme point of the closed unit ball of B(H)∗ , the predual of B(H), at which e attains its norm. A similar property holds in the wider setting of JBW∗ -triples (see subsection 1.1 for details and definitions). Namely, for each minimal tripotent e in a JBW∗ -triple, M, there exists a unique pure atom (i.e. an extreme point of the closed unit ball of M∗ ) φe at which e attains its norm and the corresponding Peirce-2 projection writes in the form P2 (e)(x) = φe (x)e for all x ∈ M (cf. [19, Proposition 4]). The mapping Umin (M ) → ∂e (BM∗ ), e 7→ φe is a bijection from the set of minimal tripotents in M onto the set of pure atoms of M . Given two minimal tripotents e and v in a JBW∗ -triple M , we define the triple transition pseudo-probability from e to v as the complex number given by T T P (e, v) = φv (e). So, the hypothesis (1.1) in Theorem 1.4 is equivalent to say that Φ preserves triple transition pseudo-probabilities. In the case of B(H), the triple transition pseudoprobability between two minimal projections is precisely the usual transition probability. We shall show that this pseudo-probability is symmetric in the sense that T T P (e, v) = T T P (v, e), for every couple of minimal tripotents e, v ∈ M . We shall also see below that the triple transition pseudo-probability between any two minimal projections p and q in a von Neumann algebra W is zero if and only if p

Maps preserving triple transition pseudo-probabilities

5

and q are orthogonal (i.e. pq = 0). The same equivalence does not necessarily hold when projections are replaced ! with tripotents ! or partial isometries, for example, the partial 10 01 isometries e = and v = are not orthogonal in M2 (C), but T T P (e, v) = 0. 00 00 This is a theoretical handicap for the triple transition pseudo-probability. However, despite Theorem 1.3 above does not hold for rank-one JB∗ -triples (cf. [18, Remark 3.6]), every (non-necessarily surjective) mapping between the lattices of tripotents of two rank-one JB∗ -triples preserving triple transition pseudo-probabilities always admits an extension to a linear and isometric triple homomorphism between the corresponding JB∗ -triples (see Proposition 2.2). This will be obtained by an application of a theorem of Ding on the extension of isometries on the unit sphere of a Hilbert space [13]. In Theorem 2.3 we establish that if M and N are atomic JBW∗ -triples and Φ : Umin (M ) → Umin (N ) is a bijective transformation preserving triple transition pseudoprobabilities between the sets of minimal tripotents, then there exists a bijective (complex) linear mapping T0 from the socle of M onto the socle of N whose restriction to Umin (M ) is Φ, where the socle of a JB∗ -triple is the subspace linearly generated by its minimal tripotents. If we additionally assume that Φ preserves orthogonality, then we prove the existence of a surjective (complex-)linear (isometric) triple isomorphism from M onto N extending the mapping Φ (cf. Corollary 2.5). Due to the just commented result, the natural question is whether every bijection between the sets of minimal tripotents in two atomic JBW∗ -triples preserving triple transition pseudo-probabilities must automatically preserve orthogonality among them. The rest of the paper is devoted to present a couple of positive answers to this problem in the case of spin and type 1 Cartan factors. Section 3 is devoted to study bijections preserving triple transition pseudo-probabilities between the sets of minimal tripotents in two spin factors. We shall show that any such bijection preserves orthogonality, and hence admits an extension to a triple isomorphism between the spin factors (see Theorem 3.2). The proof is based on an remarkable results on preservers, due to J. Chmieli´ nski, asserting that a non-vanishing mapping between two inner product spaces is linear and preserves orthogonality in the Euclidean sense if and only if it is a positive multiple of a linear isometry [11, Theorem 1]. In section 4 we also establish a positive answer to the problem stated above in the case of a bijection between the sets of minimal tripotents in two type 1 Cartan factors (see Theorem 4.4). On this occasion, our arguments run closer to those given by Moln´ar in the proof of Theorem 1.4. For this purpose we shall establish a variant of several results previously explored by M. Marcus, B.N. Moyls [35], R. Westwick [40] and M. ˇ Omladiˇc and P. Semrl [38]. We concretely prove in Theorem 4.3 that for each linear

6

A.M. Peralta

bijection Φ : soc(B(H1 , K1 )) → soc(B(H2 , K2 )) preserving rank-one operators in both directions, where H1 , H2 , K1 and K2 are complex Hilbert spaces with dimensions ≥ 2, one of the next statements holds: (a) either there are bijective linear mappings u : K1 → K2 , and v : H1 → H2 such that Φ(ξ ⊗ η) = u(ξ ⊗ η)v = u(ξ) ⊗ v(η) (ξ ∈ K1 , η ∈ H1 ); (b) or there are bijective conjugate-linear mappings u : H1 → K2 , v : K1 → H2 such that Φ(ξ ⊗ η) = u(ξ ⊗ η)∗ v = u(η ⊗ ξ)v = u(η) ⊗ v(ξ) (ξ ∈ K1 , η ∈ H1 ). Let us finish this introduction with a kind of announcement or statement of intentions, it would be desirable to find a positive argument to prove that every bijection between the sets of minimal tripotents in two atomic JBW∗ -triple automatically preserves orthogonality. Perhaps a more general point of view could provide a better understanding. At the present moment it seems a open problem. Some other additional questions also arise after this first study on triple transition pseudo-probabilities.

§ 1.1.

Definitions and terminology

The model which motivated the study of C∗ -algebras is the space B(H), of all bounded linear operators on a complex Hilbert space H. Left and right weak∗ closed ideals of B(H) are precisely subspaces of the form B(H)p and pB(H), respectively, where p is a projection in B(H). These ideals are identified with subspaces of operators of the form B(p(H), H) and B(H, p(H)). However, given two complex Hilbert spaces H and K (where we can always assume that K is a closed subspace of H), the Banach space B(H, K), of all bounded linear operators from H to K, is not, in general, a C∗ subalgebra of some B(H). Despite of this handicap, B(H, K) is stable under products of the form (1.2)

{x, y, z} =

1 (xy ∗ z + zy ∗ x) (x, y, z ∈ B(H, K)). 2

-

Closed (complex) subspaces of B(H, K) which are closed for the triple product defined in (1.2) were called J∗ -algebras by L. Harris in [27, 28]. J∗ -algebras include, in particular, all C∗ -algebras, all JC∗ -algebras, all complex Hilbert spaces, and all ternary algebras of operators. Harris also proved that the open unit ball of every J∗ -algebra enjoys the interesting holomorphic property of being a bounded symmetric domain (see [27, Corollary 2]). In [7], R. Braun, W. Kaup and H. Upmeier extended Harris’ result by showing that the open unit ball of every (unital) JB∗ -algebra satisfies the same property. If the holomorphic-property ”being a bounded symmetric domain” is employed to classify the open unit balls of complex Banach spaces, the definitive result is due to W.

Maps preserving triple transition pseudo-probabilities

7

Kaup, who in his own words “introduced the concept of JB∗ -triple and showed that every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a JB∗ -triple, and in this way, the category of all bounded symmetric domains with base point is equivalent to the category of JB∗ -triples” (see [31]). A complex Banach space E is called a JB∗ -triple if it admits a continuous triple product {·, ·, ·} : E × E × E → E, which is symmetric and bilinear in the first and third variables, conjugate-linear in the middle one, and satisfies the following axioms: (a) (Jordan identity) L(a, b)L(x, y) = L(x, y)L(a, b) + L(L(a, b)x, y) − L(x, L(b, a)y) for a, b, x, y in E, where L(a, b) is the operator on E given by x 7→ {a, b, x}; (b) L(a, a) is a hermitian operator with non-negative spectrum for all a ∈ E; (c) k{a, a, a}k = kak3 for every a ∈ E. The first examples of JB∗ -triples include C∗ -algebras and B(H, K) spaces with respect to the triple product given in (1.2), the latter are known as Cartan factors of type 1. There are six different types of Cartan factors, the first one has been introduced in the previous paragraph. In order to define the next two types, let j be a conjugation (i.e. a conjugate-linear isometry or period 2) on a complex Hilbert space H. We consider a linear involution on B(H) defined by x 7→ xt := jx∗ j. Cartan factors of type 2 and 3 are the JB∗ -subtriples of B(H) of all t-skew-symmetric and t-symmetric operators, respectively. A Cartan factor of type 4, also called a spin factor, is a complex Hilbert space M provided with a conjugation x 7→ x, where the triple product and the norm are defined by (1.3)

{x, y, z} = hx, yiz + hz, yix − hx, ziy,

and (1.4)

kxk2 = hx, xi +

p

hx, xi2 − |hx, xi|2 ,

respectively (cf. [17, Chapter 3]). The Cartan factors of types 5 and 6 (also called exceptional Cartan factors) are spaces of matrices over the eight dimensional complex algebra of Cayley numbers; the type 6 consists of all 3 × 3 self-adjoint matrices and has a natural Jordan algebra structure, and the type 5 is the subtriple consisting of all 1 × 2 matrices (see [32, 27, 29] and the recent references [22, §6.3 and 6.4], [23, §3] for more details).

8

A.M. Peralta

An element e in a JB∗ -triple E is called a tripotent if {e, e, e} = e. When a C∗ algebra is regarded as a JB∗ -triple with the triple product in (1.2), tripotents and partial isometries correspond to the same elements. If we fix a tripotent e in E, we can find a decomposition of the space in terms of the eigenspaces of the operator L(e, e) which is expressed as follows: (1.5)

E = E0 (e) ⊕ E1 (e) ⊕ E2 (e),

where Ek (e) := {x ∈ E : L(e, e)x = -k2 x} is a subtriple of E called the Peirce-k subspace (k = 0, 1, 2). Peirce-k projection is the name given to the natural projection of E onto Ek (e), and it is usually denoted by Pk (e). The Peirce-2 subspace E2 (e) is a unital JB∗ -algebra with respect to the product and involution given by x ◦e y = {x, e, y} and x∗e = {e, x, e}, respectively. A tripotent e in E is called algebraically minimal (respectively, complete or algebraically maximal ) if E2 (e) = Ce 6= {0} (respectively, E0 (e) = {0}). We shall say that e is a unitary tripotent if E2 (e) = E. The symbols U(E), Umin (E), and Umax (E) will stand for the sets of all tripotents, minimal tripotents, and complete tripotents in E, respectively. A JB∗ -triple might contain no non-trivial tripotents, that is the case of the JB∗ triple C0 [0, 1] of all complex-valued continuous functions on [0, 1] vanishing at 0. However, in a JB∗ -triple E the extreme points of its closed unit ball are precisely the complete tripotents in E (cf. [7, Lemma 4.1], [33, Proposition 3.5] or [14, Corollary 4.8]). Thus, every JB∗ -triple which is also a dual Banach space contains an abundant set of tripotents. JB∗ -triples which are additionally dual Banach spaces are called JBW∗ triples. Each JBW∗ -triple admits a unique (isometric) predual and its triple product is separately weak∗ continuous (cf. [2]). A JBW∗ -triple is called atomic if it coincides with the w∗ -closure of the linear span of its minimal tripotents. A very natural example is given by B(H), where each minimal tripotent is of the form ξ ⊗ η with ξ, η in the unit sphere of H. Every Cartan factor is an atomic JBW∗ -triple. Cartan factors are enough to exhaust all possible cases since every atomic JBW∗ -triple is an ℓ∞ -sum of Cartan factors (cf. [20, Proposition 2 and Theorem E]). The notion of orthogonality between tripotents is an important concept in the theory of JB∗ -triples. Suppose e and v are two tripotents in a JB∗ -triple E. According to the standard notation (see, for example [34, 3]) we say that e is orthogonal to u (e ⊥ u in short) if {e, e, u} = 0. It is known that e ⊥ u if and only if {u, u, e} = 0 (and the latter is equivalent to any of the next statements: L(e, u) = 0; L(u, e) = 0; e ∈ E0 (u); u ∈ E0 (e) cf. [34, Lemma 3.9]). It is worth to remark that two projections p and q in a C∗ -algebra A, regarded as a JB∗ -triple, are orthogonal if and only if pq = 0

Maps preserving triple transition pseudo-probabilities

9

(that is, they are orthogonal in the usual sense). We can also speak about orthogonality for pairs of general elements in a JB∗ -triple E. We shall say that x and y in E are orthogonal (x ⊥ y in short) if L(x, y) = 0 (equivalently L(y, x) = 0, compare [9, Lemma 1.1] for several reformulations). Any two orthogonal elements a and b in JB∗ -triple E are M -orthogonal in a strict geometric sense, that is, ka + bk = max{kak, kbk} (see [19, Lemma 1.3(a)]). Building upon the relation “being orthogonal” we can define a canonical order “≤” on tripotents in E given by e ≤ u if and only if u − e is a tripotent and u − e ⊥ e. This partial ordering is precisely the order consider by L. Moln´ar in Theorem 1.2, and it provides an important tool in JB∗ -triples (see, for example, the recent papers [25, 26, 22, 23, 21, 24] where it plays an important role). The partial order in U(E) enjoys several interesting properties; for example, e ≤ u if and only if e is a projection in the JB∗ -algebras E2 (e) (cf. [3, Lemma 3.2] or [19, Corollary 1.7] or [22, Proposition 2.4]). In particular, if e and p are tripotents (i.e. partial isometries) in a C∗ -algebra A regarded as a JB∗ -triple with the triple product in (1.2) and p is a projection, the condition e ≤ p implies that e is a projection in A with e ≤ p in the usual order on projections (i.e. pe = e). A non-zero tripotent e in E is called (order ) minimal (respectively, (order ) maximal ) if 0 6= u ≤ e for a tripotent u in E implies that u = e (respectively, e ≤ u for a tripotent u in E implies that u = e). Clearly, every algebraically minimal tripotent is (order) minimal but the reciprocal implication does not necessarily hold, for example, the unit element in C[0, 1] is order minimal but not algebraically minimal. In the C∗ algebra C0 [0, 1], of all continuous functions on [0, 1] vanishing at 0, the zero tripotent is order maximal but it is not algebraically maximal. In the setting of JBW∗ -triples these pathologies do not happen, that is, in a JBW∗ -triple order and algebraic maximal (respectively, minimal) tripotents coincide (cf. [14, Corollary 4.8] and [3, Lemma 4.7]). A triple homomorphism between JB∗ -triples E and F is a linear map T : E → F such that T {a, b, c} = {T (a), T (b), T (c)} for all a, b, c ∈ E. Every triple homomorphism between JB∗ -triples is continuous [1, Lemma 1]. A triple isomorphism is a bijective triple homomorphism. Clearly, the inclusion T (U(E)) ⊆ U(F ) holds for each triple homomorphism T , while the equality T (U(E)) = U(F ) is true for every triple isomorphism T . Every injective triple homomorphism is an isometry (see [1, Lemma 1]). Actually a deep result in the theory of JB∗ -triples, established by W. Kaup in [31, Proposition 5.5], proves that a linear bijection between JB∗ -triples is a triple isomorphism if and only if it is an isometry. Therefore, each triple isomorphism T : E → F induces a surjective isometry T |U (E) : U(E) → U (F ) which preserves orthogonality and partial order in both directions. Similar arguments prove that the mappings T |Umin (E) : Umin (E) → Umin (F ) and T |Umax (E) : Umax (E) → Umax (F ) are surjective isometries.

10

A.M. Peralta

Along this note, the unit sphere of each normed space X will be denoted by SX , and we shall write T for SC .

§ 2.

Maps preserving triple transition pseudo-probabilities between minimal tripotents

As we recalled at the introduction, the transition probability between two minimal projections p = ξ ⊗ ξ and q = η ⊗ η in B(H) is given by tr(pq) = tr(pq ∗ ) = tr(qp∗ ) = |hξ, ηi|2 . Let us observe that each minimal projection p = ξ ⊗ ξ in B(H) is bi-univocally associated with a pure normal state φp ∈ B(H)∗ (i.e. an extreme point of the normal state space) at which p attains its norm. Clearly φp is identified with the pure normal state given by φ(a) = (ξ ⊗ ξ)(a) := ha(ξ), ξi = tr(ap) (a ∈ B(H)). Thus, the transition probability between p and q is given by the identity (2.1)

tr(pq) = |hξ, ηi|2 = |φp (q)|2 = |φq (p)|2 .

For each minimal partial isometry e = ξ ⊗ η in B(H), with ξ, η unitary vectors in H, there exists a unique extreme point φe of the closed unit ball of C1 (H) = B(H)∗ such that φe (e) = 1. Actually φe is defined by φe (x) := hx(ξ), ηi = tr(e∗ x) (x ∈ B(H)). Motivated by the identity in (2.1), for each couple e, v of minimal partial isometries in B(H), we define the triple transition pseudo-probability between e and v as the scalar φe (v) –this is not a real probability, since it actually takes complex values. The question is whether we can extend this definition to the wider setting of Cartan factors and atomic JBW∗ -triples. The lacking of a positive cone in general JB∗ -triples induced us to replace the lattice of projections in B(H) by the poset of tripotents in a Cartan factor or in an atomic JBW∗ -triple in our recent study on bijections preserving the partial ordering and orthogonality between the poset of two atomic JBW∗ -triples in [18]. Here we introduce the triple transition pseudo-probability between two minimal tripotents in an atomic JBW∗ -triple. To understand well the definition we need to recall some geometric properties of JBW∗ -triples. Following [19], the extreme points of the closed unit ball, BM∗ , of the predual, M∗ , of a JBW∗ -triple M are called atoms or pure atoms. We recall that the extreme points of the convex set of all positive functionals with norm ≤ 1 in the predual of a von Neumann algebra are called pure states. The symbol ∂e (BM∗ ) will stand for the set of all pure atoms of M . By [19, Proposition 4], for each minimal tripotent e in a JBW∗ -triple M there exists a unique pure atom φe satisfying P2 (e)(x) = φe (x)e for all x ∈ M . Furthermore, the mapping Umin (M ) → ∂e (BM∗ ), e 7→ φe

Maps preserving triple transition pseudo-probabilities

11

is a bijection from the set of minimal tripotents in M onto the set of pure atoms of M . We are now in a position to introduce the key notion of this note. Definition 2.1. Let e and v be minimal tripotents in a JBW∗ -triple M . We define the triple transition pseudo-probability from e to v as the complex number given by (2.2)

T T P (e, v) = φv (e).

Observe that every triple transition pseudo-probability lies in the closed unit ball of C. Formally speaking, the triple transition pseudo-probability is not a probability because it can take complex values. However, it satisfies many interesting and natural properties. For example, by [19, Lemma 2.2] we have (2.3)

T T P (v, e) = φe (v) = φv (e) = T T P (e, v),

for every e, v ∈ Umin (M ), which is naturally expressing the property of symmetry of the triple transition pseudo-probability. If p and q are two minimal projections in a von Neumann algebra W , having in mind that φp is a norm-one functional attaining its norm at p, it follows that φp is a positive normal state on W , and hence T T P (q, p) = φp (q) is a real number in the interval [0, 1] and coincides with T T P (p, q) = φq (p). Therefore the new notion of triple transition pseudo-probability agrees with the usual transition probability in the case of minimal projections. Moln´ar’s theorem [37, Theorem 2], presented as Theorem 1.4 in the introduction, can be now restated in the following terms: Let Φ : Umin (B(H)) → Umin (B(H)) be a bijective mapping preserving triple transition pseudo-probabilities. Then Φ extends to a surjective complex-linear isometry. Inspired by Moln´ar’s result, it seems natural to study the bijections preserving the triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW∗ -triples. The first unexpected conclusion appears when dealing with rank-one JB∗ -triples. Contrary to the serious obstacles affecting bijective preservers of partial ordering in both directions and orthogonality in the case of rank-one Cartan factors cf. [18, Remark 3.6]), preservers of triple transition pseudo-probabilities between sets of minimal tripotents have an excellent behaviour in the case of rank-one Cartan factors. Let us first recall that a subset S of a JB∗ -triple E is called orthogonal if 0 ∈ / S and a ⊥ b for all a, b ∈ S. The minimal cardinal number r satisfying card(S) ≤ r for every orthogonal subset S ⊆ E is called the rank of E. Spin factors have rank 2 and the exceptional Cartan factors of type 5 and 6 have ranks 2 and 3, respectively. A JB∗ -triple has finite rank if and only if it is reflexive (cf. [8, Proposition 4.5] and [12,

12

A.M. Peralta

Theorem 6] or [6, 5]). Furthermore, if E is a JB∗ -triple of rank-one, it must be reflexive and a rank-one Cartan factor, and moreover, it must be isometrically isomorphic to a complex Hilbert space (see the discussion in [5, §3] and [32, Table 1 in page 210]). The rank of a tripotent e in a JB∗ -triple E is defined as the rank of E2 (e). It is known that for each tripotent e in a Cartan factor C we have r(e) = r(C2 (e)) = n < ∞ if and only if it can be written as an orthogonal sum of n mutually orthogonal minimal tripotents in C (see, for example, [32, page 200]). The rank theory plays a fundamental role in the different solutions to Tingley’s problem in the case of compact C∗ -algebras [39] and weakly compact JB∗ -triples [15, 16], as well as to prove that every JBW∗ -triple satisfies the Mazur–Ulam property [4, 30]. In our next result we shall apply some of the techniques developed in the just quoted results. Proposition 2.2. Let Φ : Umin (E) → Umin (F ) be a transformation preserving triple transition pseudo-probabilities, that is, T T P (Φ(u), Φ(e)) = φΦ(e) (Φ(u)) = φe (u) = T T P (u, e), for all e, u ∈ Umin (E), where E and F are two rank-one JB∗ -triples. Then Φ extends to a (complex-)linear isometric triple homomorphism from E to F . Proof. As we have seen before the statement of this proposition, we can assume that E and F are two complex Hilbert spaces regarded as type 1 Cartan factors. We observe that U(E)\{0} = Umin (E) = SE , the unit sphere of E, and U(F )\{0} = Umin (F ) = SF . Since for each e ∈ SE , φe is precisely the functional given by φe (x) = hx, ei (x ∈ E), the hypothesis on Φ is equivalent to hΦ(u), Φ(e)i = hu, ei, for all e, u ∈ Umin (E) = SE . A simple computation shows that kΦ(e) − Φ(v)k2 = hΦ(e) − Φ(v), Φ(e) − Φ(v)i = hΦ(e), Φ(e)i − hΦ(v), Φ(e)i − hΦ(e), Φ(v)i + hΦ(v), Φ(v)i = he, ei − hv, ei − he, vi + hv, vi = ke − vk2 , for all e, v ∈ S(E). That is Φ : SE → SF is an isometry. Moreover, by the assumptions on Φ we also have h−Φ(e), Φ(−e)i = −hΦ(e), Φ(−e)i = −he, −ei = 1, which proves that Φ(−e) = −Φ(e), for all e ∈ SE . An application of the solution to Tingley’s problem for Hilbert spaces established by G.G. Ding in [13, Theorem 2.2]

Maps preserving triple transition pseudo-probabilities

13

guarantees the existence of a real linear isometry T : E → F whose restriction to SE is Φ. We shall finally show that T is complex linear. As before, by the assumptions on Φ, for each λ ∈ T we also have hλΦ(e), Φ(λe)i = λhΦ(e), Φ(λe)i = λhe, λei = 1, witnessing that Φ(λe) = λΦ(e), for all e ∈ SE and λ ∈ T. The rest is clear.

□

Let us note that in the previous proposition we are not assuming that Φ is injective nor surjective. It is now time to see a handicap or a limitation of the triple transition pseudoprobability. Let p and q be two minimal projections in a von Neumann algebra W . Suppose that the transition probability between p and q, as given in (2.1), is zero, that is φp (q) = 0, or equivalently, P2 (p)(q) = pqp = 0. Since 0 = pqp = (pq)(pq)∗ , we deduce that pq = qp = 0, and thus q = (1 − p)q(1 − q) = P0 (p)(q) ⊥ p. This property does not always hold when projections are replaced with tripotents or partial isometries, for example if e and v are minimal tripotents in a Cartan factor C with v ∈ C1 (e) we clearly have ! φe (v) = 0 but!e and v are not orthogonal. A simple example can be given by 10 01 e= and v = in M2 (C). However, every (real linear) triple homomorphism 00 00 between JB∗ -triples preserves orthogonality. Despite of this handicap, we can now get a first extension of every bijective transformation preserving triple transition probabilities between the sets of minimal tripotents of two atomic JBW∗ -triples to their socles. Let us first recall some structure results for atomic JBW∗ -triples. Every JBW∗ triple M decomposes as the orthogonal sum of two weak∗ -closed ideals A and N , where A is an atomic JBW∗ -triple (called the atomic part of M ) and N contains no minimal tripotents [19, Theorem 2]. Furthermore, M∗ decomposes as the ℓ1 -sum of two norm closed subspaces A∗ –the predual of A– and N∗ –the predual of N – satisfying that A∗ is the norm closure of the linear span of all pure atoms of M and the closed unit ball of N∗ contains no extreme points [19, Theorem 1]. At this stage the reader should also get some information about elementary JB∗ triples. Let Cj be a Cartan factor of type j ∈ {1, . . . , 6}. The elementary JB∗ -triple, Kj , of type j associated with Cj is defined as follows: K1 = K(H1 , H2 ); Ki = C ∩ K(H) when C is of type i = 2, 3, and Kj = Cj in the remaining cases (cf. [8]). For each elementary JB∗ -triple of type j, its bidual space is precisely a Cartan factor of j. The socle of a JB∗ -triple E, soc(E), is the (non-necessarily closed) linear subspace of E generated by all minimal tripotents in E. For example, the socle of B(H) is the subspace, F(H), of all finite rank operators, and it is not, in general, closed. If C is a

14

A.M. Peralta

Cartan factor of finite rank (or, more generally, a reflexive JB∗ -triple), every element in C can be written as a finite linear combination of mutually orthogonal minimal tripotents (see [8, Proposition 4.5 and Remark 4.6] or [5]), and thus the socle of C is the whole C–that is soc(C) = K(C) = C. For a general Cartan factor we have w∗

∥−∥

soc(C) = K(C) and K(C) = C. In an atomic JBW∗ -triple M , the symbol K(M ) will stand for the c0 -sum of the elementary JB∗ -triples associated with the Cartan factors expressing M as an ℓ∞ -sum. Theorem 2.3. Let Φ : Umin (M ) → Umin (N ) be a bijective transformation preserving triple transition pseudo-probabilities (i.e., T T P (Φ(v), Φ(e)) = φΦ(e) (Φ(v)) = φe (v) = T T P (v, e), for all e, v in Umin (M )), where M and N are atomic JBW∗ -triples. Then there exists a bijective (complex) linear mapping T0 : soc(M ) → soc(N ) whose restriction to Umin (M ) is Φ. Proof of Theorem 2.3. Clearly, the pure atoms of M and N are norming sets for m m X X K(M ) and K(N ), respectively. Let us suppose that αi ei = βj vj ∈ soc(M ), where i=1

j=1

αi , βj ∈ C and ei , vj ∈ Umin (M ). By the hypothesis on Φ, for each ψ ∈ ∂e (BN∗ ), there exists Φ(w) = w ˜ ∈ Umin (N ) (and w ∈ Umin (M )) such that ψ = ψw˜ = ψΦ(w) . It also follows from the hypotheses that ! ! m m m X X X αi ψΦ(w) (Φ(ei )) ψ αi Φ(ei ) = ψΦ(w) αi Φ(ei ) = i=1

i=1

i=1

=

m X

m X

αi ψw (ei ) = ψw

i=1

=

m X

!

αi ei

  m X = ψw  βj vj 

i=1

βj ψw (vj ) =

j=1

m X

j=1

βj ψΦ(w) (Φ(vj ))

j=1

    m m X X = ψΦ(w)  βj Φ(vj ) = ψ  βj Φ(vj ) . j=1

j=1

The arbitrariness of ψ ∈ ∂e (BM∗ ) together with the fact that the set of pure atoms of N separates the point of K(M ) imply that m X i=1

αi Φ(ei ) =

m X

βj Φ(vj ).

j=1

P  Pm m Therefore, the mapping T0 : soc(M ) → soc(N ), T0 α e = i=1 i i i=1 αi Φ(ei ) is well-defined and linear. We further know that T0 (e) = Φ(e) for all e ∈ Umin (M ).

Maps preserving triple transition pseudo-probabilities

15

We can similarly define a linear mapping R0 : soc(N ) → soc(M ) satisfying R0 (Φ(e)) = e for all e ∈ Umin (M ) and R0 = T0−1 . Therefore T0 and R0 are bijections. □ It should be remarked that, at this stage the hypotheses of the previous Theorem 2.3 do not imply, in a simple way, that the linear mapping T0 is continuous.

 PActually,  if

n e1 , . . . , en are mutually orthogonal minimal tripotents in M , we have T0 j=1 ej =

P

n

j=1 Φ(ej ) ≤ n. We cannot get a better bound without assuming orthogonality (and hence M -orthogonality) on the minimal tripotents Φ(e1 ), . . . , Φ(en ). In this line we recall next a result by F.J. Herves and J.M. Isidro from [29]. Theorem 2.4. [29, Theorem in page 199] Let E be a finite-rank JB∗ -triple, and let T : E → E be a linear mapping (continuity is not assumed). Then the following statements are equivalent: (1) T is a triple automorphism. (2) T (Umin (E)) = Umin (E) and preserves orthogonality. We establish now a hybrid version of the previous two results. Corollary 2.5. Let Φ : Umin (M ) → Umin (N ) be a bijective transformation preserving orthogonality and triple transition pseudo-probabilities (i.e. T T P (Φ(v), Φ(e)) = φΦ(e) (Φ(v)) = φe (v) = T T P (v, e), for all e, v in Umin (M )), where M and N are atomic JBW∗ -triples. Then Φ extends (uniquely) to a surjective complex-linear (isometric) triple isomorphism from M onto N . Proof. By Theorem 2.3 there exists a linear bijection T0 : soc(M ) → soc(N ) whose restriction to Umin (M ) coincides with Φ. By hypotheses, given u, v ∈ Umin (M ) with u ⊥ v, we have Φ(u) ⊥ Φ(v). Having in mind that for each x in the closed unit ball of soc(M ) there exists a finite family {enX }n of mutually orthogonal minimal tripotents + λn en and 1 = kxk = max{λn : n} (cf. [8, in M and {λn }n in R such that x = n

Remark 4.6]). It follows from the definition of T0 that ! X X X T0 (x) = T0 λn en = λn T0 (en ) = λn Φ(en ), n

n

n

and hence kT0 (x)k = kxk, because, by hypotheses, {Φ(en )}n is a family of mutually orthogonal minimal tripotents in N . Furthermore, by the previous conclusion X {T0 (x), T0 (x), T0 (x)} = λ3n {Φ(en ), Φ(en ), Φ(en )} n

=

X n

λ3n Φ(en ) =

X n

λ3n T0 (en ) = T0 ({x, x, x}),

16

A.M. Peralta

which shows that T0 is a contractive triple isomorphism from soc(M ) onto soc(N ). We can therefore find a continuous linear extension of T0 to a continuous (isometric) linear triple isomorphism from K(M ) onto K(N ) denoted by the same symbol T0 . The bitransposed mapping T0∗∗ : K(M )∗∗ = M → K(N )∗∗ = N is a triple isomorphism whose restriction to Umin (M ) coincides with Φ. This finishes the proof of the result. □ It seems a natural (and important) question to ask whether a bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents in two atomic JBW∗ -triples also preserves orthogonality. That is, whether in Corollary 2.5 the hypothesis concerning preservation of orthogonality can be relaxed. This will be answered for spin and type 1 Cartan factors along the next sections.

§ 3.

The case of spin factors

As well as the study of those maps preserving triple transition pseudo-probabilities between the sets of minimal tripotents in two rank-one JB∗ -triples deserved its own treatment in Proposition 2.2, the case of spin factors is also worth to study by itself. Let us fix a spin factor M whose inner product, involution and triple product are given by h·, ·i, x 7→ x, and {a, b, c} = ha, bic + hc, bia − ha, c¯i¯b, respectively (cf. the definition in page 7). It is usually assumed that dim(M ) ≥ 3; actually if dim(M ) = 2, the defined structure produces C ⊕∞ C, which is not a factor (cf. [32, Remark 4.3]). The real subspace MR− = {a ∈ M : a = a ¯}, of all fixed points for the involution · is a real Hilbert space with respect to the restricted inner product ha, bi = 1/n. Let E be a linear subspace generated by {1} ∪ {fn }∞ n=1 . Note that E is a K-linear subspace of C(Y, K) which is not uniformly closed. Then 1 is a strong boundary point for E, while 1 does not satisfy the condition (ii) of Proposition 2.9 If the corresponding space E is an algebra, we have the following. Corollary 2.11. Let A be a closed subalgebra of C0 (Y, K). Let x0 ∈ Y . Then the following are equivalent.

D

38

Osamu Hatori

(i) The point x0 is a strong boundary point for A, (ii) For every open neighborhood U of x0 , and ε > 0, there exists f ∈ A such that f (x0 ) = 1 = kf k∞ and |f (x)| < ε for every x ∈ Y \ U , (iii) For every open neighborhood U of x0 , and ε > 0, there exists f ∈ A such that f (x0 ) = 1 = kf k∞ and |f (x)| < ε for every x ∈ Y \ U and Ranπ (f ) = {1}, (iv) The point x0 is a weak peak point for A Proof. By Proposition 2.8 we have that (i) and (ii) are equivalent. The rest of the proof is easily followed by Proposition 2.9 □ To study subspace E in C0 (Y, K) it is usefull to consider the addition of constant. We add constant functions in a subspace of C0 (Y, K). Definition 2.12. For a locally compact Hausdorff space Y , we denote by Y∞ = Y ∪{∞} the one-point-compactification of Y . Let E be a K-linear subspace of C0 (Y, K). For f ∈ E we denote by f˙ the unique extension of f on Y ,  f (y), y ∈ Y f˙(y) = 0, y = ∞. Then f˙ is continuous on Y∞ . We denote E˙ + K = {F ∈ C(Y∞ , K) : F = f˙ + c, f ∈ E, c ∈ K}. Then E˙ + K is a K-linear subspace of C(Y∞ , K) which contains constants. We may sometimes suppose that E is a closed subspace of E˙ + K without a confusion. It is easy to see that E˙ + K separates the points of Y∞ and has no common zeros provided that E separates the points of Y and it has no common zeros. By a routine exercise we have that E˙ + K is closed in C(Y∞ , K) if E is closed in C0 (Y, K). It is also a routine exercise to see that for F ∈ E˙ + K, F = f˙ for a f ∈ E if and only if F (∞) = 0. Lemma 2.13. Suppose that E is a K-linear subspace of C0 (Y, K). If x0 ∈ Y is a strong boundary point for E, then x0 is a strong boundary point for E˙ + K. Proof. The proof is trivial and is omitted. The converse of the above lemma does not hold in general.

□

The Mazur-Ulam property and point-separation property

39

Example 2.14. Let E be the same space defined in Example 2.6. Then 1 is not a strong boundary point for E. On the other hand, 1 is a strong boundary point for E˙ + K. The converse of Lemma 2.13 holds for a closed subalgebra of C0 (Y, K). Proposition 2.15. Suppose that A is a closed subalgebra of C0 (Y, K). A point x0 ∈ Y is a strong boundary point for A if and only if it is a strong boundary point for A˙ + K. Proof. Suppose that x0 ∈ Y is a strong boundary point for A˙ + K. For any open neighborhood U of x0 in Y , U may be considered as an open neighborhood of x0 in Y∞ . Hence there exists a function F ∈ A˙ + K such that F (x0 ) = 1 = kF k∞ and |F | < 1 on Y∞ \ U . We note that ∞ 6∈ U . Hence we may suppose that |F (∞)| < 1. Put π : {z ∈ C : |z| ≤ 1} → {z ∈ C : |z| ≤ 1} by π(z) =

1 − F (∞) z − F (∞) , · 1 − F (∞) 1 − F (∞)z

z ∈ {z ∈ C : |z| ≤ 1}

if K = C. We infer that π(F (∞)) = 0 and π(1) = 1. As π is uniformly approximated by analytic polynomials (in fact, π(rz) for any 0 < r < 1 is uniformly approximated by the Taylor expansion on {z ∈ C : |z| ≤ 1}, and π(rz) → π(z) uniformly on {z ∈ C : |z| ≤ 1}) and as A˙ + C is uniformly closed algebra, we have π ◦ F ∈ A˙ + C. If K = R, then put π : [−1, 1] → [−1, 1] by   1 t − F (∞) , −1 ≤ t ≤ F (∞) 1+F (∞) π(t) = 1+F (∞) 1  t − F (∞) , F (∞) ≤ t ≤ 1. 1−F (∞)

1−F (∞)

Then π(F (∞)) = 0 and π(1) = 1. By the Weierstrass approximation theorem π is uniformly approximated by polynomials on [−1, 1]. Hence π ◦ F ∈ A˙ + R. In any case we have that π ◦ F ∈ A˙ + K, π ◦ F (∞) = 0. Thus there exists f ∈ A such that F = f˙. Then we have that kf k∞ = kF k∞ = 1 and f (x) = π ◦ F (x) = π(1) = 1. As |π(z)| < 1 for any |z| < 1, we have |π ◦ F | < 1 on Y∞ \ U . Therefore |f | < 1 on Y \ U . It follows that x0 is a strong boundary point for A. The converse statement is just Lemma 2.13 □

§ 2.3.

ˇ The Choquet boundary and the Silov boundary

The Choquet boundary was first mentioned by that name in a paper of Bishop and de Leeuw [7, p.306]. Definitions of Choquet boundary differ from case to case, although they are equivalent in the possible situation where the definitions can be applied. A

40

Osamu Hatori

definition of the Choquet boundary for K-linear subspace of C(X, K), for a compact Hausdorff space X, which contains constant functions is described by Phelps in [37, Section 6]. Let E be a K-linear subspace of C(X, K). Suppose that 1 ∈ E. The state space of E is K(E) = {ϕ ∈ E ∗ : ϕ(1) = 1 = kϕk} (cf. [37, p.27]). The Choquet boundary in [37] is defined as the set of all x ∈ Y such that the point evaluation τx is in ext(K(E)), the set of all extreme points of the state space. Since it is easy to see that T ext(K(E)) = ext(Ball(E ∗ )), the two definitions of the Choquet boundary (Definition 2.16 and Definition in [37, p.29]) are equivalent provided that Y is compact and 1 ∈ E. In section 8 of [37] Phelps describs an equivalent form of the Choquet boundary for a uniform algebra in terms of strong boundary points by referring a theorem of Bishop and de Leeuw. Browder [12, Section 2-2] exhibits a definition of the Choquet boundary for a uniform algebra in terms of mesures, which is also equivalent to that exhibited in Definition 2.16 in the case of a uniform algebra. Rao and Roy [43, p.176] defines the Choquet boundary for a uniformly closed complex linear subspace of C(X, C) which separates the points of a compact Hausdorff space X, in a similar way as our Definition 2.16. We are aware of the fact that some results in this subsection are a part of folklore, but for the sake of a self-contained exposition we have included as many proofs as possible of all the results stated. 2.3.1. Definition of the Choquet boundary. We recall the notion of the Choquet boundary for a K-subspace of C0 (Y, K) from [24, Definition 2.3.7], which was stated by Novinger [36, p.274]. See also [44]. For a K-linear subspace E of C0 (Y, K) and x ∈ Y , τx denotes the point evaluation at x, that is, τx : E → K such that τx (f ) = f (x) for f ∈ E. If E contains constants, then kτx k = 1. In general, τx ∈ Ball(E ∗ ) and kτx k needs not be 1. For example, put ¯ : f (0) = 0}|(D ¯ \ {0}), E = {f ∈ P (D) ¯ is the disk algebra on the closed unit disk D ¯ in the complex plane. Let where P (D) x ∈ D, the open disk. Then kτx k = |x| by the Schwarz lemma. We define the Choquet boundary for a K-linear subspace which needs not to be closed, not to separate the points of the underlying space, may have common zeros. Definition 2.16. Suppose that E is a K-linear subspace of C0 (Y, K). The Choquet boundary for E denoted by Ch(E) is the set of all x ∈ Y such that the point evaluation τx is in ext(Ball((E, k · k∞)∗ )), the set of all extreme points of Ball((E, k · k∞ )∗ ), where (E, k · k∞ ) denotes the normed linear space E with the uniform norm k · k∞ . Note that even if E is a Banach space with some norm other than the uniform one, we consider the space (E, k · k∞ ) to define the Choquet boundary.

The Mazur-Ulam property and point-separation property

41

2.3.2. The representing measures and the Arens-Kelley theorem revisited It is crucial for the foregoing discussion on the Choquet boundaries that measure theoretic arguments should be concerned. We begin by recalling the representing measures for bounded linear functionals. See [37]. Definition 2.17. Let E be a K-linear subspace of C0 (Y, K). Suppose that ϕ ∈ E . We say that a complex regular Borel measure m on Y is a representing measure for ϕ if Z ∗

ϕ(f ) =

f dm,

f ∈E

and kmk = kϕk, where kmk = |m|(Y ) is the total valuation of m. Existence of a representing measure for any ϕ ∈ E ∗ is as follows: let ϕ ∈ E ∗ . By the Hahn-Banach extension theorem there is Φ ∈ C0 (Y, K)∗ which extends ϕ and kϕk = kΦk. Then by the Riesz-Kakutani theorem, there exists a complex regular Borel R measure m such that Φ(f ) = f dm for every f ∈ C0 (Y, K) and kmk = kΦk. Hence m is a representing measure for ϕ. Recall that the support supp(m) of a complex regular Borel measure m is the set {x ∈ Y : |m|(G) > 0 for every open neighborhood G of x}, where |m| is the total valuation measure of m. Versions of the Arens-Kelley theorem, which characterizes extreme points of the unit ball in the dual space of a subspace of C0 (Y, K), have been obtained by a variety of authors. The Arens-Kelley theorem and the following corollary are well known, however for the sake of a self-contained exposition and for the convenience of the readers we include complete proofs. For x ∈ Y we denote the point mass at x by Dx : Dx is a complex regular Borel measure on Y such that Dx ({x}) = 1 = kDx k. The Arens-Kelley theorem . Suppose that ϕ ∈ ext Ball(C0 (Y, K)∗ ). Then there exists a unique x ∈ Y and λ ∈ T such that ϕ = λτx . The representing measure for ϕ is only λDx . Conversely, λτx is an extreme point of Ball(C0 (Y, K)∗ ) for every x ∈ Y and λ ∈ T. Proof. Let ϕ ∈ ext Ball(C0 (Y, K)∗ ) and m a representing measure for ϕ. Let y ∈ supp(m) arbitrary. Suppose that |m|(U ) = 1 for every open neighborhood U of y. By the regularity of m we infer that |m|({y}) = 1 = kmk, hence m = λDy for unimodular complex number λ. Suppose that there exists an open neighborhood U0 of y with |m|(U0 ) < 1. As y ∈ supp(m), 0 < |m|(U0 ) holds. By the definition of |m| we have X |m|(U0 ) = sup{ |m(Gj )| : Gj is a Borel set, ∪j Gj = U0 , Gi ∩ Gj = ∅ for i 6= j},

42

Osamu Hatori

hence there is Gj such that m(Gj ) 6= 0. By the (inner) regularity of m, there exists a compact subset L ⊂ Gj ⊂ U0 with 0 < |m(L)|. Also there exists an open set V with L ⊂ V ⊂ U0 such that |m|(V \ L) < |m(L)|/2. By the Urysohn’s lemma there exists f0 ∈ C0 (Y, R) such that 0 ≤ f0 ≤ 1, f0 = 1 on L, and f0 = 0 on Y \ V . Then we have 0 < |m(L)| ≤ |m|(L) ≤ |m|(V ) ≤ |m|(U0 ) < 1. Hence we also have 0 < |m|(Y \ V ) < 1. Put Z 1 gdm, ϕ1 (g) = |m|(V ) V and

1 ϕ2 (g) = |m|(Y \ V )

g ∈ C0 (Y, K)

Z gdm, Y \V

g ∈ C0 (Y, K).

As m is a representing measure for ϕ we have ϕ = |m|(V )ϕ1 + |m|(Y \ V )ϕ2 , where |m|(V ) + |m|(Y \ V ) = kmk = 1. As ϕ ∈ ext Ball(C0 (Y, K)∗ ) we have ϕ = ϕ1 = ϕ2 . Then Z 1 f0 dm = 0 ϕ(f0 ) = ϕ2 (f0 ) = |m|(Y \ V ) Y \V since f0 = 0 on Y \ V . On the other hand Z 1 1 |ϕ(f0 )| = |ϕ1 (f0 )| = f0 dm ≥ |m|(V ) V |m|(V ) ≥

Z |

f0 dm| − | L

!

Z V \L

f0 dm|

1 |m(L)| (|m(L)| − |m|(V \ L)) > > 0, |m|(V ) 2|m|(V )

hence ϕ(f0 ) 6= 0, which is a cotradiction. Suppose that m is a representing measure for λτx . Let U be an open neighborhood of x. Then by the Urysohn’s lemma there exists f ∈ C0 (Y, K) such that 0 ≤ f ≤ 1 on Y , f (x) = 1, and f = 0 on Y \ U . Since Z Z 1 = f (x) = τx (f ) = | f dm| ≤ |f |d|m| ≤ |m|(U ) ≤ 1, U

we see that supp(m) ⊂ U . As U can be arbitrary, we see that supp(m) = {x}. Thus we infer that m = λDx . Suppose conversely that x0 ∈ Y and λ0 ∈ T and ϕ0 = λ0 τx . Suppose that ϕ0 = (ϕ1 + ϕ2 )/2 for ϕj ∈ Ball(C0 (Y, K)∗ ), j = 1, 2. Let µ1 be a representing measure for ϕ1 . We prove that supp(µ1 ) = {x0 }. Suppose not. As the support of a regular measure is not empty, there exists y ∈ supp(µ1 ) \ {x0 }. Then by the Urysohn’s lemma there exists f ∈ C0 (Y, R) such that f (y) = 0 ≤ f ≤ 1 = f (x0 ). Put U = {z ∈ Y : f (z) < 1/2}. Then U is an open neighborhood of y, and 0 < |µ1 |(U ) < 1 since y ∈ supp(µ1 ). Thus Z Z 1 |ϕ1 (f )| ≤ |f |d|µ1 | + |f |d|µ1 | ≤ |µ1 |(U ) + |µ1 |(Y \ U ) < 1. 2 U Y \U

The Mazur-Ulam property and point-separation property

43

It follows that 1 = |ϕ0 (f )| ≤ (|ϕ1 (f )| + |ϕ2 (f )|)/2 < 1, which is a contradiction proving that supp(µ1 ) = {x0 }. We infer that ϕ1 = λ1 τx0 for λ1 ∈ T. In the same way we have that ϕ2 = λ2 τx0 for λ2 ∈ T. Since λ0 = ϕ0 (f ) = (ϕ1 (f ) + ϕ2 (f ))/2 = (λ1 + λ2 )/2 and |λj | = 1 for j = 0, 1, 2 we infer that λ0 = λ1 = λ2 . Thus ϕ1 = ϕ2 = ϕ0 . We concluded that ϕ0 ∈ ext Ball(C0 (Y, K)∗ ). □ Corollary 2.18. Suppose that E is a K-linear subspace of C0 (Y, K). Suppose ∗ that ϕ ∈ ext Ball(E ). Then there exist y ∈ Y and λ ∈ T such that ϕ = λτy . Proof. Put S = {φ ∈ C0 (Y, K)∗ : φ is a Hahn-Banach extension of ϕ}. Then S is a non-empty weak∗ -closed convex subset of C0 (Y, K). Then the KreinMilman theorem asserts that there exists a Φ ∈ ext(S). Then Φ is an extreme point of Ball(C0 (Y, K)∗ ). In fact, suppose that Φ = (Φ1 + Φ2 )/2 for Φ1 , Φ2 ∈ Ball(C0 (Y, K)∗ ). Then ϕ = Φ|E = (Φ1 |E + Φ2 |E)/2, and Φj |E ∈ Ball(E ∗ ) for j = 1, 2. As ϕ is an extreme point of Ball(E ∗ ), we have Φ1 |E = Φ2 |E = ϕ. Since 1 = kϕk = kΦj |Ek ≤ kΦj k = 1, we have kΦj k = 1 for j = 1, 2. Hence Φj ∈ S for j = 1, 2. As Φ is an extreme point in S, we have Φ = Φ1 = Φ2 . Thus Φ ∈ ext(C0 (Y, K)∗ ). By the Arens-Kelley theorem there exists y ∈ Y and λ ∈ T such that Φ = λDy . Thus we see that ϕ = Φ|E = λDy . □ Note that λτx needs not to be an extreme point of Ball(E ∗ ) in general. In fact, ¯ ∗ ) for the disk algebra on the closed unit disk D ¯ in the complex τ0 6∈ ext Ball(P (D) plane. 2.3.3. Relationship among three properties about a point x : (i) being a strong boundary point; (ii) being in the Choquet boundary; (iii) the representing measure for the point evaluation at x is unique. Let E be a K-linear subspace of C0 (Y, K) and x ∈ Y . We study the relationship of the following (i), (ii) and (iii) : (i) x is a strong boundary point for E, (ii) x ∈ Ch(E),

44

Osamu Hatori

(iii) the representing measure for τx is only Dx . We recall the definition of the boundary. Definition 2.19. Suppose that E is a K-linear subspace of C0 (Y, K). A subset L of Y is said to be a boundary if for each f ∈ E there exists a point x ∈ L such that |f (x)| = kf k∞ . The following may be well known, for example, [37, Proposition 6.3] states about the case where E contains 1 and Y is compact. However for the sake of a self-contained exposition and for the convenience of the readers we include many proofs as possible. Proposition 2.20. Suppose that E is a K-linear subspace of C0 (Y, K). Then the Choquet boundary Ch(E) is a boundary. Proof. Let f ∈ E. We may assume that kf k∞ = 1. Then there exists y ∈ Y such that |f (y)| = 1. Put L = {ϕ ∈ Ball(E ∗ ) : ϕ(f ) = f (y)}. As τy ∈ L, L is non-empty weak*-closed convex subset of Ball(E ∗ ). The Krein-Milman theorem asserts that there exists ϕ0 ∈ ext L. Then ϕ0 is an extreme point of Ball(E ∗ ). In fact, suppose that ϕ0 = (ϕ1 + ϕ2 )/2 for ϕ1 , ϕ2 ∈ Ball(E ∗ ). Then by 1 = |f (y)| = |ϕ1 (f ) + ϕ2 (f )|/2 ≤ (|ϕ1 (f )| + |ϕ2 (f )|)/2 ≤ 1 we have ϕ1 (f ) = ϕ2 (f ). By ϕ0 = (ϕ1 + ϕ2 )/2 we infer that f (y) = ϕ0 (f ) = ϕ1 (f ) = ϕ2 (f ). Thus ϕ1 , ϕ2 ∈ L. As ϕ0 ∈ ext L we have ϕ0 = ϕ1 = ϕ2 . Thus ϕ0 ∈ ext Ball(E ∗ ). By Corollary 2.18 there exists x ∈ Y and a unimodular complex number λ such that ¯ 0 = τx is an extreme point of Ball(E ∗ ) ϕ0 = λτx on E. Note that x ∈ Ch(E) since λϕ for λ is a unimodular complex number. We have |f (x)| = |λτx (f )| = |ϕ0 (f )| = |f (y)| = kf k. □

Proposition 2.21. Let E be a K-linear subspace of C0 (Y, K). Suppose that x ∈ Y is a strong boundary point for E. Then the representing measure for τx is only Dx Proof. Suppose that x ∈ Y is a strong boundary point for E and m is its representing measure. Let U be an open neighborhood of x. Since x is a strong boundary point, there is a function f ∈ E with f (x) = 1 = kf k∞ and |f | < 1 on Y \ U . Thus 1 = |τx (f )| ≤ kτx k ≤ 1, so we have 1 = kτx k = kmk. We prove supp (m) = {x}. Suppose contrarily that there exists y ∈ supp(m) \ {x}. Then there exists an open neighborhood V of y and an open neighborhood W of x such

The Mazur-Ulam property and point-separation property

45

that W ∩V = ∅. There exists g ∈ E such that g(x) = 1 = kgk∞ , |g| < 1 on Y \W . Since Y \ W is closed set, there exists δ > 0 such that |g| ≤ 1 − δ on Y \ W . As W ∩ V = ∅ we R have |g| ≤ 1 − δ on V . Since y ∈ supp(m) we have |m|(V ) > 0. Since τx (g) = gdm, we have Z Z 1 = |τx (g)| ≤ |g|d|m| + |g|d|m| ≤ (1 − δ)|m|(V ) + |m|(Y \ V ) < 1, Y \V

V

which is a contradiction proving that supp(m) \ {x} = ∅. As m is a regular measure, supp(m) is not empty, so supp(m) = {x}. Thus m = λDx for some unimodular complex number λ. As Z Z 1 = g(x) = τx (g) = gdm = gd(λDx ) = λg(x), we have that λ = 1 and m = Dx

□

Proposition 2.22. Let E be a K-linear subspace of C0 (Y, K). Let x ∈ Y . Suppose that the representing measure for τx is only Dx . Then x ∈ Ch(E). Proof. Suppose that x ∈ Y and the representing measure for τx is only Dx . Let τx = (ϕ1 + ϕ2 )/2 for ϕ1 , ϕ2 ∈ Ball(E ∗ ). Suppose that mj is a representing measurer for ϕj for j = 1, 2. Then (m1 + m2 )/2 is a representing measure for τx . Thus Z |τx (f )| = f d(m1 + m2 )/2 ≤ kf k∞ k(m1 + m2 )/2k for every f ∈ E. Thus 1 = kτx k ≤ k(m1 + m2 )/2k ≤ (km1 k + {m2 k)/2 = 1. Hence k(m1 + m2 )/2k = kτx k, so (m1 + m2 )/2 is the representing measure for τx . Thus Dx = (m1 + m2 )/2 and 1 = Dx ({x}) = (m1 ({x}) + m2 ({x}))/2. As |mj ({x})| ≤ 1 for j = 1, 2, we have that mj ({x}) = 1 for j = 1, 2. As kmj k = 1 for j = 1, 2 we infer that m1 = m2 = Dx and τx = ϕ1 = ϕ2 . We conclude that τx ∈ ext(Ball(E ∗ )), so x ∈ Ch(E). □ The following corollary is straightforward from Propositions 2.21, 2.22. Corollary 2.23. Suppose that E is a K-linear subspace of C0 (Y, K). Suppose that x ∈ Y is a strong boundary point. Then x ∈ Ch(E).

46

Osamu Hatori

¯ be the disk The converse of Proposition 2.22 does not hold in general. Let P (D) ¯ Let algebra on the closed unit disk D. ¯ : f (i) = if (1)}. E = {f ∈ P (D) ¯ C) which separates the points Then E is a uniformly closed C-linear subspace of C(D, ¯ and has no common zeros. In fact f (z) = z ∈ E separates the points in D ¯ and of D ¯ with y 6= 0. Put g(z) = (z − 1)(z − i). Then g ∈ E and f (y) = y 6= 0 for y ∈ D ¯ Furthermore we have the following. g(0) = i 6= 0. Thus E has no common zeros on D. ¯ : f (i) = if (1)}. Then 1 ∈ Ch(E), and Proposition 2.24. Let E = {f ∈ P (D) ¯ and −iDi are representing measures for τ1 . D1 , the point mass at the point 1 ∈ D, Proof. Suppose that τ1 = (p + q)/2 for some p, q ∈ Ball(E ∗ ). As τ1 (z) = 1, we infer that kτ1 k = 1. Since 1 = τ1 (z) = (p(z) + q(z))/2 and |p(z)| ≤ 1, |q(z)| ≤ 1 we infer that p(z) = q(z) = 1. Hence kpk = kqk = 1. Let mp be a representing measure for p and mq a representing measure for q. We show that supp(mp ) ⊂ {1, i}. Suppose not; suppose that there exists y ∈ supp(mp ) \ {1, i}. ¯ (cf. [12, Suppose that |y| < 1. As {1, i} is a peak interpolation set for P (D) ¯ such that f (1) = 1, = kf k p.111] or [27, Lemma 4.1]), there is a function f ∈ P (D) and f (i) = i. By the maximum absolute value principle for analytic functions, we infer that |f (y)| < 1. Hence there is a positive integer n such that |f 4n+1 (y)| < 1/2. As if 4n+1 (1) = f 4n+1 (i) we have f 4n+1 ∈ E. Then put h = f 4n+1 . Suppose that |y| = 1. As {1, i, y} is a peak interpolation set [12, p.111], there exists ¯ such that f (1) = 1 = kf k, h(i) = i, and |h(y)| < 1/2. h ∈ P (D) Let Uy be an open neighborhood of y such that |h| < 1/2 on Uy . Since y ∈ supp(mp ) and the measure mp is regular, we have 0 < |mp |(Uy ). Then we get Z Z 1 ¯ \ Uy ) < 1. |p(h)| ≤ | |h|d|mp | + |h|d|mp | ≤ |mp |(Uy ) + |mp |(D 2 ¯ y Uy D\U Since |q(h) ≤ 1, we get 1 = |τ1 (h)| ≤ (|p(h) + q(h)|)/2 < 1, which is a contradiction proving that supp(mp ) ⊂ {1, i}. Then we infer that there exists two complex numbers λp and µp with |λp | + |µp | = 1 such that mp = λp D1 + µp Di In the same way there exists two complex number λq and µq with |λq | + |µq | = 1 such λ +λ µ +µ that mq = λq D1 + µq Di . Hence we obtain that p 2 q D1 + p 2 q Di is a representing measure for p+q 2 = τ1 . Thus   λp + λq µp + µq λp + λq µp + µq f (1) = f (1) + f (i) = +i f (1) 2 2 2 2

The Mazur-Ulam property and point-separation property

47

for every f ∈ E. Hence (2.2)

λp + λq µp + µq λp + iµp λq + iµq +i = + = 1. 2 2 2 2

Since

λp + iµp |λp | + |µp | 1 ≤ = 2 2 2

and

λq + iµq |λq | + |µq | 1 ≤ = , 2 2 2

we infer from (2.2) that λp + iµp = 1. Thus p(f ) = λp f (1) + µp f (i) = (λp + iµp )f (1) = f (1) = τ1 (f ) for every f ∈ E. We have that τ1 = p, so τ1 ∈ ext(Ball(E ∗ )). We conclude that 1 ∈ Ch(E). It is evident that D1 and −iDi are different representing measures for τ1 . □ The strong separation condition ensures uniqueness of the representing measure for the point evaluation at a Choquet boundary point. In fact, we have Proposition 2.25. ing is equivalent.

Let E be a K-linear subspace of C0 (Y, K). Then the follow-

(i) E strongly separates the points of Ch(E), (ii) for every x ∈ Ch(E), the representing measure for τx is only Dx . Proof. We prove (i) implies (ii). Suppose that x ∈ Ch(E) and m a representing measure for τx . Note that kτx k = 1 since τx ∈ ext Ball(E ∗ ). We have kmk = 1. Let y ∈ supp(m). We prove that for every open neighborhood U of y we have |m|(U ) = 1. (If it were proved, then m = λDy for some complex number λ of modulus 1 since m is a regular measure. Then Z Z (2.3) f (x) = τx (f ) = f dm = f dλDy = λf (y) = λτy (f ) ¯ x . Since τx is an extreme point of Ball(E ∗ ), for every f ∈ E. Then we have τy = λτ τy is also an extreme point of Ball(E ∗ ). Thus y ∈ Ch(E). By the condition (i) that E strongly separates the points of Ch(E), we have from (2.3) that λ = 1 and x = y. Thus m = Dx follows.) Suppose not: suppose that there is an open neighborhood U0 of y such that |m|(U0 ) 6= 1. Then |m|(U0 ) < 1 as kmk = 1. As m is regular and y ∈ supp(m), we have 0 < |m|(U0 ). Since |m|(Y ) = kmk = 1, |m|(Y \ U0 )| > 0. Put Z 1 f dm, f ∈ E, φ1 (f ) = |m|(U0 ) U0

48

Osamu Hatori

1 φ2 (f ) = |m|(Y \ U0 )

Z f dm, Y \U0

f ∈ E.

It follows that φ1 , φ2 ∈ Ball(E ∗ ) and τx = |m|(U0 )φ1 +|m|(Y \U0 )φ2 , where m|(U0 ) > 0, |m|(Y \ U0 ) > 0 and |m|(U0 ) + |m|(Y \ U0 ) = 1. As τx ∈ ext Ball(E ∗ ), we have that τx = φ1 . In the same way we have Z 1 τx (f ) = f dm, f ∈ E |m|(V ) V for any open neighborhood V of y with V ⊂ U0 . Since f is continuous, for every ε > 0, there exists an open neighborhood Vε of y such that |f − f (y)| < ε on Vε . Hence 1 |f (y) − f (x)| ≤ |m|(Vε )

Z |f (y) − f |d|m| ≤ ε. Vε

Hence f (y) = f (x) for every f ∈ E, so τy = τx . As τx is an extreme point of Ball(E ∗ ), so is τy . Hence y ∈ Ch(E). Since E strongly separates the points of Ch(E), we have that x = y. It follows that for any y ∈ supp(m), y coincides with x, that is, supp(m) = {x}, which is a contradiction since we assume that |m|(U0 ) < 1. We conclude that |m|(U ) = 1 for any open neighborhood U of y. We prove (ii) implies (i) by reductio ad absurdum. Suppose that E does not strongly separate the points of Ch(E): there exists a pair x and y of different points in Ch(E) such that the equation |f (x)| = |f (y)| holds for every f ∈ E. As τx ∈ ext Ball(E ∗ ), kτx k = 1 holds, so there exists f0 ∈ E such that f0 (x) = 1. Then there exists a complex number λ0 with |λ0 | = 1 such that f0 (y) = λ0 f0 (x) = λ0 . For any f ∈ E with f (x) 6= 0 there exists a complex number λf of unit modulus such that f (y) = λf f (x). As (f /f (x) + f0 )(x) = 2, we have λf + λ0 = (f /f (x) + f0 )(y) = λf /f (x)+f0 (f /f (x) + f0 )(x) = 2λf /f (x)+f0 for every f ∈ E with f (x) 6= 0. As |λf | = |λ0 | = |λf /f (x)+f0 | = 1 we infer that λf = λ0 . Thus f (y) = λ0 f (x) for every f ∈ E with f (x) 6= 0. This equation also holds for f ∈ E with f (x) = 0. We conclude that f (y) = λ0 f (x) for every f ∈ E. Thus λ¯0 Dy is a representing measure for τx . As x 6= y we have at least two representing measures Dx and λ¯0 Dy for τx . □ Corollary 2.26. Let E be a K-linear subspace of C0 (Y, K). Suppose that E strongly separates the points of Ch(E). Then x ∈ Y is in the Choquet boundary if and only if the representing measure for τx is only Dx . Proof. It is straightforward from Propositions 2.22, 2.25.

□

The Mazur-Ulam property and point-separation property

49

If E is a subspace of C(X, K) which contains constants or E is a subalgebra of C0 (Y, K) which separates the points of Y , then the Choquet boundary points are characterized by the uniqueness of the representing measures for the corresponding point evaluations. Corollary 2.27. Supposes that X is a compact Hausdorff space and E is a Klinear subspace of C(X, K) which separates the points of X and contains constants. Then x ∈ Y is in the Choquet boundary if and only if the representing measure for τx is only Dx . Proof. Proposition 2.3 asserts that E strongly separates the points of X. Then by Corollary 2.26 we have the conclusion. □ Suppose that E is a subalgebra of C0 (Y, K) which separates the Corollary 2.28. points of Y . Then x ∈ Y is in the Choquet boundary if and only if the representing measure for τx is only Dx . Proof. Proposition 2.3 asserts that E strongly separates the points of X. Then by Corollary 2.26 we have the conclusion. □ We summarize the results to generalize a theorem of Bishop and de Leeuw on a characterization of the Choquet boundary for uniform algebras [37, p. 39], [12, Theorem 2.2.6] (cf. [42, Theorem 2.1], [43, Theorem 9]). Theorem 2.29. Suppose that A is a closed K-subalgebra of C0 (Y, K) which separates the points of Y and has no common zeros. Let x ∈ Y .The following are equivalent. (i) x ∈ Ch(A), (i′ ) x ∈ Ch(A˙ + K), (ii) x is a strong boundary point for A, (ii′ ) x is a strong boundary point for A˙ + K, (iii) the representing measure for the point evaluation τx on A is only Dx , (iv) there exists a pair of 0 < α < β ≤ 1 such that for every open neighborhood U of x there exists a function f ∈ A such that kf k∞ ≤ 1, |f (x)| ≥ β, and |f | < α on Y \ U, (v) for every pair of 0 < α < β ≤ 1 and for every open neighborhood U of x there exists a function f ∈ A such that kf k∞ ≤ 1, |f (x)| ≥ β, and |f | < α on Y \ U .

50

Osamu Hatori

Proof. If K = R, then by the Stone-Weierstrass theorem, A = C0 (Y, R). It follows that the conditions in (i) through (v) hold for every x ∈ Y . Suppose that K = C. (i) ↔ (iii) is just Corollary 2.28. (ii) → (i) is Corollary 2.23. (ii) ↔ (ii′ ) is Proposition 2.15. Since A˙ + C is a uniform algebra on Y∞ , (i′ ) → (ii′ ) follows from the Bishop-de Leeuw theorem (cf. [37, p.39], [12, Theorem 2.2.6]). We prove (i) → (i′ ). Let x ∈ Ch(A). To distinguish the point evaluations on A and A˙ + C, denote the point evaluation on A˙ + C by τ˙x : A˙ + C → C. The point evaluation on A is denoted τx as usual. Suppose that τ˙x = (p + q)/2, where p, q ∈ Ball((A˙ + C)∗ ). As 1 = τ˙x (1) = (p(1) + q(1))/2 and |p(1)| ≤ 1, |q(1)| ≤ 1 we infer that p(1) = q(1) = 1. Hence for each f˙ + λ ∈ A˙ + C we have τx (f ) + λ = τ˙x (f˙ + λ) = (p(f˙ + λ) + q(f˙ + λ))/2 = (p(f˙) + q(f˙))/2 + λ. Define p′ : A → C by p′ (g) = p(g) ˙ for g ∈ A, we have p′ ∈ Ball(A∗ ). In the same way q ′ can be defined and q ′ ∈ Ball(A∗ ). By the above equality we have τx = (p′ + q ′ )/2. As τx ∈ ext(Ball(A∗ )), p′ = q ′ = τx . It follows that p = q = τ˙x , proving that τ˙x ∈ ext(Ball(A˙ + C)∗ ), so x ∈ Ch(A˙ + C). ˙ ˙ We prove (i′ ) → (i). Suppose that x ∈ Ch(A+C), in other wards, τ˙x ∈ ext(Ball(A+ C)∗ ). We have already proved that (i′ ) implies (ii′ ) and (ii′ ) implies (ii). Hence x is ˙ = 1. Let ∆x a strong boundary point for A. Thus we infer that 1 = kτx k = |τ˙x |Ak ˙ We show that ∆x = τ˙x . By the denote a Hahn-Banach extension on A˙ + C of τ˙x |A. Riesz-Kakutani theorem there exists mx of a representing measure of ∆x on Y∞ . Note R ˙ = 1 and ∆x (f˙ + λ) = (f˙ + λ)dmx for f˙ + λ ∈ A˙ + C. that kmx k = k∆x k = kτ˙x |Ak R Note also that 1 = τ˙x (1) = 1dmx ensures that mx is a probability measure. As x is a strong boundary point for A, by Proposition 2.9 there exists a family {Kα } of peak T sets for A such that α Kα = {x}. Denote fα ∈ A the corresponding peaking function for Kα . By the bounded convergence theorem for the probability measure mx we get Z n 1 = ∆x (fα ) = fαn dmx → mx (Kα ) as n → ∞ since fα = 1 on Kα and |fα | < 1 on Y \Kα . We see that mx (Kα ) = 1 for every peak set Kα for A which contains x. Let U be an open neighborhood of x in Y . Although Y needs not be compact, by considering that U and Kα′ s are subsets of compact space Tn Y∞ , there exists a finite number of Kα1 , . . . , Kαn such that U ⊃ j=1 Kαj . Then Tn we have 1 = mx ( j=1 Kαj ) ≤ mx (U ) ≤ 1. As U is arbitrary open neighborhood of x, we get mx ({x}) = 1 since mx is a regular measure. Thus mx = Dx . Hence R ∆x (f˙ + λ) = f˙ + λdDx = τ˙x (f˙ + λ) for every f˙ + λ ∈ A˙ + C. We get that ∆x = τ˙x . We prove that τx ∈ ext(Ball(A∗ )). Suppose that τx = (p + q)/2 for some p, q ∈ Ball(A∗ ). Let pˇ : A˙ → C be defined as pˇ(f˙) = p(f ), f˙ ∈ A˙ and qˇ : A˙ → C be defined as

The Mazur-Ulam property and point-separation property

51

˙ Note that it is well defined since f 7→ f˙ is a bijection from A onto qˇ(f˙) = q(f ), f˙ ∈ A. ˙ Note also that A.     pˇ + qˇ p + q (f˙) = (f ) = τx (f ) = τ˙x (f˙), f ∈ A, 2 2 so τ˙x |A˙ = (ˇ p + qˇ)/2. Let p˙ and q˙ be Hahn-Banach extensions of pˇ and qˇ on A˙ + C ˙ we have respectively. As (p˙ + q)/2 ˙ is an extension of (ˇ p + qˇ)/2 = τ˙x |A, 1 = k(p + q)/2k = k(ˇ p + qˇ)/2k ≤ k(p˙ + q)/2k ˙ ≤ (kpk ˙ + kqk)/2 ˙ = 1. ˙ By the result Thus k(p˙ + q)/2k ˙ = 1, so (p˙ + q)/2 ˙ is a Hahn-Banach extension of τ˙x |A. in the previous paragraph that a Hahn-Banach extension of τ˙x |A˙ is always τ˙x we have τ˙x = (p˙ + q)/2. ˙ As τ˙x ∈ ext(Ball(A˙ + C∗ )) we have p˙ = q˙ = τ˙x . For every f ∈ A we have τ˙x (f˙) = τx (f ), p( ˙ f˙) = pˇ(f˙) = p(f ), and q( ˙ f˙) = qˇ(f˙) = q(f ) for every f ∈ A, we have τx (f ) = p(f ) = q(f ) for every f ∈ A. We conclude that p = q = τx , τx ∈ ext(Ball(A∗ )). It follows that x ∈ Ch(A). Suppose that iv) holds. We prove (ii′ ). For every open neighborhood (as a subset of Y∞ ) U of x there exists a function f˙ ∈ A˙ ⊂ A˙ + C such that kf˙k∞ ≤ 1, |f (x)| ≥ β, and |f | < α on Y∞ \ U . Then by [12, Theorem 2.3.4] we have x is a strong boundary point for A˙ + C; (ii′ ) holds. Suppose that (ii′ ) holds. We prove (iv). Let U be an open neighborhood of x. As we have already pointed out that (ii′ ) is equivalent to (ii), there exists f ∈ A such that f (x) = 1 = kf k∞ and |f | < 1 on Y \ U . As Y \ U is closed, there exists δ < 1 such that |f | < δ on Y \ U . Put α = δ and β = (1 + α)/2. Then we have that kf k = 1 = f (x) > β and |f | < α on Y \ U ; iv) holds. (v) → (iv) is trivial. We prove (ii) → (v). Suppose that x is a strong boundary point for A. Let α, β be any pair such that 0 < α < β ≤ 1. Let U be an arbitrary open neighborhood of x. Then there exists f ∈ A such that f (x) = 1 = kf k and |f | < 1 on Y \ U . As Y \ U is closed, there exists δ < 1 such that |f | < δ on Y \ U . For a sufficiently large positive integer n the inequality δ n < α holds. Thus f n ∈ A satisfies β ≤ |f n (x)| = 1 = kf n k∞ and |f n | < α on Y \ U . □ Even if E is a strongly separating K-linear subspace of C0 (Y, K), a point y ∈ Ch(E˙ + K) needs not be a point in Ch(E). Example 2.30. Let E be the space defined in Example 2.6. Then 1 6∈ Ch(E) while 1 ∈ Ch(E˙ + K). The reason is as follows. The space E is strongly separating.

52

Osamu Hatori

Hence t ∈ (0, 1] is in Ch(E) (resp. Ch(E˙ + K)) if and only if the representing measure for τt is unique. It is trivial that D1 and −D 14 are both representing measure for τ1 . Thus 1 6∈ Ch(E). On the other hand 1 is a strong boundary point for E˙ + K. Thus 1 ∈ Ch(E˙ + K). ˇ 2.3.4. The Silov boundary ˇ According to [3, p.80] the existence of Silov boundary for a subalgebra of C(X, K) ˇ which separates the points of X and contains constant is given by Silov [45]. A simple ˇ proof of the existence of the Silov boundary for a K-linear subspace of C(X, K) which separates the points of X and contains constants for a compact Hausdorff space is exhibited by Bear [5]. In [37, Proposition 6.4] Phelps showed that the closure of the ˇ ˇ Choquet boundary is the Silov boundary. For further references on Silov boundary see [3, 8, 9, 30, 43] for example. Definition 2.31. Let E be a K-linear subspace of C0 (Y, K). We say that a ˇ closed subset K of Y is a Silov boundary for E if it is the smallest closed boundary in the sense that it is a boundary for E and K ⊂ L for any closed boundary (a boundary for E which is a closed subset of Y ) L for E. Araujo and Font [3, Theorem 1] proved that the closure of the Choquet boundary ˇ for E is the Silov boundary for E if E is a K-linear subspace of C0 (Y, K) which strongly separates the points of Y . The following slightly generalizes it. Proposition 2.32. Let E be a K-linear subspace of C0 (Y, K). If E strongly separates the points of Ch(E), then the closure Ch(E) of the Choquet boundary in Y is ˇ the Silov boundary for E. Proof. Let K be a closed boundary for E. Let x ∈ Ch(E). We prove that x ∈ K. As K is a boundary, the restriction E|K of E on K is uniformly closed K-subspace of C0 (K, K). The restriction map T : E → E|K by T (f ) = f |K, f ∈ E is a bijection and an isometry. We define τ˙x : E|K → K by τ˙x (F ) = τx (T −1 (F )), F ∈ E|K. Then T ∗ ◦ τ˙x (f ) = τ˙x (T (f )) = τx (T −1 (T (f ))) = τx (f ),

f ∈ E,

so we infer that T ∗ ◦ τ˙x = τx on E. We prove that τ˙x ∈ ext(Ball(E|K)∗ ). Suppose that τ˙x = (p+q)/2 for p, q ∈ (E|K)∗ . Then T∗ ◦ p + T∗ ◦ q . τx = T ∗ ◦ τ˙x = 2 As τx ∈ ext E ∗ we have T ∗ ◦ p = T ∗ ◦ q = τx . Hence p = q = τ˙x since T ∗ is a bijection. It follows that τ˙x ∈ ext(Ball(E|K)∗ ).

The Mazur-Ulam property and point-separation property

53

By Corollary 2.18 there exist y ∈ K and λ ∈ T such that τ˙x = λτy |(E|K), where τy |(E|K) : E|K → K by (τy |(E|K))(F ) = F (y) for each F ∈ E|K. It is well defined since y ∈ K. We have (T ∗ ◦ (τy |(E|K)))(f ) = (τy |(E|K))(T (f )) = (τy |(E|K))(f |K) = f (y) = τy (f ),

f ∈ E,

so τx (f ) = (T ∗ ◦ τ˙x )(f ) = (T ∗ ◦ (λτy )|(E|K)))(f ) = λτy (f ),

f ∈ E.

Thus τx = λτy on E. It follows that Dx and λDy are representing measures for τx . By Proposition 2.25 we see that a representing measure for τx is only Dx since we assume E strongly separates the points of Ch(E). We conclude that x = y and λ = 1. Thus x ∈ K, Ch(E) ⊂ K, so the closure Ch(E) of Ch(E) is a subset of K. As Ch(E) is a ˇ boundary for E (Proposition 2.20) so is Ch(E). We conclude that Ch(E) is the Silov boundary. □ ˇ It is not always the case that the Silov boundary exists. A simple example is as follows. Example 2.33. Let E = {f ∈ C(T, C) : f (λ) = λf (1), λ ∈ T}. Then E is C-linear subspace of C(T, C) which separates the points in T. It is easy to see that Ch(E) = T. On the other hand {λ} is a closed boundary for E, for each λ ∈ T. Thus there is no smallest closed boundary for E. ˇ Even if the Silov boundary exists, it needs not coincide with the closure of the Choquet boundary. Example 2.34. Let X = [0, 1]∪{2}. Let E = {f ∈ C(X, K) : f (2) = −f (1)}. It ˇ is evident that [0, 1] is the Silov boundary and Ch(E) = X. Note that Ch(E) separates, but does not strongly separate, 1 and 2. Note also that −D1 and D2 are representing measures for τ2 Corollary 2.35. Let E be a K-linear subspace of C0 (Y, K) which separates the points of Y , If Y is compact and E contains constants, or E is a subalgebra of C0 (Y, K), ˇ boundary. then the closure Ch(E) of Ch(E) in Y is the Silov Proof. If Y is compact and E contains constants, then E strongly separates the points of Y by Proposition 2.3. If E is a subalgebra of C0 (Y, K), then by Proposition 2.3 asserts that E strongly separates the points of Y . Hence by Proposition 2.32 we have the conclusion. □ Note that the case of 1 ∈ E is described in [37, Proposition 6.4]. Note also that if Y is compact and E is a strongly separating space, then [43, Proposition 6] described the above corollary.

54

Osamu Hatori

The following is a well known example that shows the Choquet boundary needs ˇ not to be closed even if the Silov boundary exists. ¯ : f (0) = f (1)}|T, where p(D) ¯ is the disk Example 2.36. Let A = {f ∈ P (D) ¯ Then A is a uniform algebra on the unit circle T. algebra on the closed unit disk D. Then Ch(A) = T \ {1} since every point λ ∈ T \ {1} is a peak point with the peaking function (z + λ)/2 while 1 is not a peak point by the maximum absolute value principle ˇ for analytic functions. Note that the Silov boundary is T.

§ 3. § 3.1.

C-rich spaces, lush spaces and extremely C-regular spaces C-richness, lushness, the numerical index and the Mazur-Ulam property.

A C-rich subspace was introduced by Boyko, Kadets, Mart´ın and Werner [11]. Definition 3.1 ([11]). A closed K-linear subspace E of C(X, K), K = C or R, is called C-rich if for every nonempty open subset U of X and ε > 0, there exists a positive function hε of norm 1 with support inside U such that the distance from hε to E is less than ε. Suppose that X is a compact Hausdorff space without isolated points. Suppose also Tn that p1 , . . . , pn ∈ C(X, K)∗ . Then E = j=1 p−1 j (0) is a C-rich subspace of C(X, K) [11, Proposition 2.5]. Furthermore, if X is perfect, then every subspace of C(X, K) of codimension finite is C-rich since in this case C-richness is equivalent to richness [29, Proposition 1.2]. Another example is a uniform algebra. Recall that a uniform algebra A on a compact Hausdorff space X if A is a closed subalgebra of C(X, C) which contains constants and separates the points of X. Proposition 3.2. Let A be a uniform algebra on a compact Hausdorff space X ˇ and S the Silov boundary. Then A|S = {f ∈ C(S, C) : F = f on S for some F ∈ A} is a C-rich C-subspace of C(S, C). Proof. Let U be an open subset of S and ε0 > 0 arbitrary. We may suppose √that U is a proper subset of S. Let 0 < ε < min{1/2, ε0 /( 5 + 1)}. It is known that the Choquet boundary Ch(A) for A is dense in S and each point x ∈ Ch(A) is a strong boundary point [12] (cf. Theorem 2.29 and Corollary 2.35). Hence there exist p ∈ U ∩ Ch(A) and f ∈ A|S such that f (p) = 1 = kf k∞ and |f | < 1 on S \ U . Since A is closed under the multiplication, we may suppose that |f | < 1/2 on S \ U . Put ¯ : Re z ≥ 0, | Im z| ≤ ε}. ∆ = {z ∈ D

The Mazur-Ulam property and point-separation property

55

Then by the well known Carath´eodory theorem (cf. [41]) there is a homeomorphism ¯ → ∆ such that πε is analytic from D onto the interior of ∆. We may assume πε : D ¯ : |z| < 1/2}) ⊂ {z ∈ ∆ : Re z < ε}. As πε is that πε (1) = 1 and πε ({z ∈ D ¯ we have g = πε ◦ f ∈ A|S. uniformly approximated by analytic polynomials on D, Note that 0 ≤ Re g ≤ 1 on S. By Urysohn’s lemma there exists a continuous function h : S → [0, 1] such that  0, if Re g(y) ≤ ε, h(y) = 1, if Re g(y) ≥ 2ε. If y ∈ S \ U , then |f (y)| < 1/2, so Re g(y) = Re(πε ◦ f )(y) < ε. Thus hg = 0 on S \ U . As g(p) = πε ◦ f (p) = 1 we have that hg(p) = 1. As kgk∞ = g(p) = 1 = h(p) = khk∞ √ we have khgk∞ = 1. We show that khg − gk ≤ 5ε. Let y ∈ S. If Re g(y) ≥ 2ε, then h(y) = 1. Hence (hg − g)(y) = 0. Suppose that Re g(y) ≤ 2ε. As 0 ≤ h(y) ≤ 1, √ |h(y) − 1| ≤ 1. As g(y) ∈ ∆ and Re g(y) ≤ 2ε we infer that |g(y)| ≤ 5ε. Hence |(hg − g)(y)| = |g(y)||h(y) − 1| ≤

√ 5ε.

As Re g ≥ 0 on X, we have 0 ≤ h Re g ≤ 1. Put h0 = h Re g. Then h0 is a positive function of norm 1 since h0 (p) = 1. As h = 0 on S \ U , we have h0 = 0 on S \ U . Thus the support of h0 is inside of U . We have kh0 − gk∞ ≤ khg − gk∞ + khk∞ k Re g − gk∞ ≤

√ 5ε + ε < ε0 .

Thus d(h0 , A|S) < ε0

□

In the rest of the section B for a subset B ∈ C0 (Y, K) denotes the uniform closure of B. In the same way as the proof of Proposition 3.2 we see the following. Proposition 3.3. Suppose that A is a uniform algebra on a compact Hausdorff ˇ space X and S its Silov boundary. Then Re A|S is a C-rich R-subspace of C(X, R). Proof. Let U be an open subset U of S and ε0 > 0 arbitrary. In fact, a given U and ε0 > 0, h0 and g ∈ A are the same functions as in the proof of Proposition 3.2 we have kh0 − Re gk∞ ≤ ε0 . □ A lush space was introduced by Boyko, Kadets, Mart´ın and Werner [11]. Definition 3.4. Let B be a K-Banach space. Let δ > 0 and p ∈ S(B ∗ ). The slice denoted by SL(Ball(B), p, δ) is {a ∈ Ball(B) : Re p(a) > 1 − δ}.

56

Osamu Hatori

B is said to be K-lush if for every a, b ∈ S(B) and ε > 0, there exists q ∈ S(B ∗ ) such that b ∈ SL(Ball(B), q, ε) and d(a, co(TSL(Ball(B), q, ε))) < ε. where co(·) stands the convex hull. Boyko, Kadets, Mart´ın and Werner [11, Theorem 2.4] proved that a C-rich Ksubspace of C(X, K) for a compact Hausdorff space X is K-lush. Tan, Huang and Liu introduced a local-Gl-space in [50], which is a real Banach space. They proved that R-lush space is a local-GL-space [50, Example 3.6] and every local-GL-space has the Mazur-Ulam property. Let us briefly recall that a real Banach space B is GL-space if for every a ∈ S(B) and every 0 < ε < 1 there exists a p ∈ S(B ∗ ) such that d(b, SL(Ball(B), p, ε)) + d(−b, SL(Ball(B), p, ε)) < 2 + ε for all b ∈ S(B). A real Banach space B is said to be a local GL-space if for every separable subspace E of B , there exists a GL-subspace E ′ such that E ⊂ E ′ ⊂ B. Corollary 3.5. Every uniform algebra is C-lush. The uniform closure of the real part of a uniform algebra is R-lush and consequently it has the Mazur-Ulam property. Before proving Corollary 3.5 we show two lemmas to prove it. Lemma 3.6. Let A be a uniform algebra on a compact Hausdorff space X and S a boundary for A. Then k Re f k∞(X) = k Re f k∞(S) for every f ∈ A. Proof. Suppose that k Re f0 k∞(X) 6= | Re f0 k∞(S) for some f0 ∈ A, whence we have k Re f0 k∞(X) > | Re f0 k∞(S) . There exists y0 ∈ X such that | Re f0 (y0 )| = k Re f0 k∞(X) . We may assume Re f0 (y0 ) > 0. (If Re f0 (y0 ) < 0, then replace f0 by −f0 .) Hence Re f0 (y0 ) > k Re f0 k∞(S) , and k exp Re f0 k∞(X) ≥ exp Re f0 (y0 ) > exp k Re f0 k∞(S) ≥ k exp Re f0 k∞(S) . As | exp f0 (y)| = exp Re f0 (y) for every y ∈ X, we have k exp f0 k∞(X) = k exp Re f0 k∞(X) and k exp f0 k∞(S) = k exp Re f0 k∞(S) . Hence we get k exp f0 k∞(X) > k exp f0 k∞(S) , which is against that S is a boundary and exp f0 ∈ A. Thus we have that k Re f k∞(X) = k Re f k∞(S) for every f ∈ A.

□

The Mazur-Ulam property and point-separation property

57

Lemma 3.7. Let A be a uniform algebra on a compact Hausdorff space X and S a boundary for A. Then we have Re A|S = Re A|S. Proof. Since the inclusion Re A|S ⊃ Re A|S is obvious, we need to prove the reverse inclusion. Let u ∈ Re A|S arbitrary. Then there exists a sequence {un } in Re A|S such that kun − uk∞(S) → 0 as n → ∞. Then by the axion of choice there is a sequence {Un } in Re A such that Un |S = un for every positive integer n. Then we have by Lemma 3.6 that kun − um k∞(S) = kUn − Um k∞(X) for every n and m. Hence {Un } is a Cauchy sequence since so is the sequence {un }. There is U ∈ Re A such that kUn − U k∞(X) → 0 as n → ∞. Then kun − U |Sk∞(S) ≤ kUn − U k∞(X) → 0 as n → ∞, so that u = U |S ∈ Re A|S. We conclude that Re A|S ⊂ Re A|S.

□

ˇ Proof of Corollary 3.5 . Let A be a uniform algebra on X and S the Silov boundary for A. Then by Proposition 3.2, A|S is C-rich C-subspace of C(S, C). Then by [11, Theorem 2.4] we see that A|S is C-lush. Note that lushness is invariant under the isometries by definition of lushness. Therefore A is C-lush since S is a closed boundary for A and the restriction map is obviously an isometry of A onto A|S by definition of a boundary. By Proposition 3.3 Re A|S is C-rich. Then [11, Theorem 2.4] ensures that Re A|S ˇ is R-lush. As the Silov boundary is a boundary, we have Re A|S = Re A|S by Lemma 3.7. Hence Re A|S is R-lush. Consider the restriction map I : Re A → Re A|S. As k Re f k∞(X) = k Re f k∞(S) for every f ∈ A, the map I is a surjective isometry. Since lushness is invariant under the isometries, we see that Re A is R-lush. Tan, Huang and Liu proved that R-lush space is a local-GL-space [50, Example 3.6] and every local-GLspace has the Mazur-Ulam property [50, Theorem 3.8]. Hence Re A has the Mazur-Ulam property. □

Recall that a uniform algebra A on a compact Hausdorff space X is a Dirichlet algebra provided that Re A = C(X, R). Several uniform algebras including the disk algebra on the unit circle is a Dirichlet algebra. Uniform algebras needs not be Dirichlet in many cases. The ball algebra and the polydisk algebra on the ball and the polyˇ disk of dimension 2 or greater are not Dirichlet algebras even on the Silov boundaries respectively. For further information see [12, 25, 46]. Let ¯ R) : u is harmonic on D}. E = {u ∈ C(D, By solving the Dirichlet problem, any real-valued continuous function on T is extended ¯ which is harmonic on D. Hence E|T = C(T, R) and E is to a continuous function on D

58

Osamu Hatori

isometric to E|T since every function in E takes the maximum value and the minimum value on T as it is harmonic on D. By a theorem of Fang and Wang [23, Theorem 3.2] the Banach space C(T, R) has the Mazur-Ulam property, hence E has the Mazur-Ulam ¯ be the disk algebra property. This is also proved by Corollary 3.5 as follows. Let A(D) ¯ It is trivial that Re A(D) ¯ ⊂ E. Since the uniform limit of a on the closed unit disk D. sequence of harmonic functions are harmonic, we infer that E is uniformly closed. Thus ¯ ¯ ⊂ E. Conversely suppose that U ∈ E. As the restriction A(D)|T we have Re A(D) (the disk algebra on the unit circle) is a Dirichlet algebra on T. There is a sequence {Un } ¯ such that kUn |T − U |Tk∞(T) → 0 as n → ∞. As U and every of functions in Re A(D) Un are harmonic on the open unit disk D, we have by the maximum value principle of harmonic functions that kUn − U k∞(D) ¯ = kUn |T − U |Tk∞(T) → 0 ¯ ¯ = E. By as n → ∞. We have proved that U ∈ Re A(D). It follows that Re A(D) Corollary 3.5 we have that E has the Mazur-Ulam property. In general we have the following. Corollary 3.8. Suppose that A(Ω) is a uniform algebra on a compact subset Ω of the complex plane C which consists of complex-valued continuous functions on Ω which is analytic on the interior of Ω. Then every function in Re A(Ω) is harmonic on the interior of Ω. In particular, Re A(Ω) has the Mazur-Ulam property. Proof. The uniform limit of a sequence of harmonic functions is harmonic. Hence every function in Re A(Ω) is harmonic on the interior of Ω. By Corollary 3.5 we have the conclusion. □ It seems not to be known if a C-lush space has the complex Mazur-Ulam property or not. We proved that a uniform algebra has the complex Mazur-Ulam property in [26]. According to [22] the numerical index of a Banach space was introduced by Lumer in 1968. For the algebra of all bounded linear operators L(B) on a Banach space B, the numerical index is n(B) = inf{ν(A) : A ∈ L(B), kAk = 1}, where ν(A) is the numerical radius given by ν(A) = sup{|p(A(a))| : a ∈ S(B), p ∈ S(B ∗ ), p(a) = 1}. Boyko, Kadets, Mart´ın and Werner [11, Proposition 2.2] showed that the numetrical index of a lush space is 1. Hence we see that

The Mazur-Ulam property and point-separation property

59

Corollary 3.9. Let A be a uniform algebra. Then n(A) = n(Re A) = 1. Let ¯ E = {u ∈ C(D, R) : u is harmonic on D}. Then n(E) = 1. Proof. By Corollary 3.5 A is C-lush and Re A is R-lush. By Proposition 2.2 in [11] we have the conclusion. ¯ where A(D) ¯ is As we have shown (just before Corollary 3.8) that E = Re A(D), the disk algebra on the closed unit disk. Hence we have n(E) = 1. □

§ 3.2.

Extremely regular spaces and extremely C-regular spaces.

The concept of an extremely regular space was given by Cengiz [16]. Extremely C-regular spaces were introduced by Fleming and Jamison [24, Definition 2.3.9]. Definition 3.10. A K-linear subspace E of C0 (Y, K) is said to be extremely Cregular (resp. regular) if for each x in the Choquet boundary1 Ch(E) (resp. x ∈ Y ) satisfies the condition that for each ε > 0 and each open neighborhood U of x, there exists f ∈ E such that f (x) = 1 = kf k∞ , and |f | < ε on Y \ U . We may say that a K-linear subspace E of C0 (Y, K) is extremely C-regular if every point in the Choquet boundary is a strong boundary point in the sense of Fleming and Jamison. Suppose that m is a complex regular Borel continuous measure on Y . Then E = R {f ∈ C0 (Y, C) : f dm = 0} is an extremely regular closed subspace of C0 (Y, C) (see [16, Theorem]). Theorem 3.11. Suppose that E is a uniformly closed extremely C-regular Klinear subspace of C0 (Y, K). The following are equivalent. (i) x ∈ Ch(E), (ii) x is a strong boundary point for E, (iii) the representing measure for the point evaluation τx on E is only Dx , Proof. Since E is extremely C-regular, E strongly separates the points in Ch(E). By Corollary 2.26 we have that (i) ↔ (iii). The implication (i) → (ii) also follows from the definition of the extremely C-regularity. Suppose that (ii) holds. Then by Corollary 2.23 (i) holds. □ Abrahamsen, Nygaard and P˜oldvere [1] introduced a somewhat regular subspaces of C0 (Y, K), which is a generalization of extremely regular subspaces. 1 The

definition is given in Definition 2.16

60

Osamu Hatori

Definition 3.12 (Definition 2.1 in [1]). We call a K-linear subspace E of C0 (Y, K) somewhat regular, if for every non-empty open subset V of Y and 0 < ε, there exists f ∈ E such that there exists x0 ∈ V with f (x0 ) = 1 = kf k∞ and |f | ≤ ε on Y \ V . Proposition 3.13. A closed subalgebra A of C0 (Y, C) which separates the points of Y and has no common zeros is an extremely C-regular subspace of C0 (Y, C). In particular, a uniform algebra on a compact Hausdorff space X is an extremely C-regular subspace of C(X, C). If the Choquet boundary Ch(A) is closed in Y , then A| Ch(A) is ˇ an extremely regular subspace of C0 (Ch(A), C). Let S be the Silov boundary for A. Then A|S is a somewhat regular subspace of C0 (S, C). Proof. Let x ∈ Ch(A). Suppose that U is an open neighborhood of x. Letting β = 1 and ε = α for (v) of Theorem 2.29 we assert that there exists g ∈ A such that |g(x)| = 1 = kgk∞ and |g| < ϵ on Y \ U . Then f = g(x)g is the required function which proves that A is extremely C-regular. If Ch(A) is closed, then by the definition of extreme regularity, we have that A| Ch(A) is an extremely regular subspace. ˇ Let S be a Silov boundary. We prove that A|S is somewhat regular. Let V be a non-empty subset of S and 1 > ε > 0 arbitrary. By Corollary 2.35 there exists x0 ∈ Ch(A) ∩ V . Then there exists f ∈ A|S such that f (x0 ) = 1 = kf0 k∞ and |f | ≤ ε. Thus A|S is somewhat regular. □ Proposition 3.14. Let A be a uniform algebra on a compact Hausdorff space X. Then the space of the real parts Re A of A is an extremely C-regular subspace of CR (X). Proof. We prove that Ch(Re A) = Ch(A). Let p ∈ Ch(A). As p is a strong boundary point (see Theorem 2.29 and preceding comments), for every open neighborhood of p and ε > 0 there exists a function f ∈ A such that f (p) = 1 = kf k∞ and |f | < ε on X \ U We asserts that (3.1) Re f (p) = 1 ≤ k Re f k∞ ≤ kf k∞ = 1 and | Re f (x)| ≤ |f (x)| < ε for every x ∈ X \ U Thus p is a strong boundary point and by Corollary 2.23 we infer that p ∈ Ch(Re A). Suppose conversely that p ∈ X \ Ch(A). Then there is a representing measure µ 6= Dp on X for the point evaluation τp . Note that µ is a probability measure (cf. [12, p.81]). Then we have Z Z Re gdµ = Re

gdµ = Re g(p)

The Mazur-Ulam property and point-separation property

61

for all g ∈ A. It means that µ is a representing measure for τp which is not Dp . By Corollary 2.27 we have that p ∈ X \ Ch(Re A). By (3.1) we see that Re A is an extremely C-regular subspace of CR (X). □ Suppose that E is an extremely C-regular subspace of C0 (Y, K) Proposition 3.15. ¯ of E is also an for a locally compact Hausdorff space Y . Then the uniformly closure E extremely C-regular subspace of C0 (Y, K). ¯ k · k∞ )∗ with Proof. Every ϕ ∈ (E, k · k∞ )∗ is uniquely extended to ϕ¯ ∈ (E, ¯ Then the map ϕ → ϕ¯ from (E, k · k∞ )∗ onto (E, ¯ k · k∞ )∗ is a surjective kϕk = kϕk. ¯ k · k∞ )∗ , we see that τy is K-linear isometry. Since τ¯y for τy ∈ (E, k · k∞ )∗ is τy ∈ (E, ¯ k · k∞ )∗ )) for y ∈ Y . Hence in ext(Ball((E, k · k∞ )∗ )) if and only if τy is in ext(Ball((E, ¯ The rest of the proof is clear. Ch(E) = Ch(E). □ Applying Propositions 3.14 and 3.15 we have Corollary 3.16. For a uniform algebra A on a compact Hausdorff space X, the uniform closure Re A of the real parts Re A of A is an extremely C-regular subspace of CR (X).

§ 3.3.

Some properties of closed subalgebras of C0 (Y, C).

In this subsection Y is an infinite locally compact Hausdorff space. Abrahamsen, Nygaard and P˜oldvere [1] showed that extremely regular spaces play a role in recent theory of Banach spaces by exhibiting that they involve the Daugavet property, the symmetric strong diameter 2 property and so on under some additional assumptions. We say B has the symmetric strong diameter 2 property (SSD2P) if for every ε > 0 and every finite collection of slices S1 , . . . , Sm , there exists xi ∈ Si for i = 1, . . . , m and y ∈ Ball(B) with kyk > 1 − ε and xi ± y ∈ Si for all i = 1, . . . , m [1, Definition 1.3]. Recall that a Banach space is almost square (ASQ) if for any finite number of x1 , . . . , xn ∈ S(B), there exists a sequence {yk } ⊂ Ball(B) such that kxi ± yk k → 1 and kyk k → 1 as k → ∞ for all 1 ≤ j ≤ n [1, definition 1.3]. Recall that a Banach space B has the Daugavet property if every rank-one operator A on B satisfies that k1 + Ak = 1 + kAk [1, Definition 1.4]. Recall that a linear surjection T : N1 → N2 for normed linear space N1 and N2 is called an ε-isometry if (1 − ε)kak ≤ kT (a)k ≤ (1 + ε)kak for every a ∈ N1 [1]. We have Corollary 3.17. Let A be a closed subalgebra of C0 (Y, C) which separates the points of Y and has no common zeros. Then we have

62

Osamu Hatori

(1) A has the SSD2P, ˇ (2) A is ASQ if the Silov boundary is non-compact, ˇ (3) A has the Daugavet property if the Silov boundary of A is perfect, (4) A contains an ε-isometric copy of c0 if 0 < ε < 1. ˇ Proof. Put A0 = A|S, where S is the Silov boundary. Then the restriction map is a surjective isometry from A onto A0 . The SSD2P, ASQ, the Daugavet property and to contain an ε-isometric copy of c0 are inherited by an isometry, so it is enough to prove results for A0 . By Proposition 3.13 that A0 is a somewhat regular. Then by [1, Theorems 2.2, 2.5, 2.6, 3.1] we have the conclusions. Note that Wojtaszczyk [57, Theorem 2] proved that the Daugavet equation (DE) : k1 + Ak = 1 + kAk holds for a weakly compact operator A on a uniform algebra on X such that the strong boundary points are dense in X and X has no isolated points. As the strong boundary point coincides with the Choquet boundary points (Theorem ˇ 2.29) and they are dense in the Silov boundary (Proposition 2.35), the hypothesis on the uniform algebra in the theorem of Wojtaszczyk can be seen that A is a uniform algebra ˇ on a perfect X, where X is the Silov boundary for A. As A and A|S are isometric, ˇ where A is a uniform algebra and S is the Silov boundary for A, Wojtaszczyk in fact proved that the Daugavet equation holds for a weakly compact operators on a uniform ˇ algebra of which Silov boundary is perfect. At the end of the section we note that C-richness implies the somewhat regularity. Proposition 3.18. Let E be a K-linear subspace of C0 (Y, K). Suppose that for every nonempty open subset U of Y and ε > 0, there exists a function hε ∈ C0 (Y, R) such that 0 ≤ hε ≤ 1 = khk∞ with support inside U and that the d(hε , E) < ε. Then E is somewhat regular. We also have (1) E has the SSD2P, (2) E is ASQ if Y is non-compact, (3) E has the Daugavet property if Y is perfect, (4) E contains an ε-isometric copy of c0 if 0 < ε < 1. In particular, if Y is compact (hence E is C-rich), then E is somewhat regular and (1), (3) and (4) hold.

□

The Mazur-Ulam property and point-separation property

63

Proof. Let U be a nonempty open subset of X and ε > 0 arbitrary. We prove that there exists f ∈ E and x0 ∈ U such that f (x0 ) = 1 = kf k and |f | ≤ ε. To prove it we may assume that ε < 1. Put ε0 = ε/(1 + ε). Then there exists h ∈ C0 (Y, R) such that 0 ≤ h ≤ 1 on Y , h = 0 on Y \ U , khk∞ = 1, and the distance between h and E is less than ε0 . Hence there exists g ∈ E such that kh − gk∞ < ε0 . Then kgk∞ > khk∞ − ε0 = 1 − ε0 . As h = 0 on Y \ U , we infer that |g| < ε0 on Y \ U . Choose x0 ∈ Y so that kgk∞ = |g(x0 )|. Put f = g/g(x0 ). Then f ∈ E, f (x0 ) = 1 = kf k∞ and |f | < ε0 /(1 − ε0 ) = ε on Y \ U . As U and ε are arbitrary, we have that E is somewhat regular. Then by [1, Theorems 2.2, 2.6, 3.1] we have the conclusion. □

§ 4. § 4.1.

Sets of representatives

Is the homogeneous extension linear?

Let B be a real or complex Banach space. Any singleton {a} of a ∈ S(B) is convex subset of S(B). Applying Zorn’s lemma, there exists a maximal convex subset of S(B) which contains {a}. Hence S(B) is a union of all maximal convex subsets of S(B). We denote the set of all maximal convex subsets of S(B) by FB . Suppose that the map T : S(B1 ) → S(B2 ) is a surjective isometry with respect to the metric induced by the norm, where B1 and B2 are both real Banach spaces or both complex Banach spaces. The homogeneous extension Te : B1 → B2 of T is defined as    kakT a , 0 6= a ∈ B1 ∥a∥ Te(a) = 0, a = 0. By the definition Te is a bijection which satisfies kTe(a)k = kak for every a ∈ B1 and it is positively homogeneous. The Tingley’s problem asks if Te is real-linear or not. In [26] we introduced a set of representatives which plays a role in study on complex Mazur-Ulam property. For the convenience of the readers we recall it here. Suppose that F ∈ FB for a real or complex Banach space B. It is well known that there exists an extreme point p in the closed unit ball Ball(B ∗ ) of the dual space B ∗ of B such that F = p−1 (1) ∩ S(B) (cf. [52, Lemma 3.3], [27, Lemma 3.1]). Recall that ext(Ball(B ∗ )) denotes the set of all extreme points of Ball(B ∗ ). Put Q = {q ∈ ext(Ball(B ∗ )) : q −1 (1) ∩ S(B) ∈ FB }. We define an equivalence relation ∼ in Q. Recall that we write T = {z ∈ C : |z| = 1} if B is a complex Banach space, where C denotes the space of all complex numbers, and T = {±1} if B is a real Banach space. In Definition 4.1 through Definition 4.4 B is a real or complex Banach space.

64

Osamu Hatori

Definition 4.1 (Definition 2.1 in [26]). Let p1 , p2 ∈ Q. We denote p1 ∼ p2 if −1 there exists γ ∈ T such that p1 (1) ∩ S(B) = (γp2 )−1 (1) ∩ S(B). Note that γp ∈ Q if γ ∈ T and p ∈ Q. It is a routine argument to show that the binary relation ∼ is an equivalence relation on Q. Definition 4.2 (Definition 2.3 in [26]). A set of all representatives with respect to the equivalence relation ∼ is simply called a set of representatives for FB . Note that a set of representatives exists due to the axiom of choice. Note also that a set of representatives P for FB is a norming family for B in the sense that kak = supp∈P |p(a)| for a ∈ B. Hence it is a uniqueness set for B. Lemma 4.3 (Lemma 2.5 in [26]). Let P be a set of representatives for FB . For F ∈ FB there exists a unique (p, λ) ∈ P × T such that F = {a ∈ S(B) : p(a) = λ}. Conversely, for (p, λ) ∈ P × T we have {a ∈ S(B) : p(a) = λ} is in FB . For each set of representatives P , Lemma 4.3 gives a bijective correspondence between FB and P × T. Definition 4.4 (Definition 2.6 in [26]). {a ∈ S(B) : q(a) = λ}. A map

For (q, λ) ∈ Q × T, we denote Fq,λ =

IB : FB → P × T is defined by IB (F ) = (p, λ) for F = Fp,λ ∈ FB . By Lemma 4.3 the map IB is well defined and bijective. An important theorem of Cheng, Dong and Tanaka states that a surjective isometry between the unit spheres of Banach spaces preserves maximal convex subsets of the unit spheres. This was first exhibited by Cheng and Dong in [15, Lemma 5.1] and a complete proof was given by Tanaka [51, Lemma 3.5]. In the following T : S(B1 ) → S(B2 ) is a surjective isometry between both real Banach spaces or both complex Banach spaces B1 and B2 . We denote by Pj a set of representatives for FBj for j = 1, 2. Applying the theorem of Cheng, Dong and Tanaka, a bijection T : FB1 → FB2 is well defined. Definition 4.5 (Definition 2.7 in [26]). The map T : FB1 → FB2 is defined by T(F ) = T (F ) for F ∈ FB1 . The map T is well defined and bijective. Put −1 Ψ = IB2 ◦ T ◦ IB : P1 × T → P2 × T. 1

Define two maps ϕ : P1 × T → P2

The Mazur-Ulam property and point-separation property

65

and τ : P1 × T → T by (4.1)

Ψ(p, λ) = (ϕ(p, λ), τ (p, λ)), (p, λ) ∈ P1 × T.

If ϕ(p, λ) = ϕ(p, λ′ ) for every p ∈ P1 and λ, λ′ ∈ T we simply write ϕ(p) instead of ϕ(p, λ). An equivalent form of (4.1) is as follows: (4.2)

T (Fp,λ ) = Fϕ(p,λ),τ (p,λ) ,

(p, λ) ∈ P1 × T.

Note that (4.3)

ϕ(p, −λ) = ϕ(p, λ), τ (p, −λ) = −τ (p, λ)

for every (p, λ) ∈ P1 × T (cf. [26]). The reason is as follows. First it is well known that T (−F ) = −T (F ) for every F ∈ FB1 (cf. [34, Proposition 2.3]). Hence Fϕ(p,−λ),τ (p,−λ) = T (Fp,−λ ) = T (−Fp,λ ) = −T (Fp,λ ) = −Fϕ(p,λ),τ (p,λ) = Fϕ(p,λ),−τ (p,λ) for every p ∈ P1 since Fp,−λ = −Fp,λ by the definition of Fp,λ . Since the map IB2 is a bijection we have (4.3). Rewriting (4.2) we get an essential equation in our argument. (4.4)

ϕ(p, λ)(T (a)) = τ (p, λ),

a ∈ Fp,λ .

Looking at this equation we will prove that the homogeneous extension Te of T is reallinear. Before describing a precise argument in the later subsections, we exhibit a rough picture of the argument. Under the Hausdorff distance condition which will be given in Definition 4.6, we have ϕ(p, λ) = ϕ(p, λ′ ), p ∈ P1 for every λ and λ′ in T, and τ (p, λ) = τ (p, 1) ×

 λ,

for some p ∈ P1

λ,

for other p’s

for λ ∈ T. We get from (4.4), under the Hausdorff distance condition on B1 , that  p(a), for some p ∈ P 1 (4.5) ϕ(p)(T (a)) = τ (p, 1) × p(a), for other p’s

66

Osamu Hatori

for a ∈ Fp,p(a) . We emphasize that (4.5) holds for a ∈ S(B1 ) with |p(a)| = 1. It is crucial to prove (4.5) for all a ∈ S(B1 ). If the equation (4.5) holds for any a ∈ S(B1 ) without the restriction that a ∈ Fp,p(a) , then applying the definition of Te we get  p(a), for some p ∈ P 1 (4.6) ϕ(p)(Te(a)) = τ (p, 1) × - - p(a), for other p’s for every a ∈ B1 , from which we infer that ϕ(p)(Te(a + rb)) = ϕ(p)(Te(a)) + ϕ(p)(rTe(b)) for every pair a, b ∈ B1 and every real number r. By a further consideration, we will conclude that Te is real-linear. It means that we will arrive at the final positive solution for Tingley’s problem if (4.5) holds for all a ∈ S(B1 ). We will apply the version of additive Bishop’s lemma (Proposition 5.5) to prove the equation (4.5) for any a ∈ S(B1 ).

§ 4.2.

The Hausdorff distance condition.

Recall that the Hausdorff distance dH (K, L) between non-empty closed subsets K and L of a metric space with metric d(·, ·) is defined by dH (K, L) = max{sup d(a, L), sup d(b, K)}. a∈K

b∈L

Definition 4.6 (Definition 3.2 in [26]). Let B be a complex Banach space and P a set of representatives for FB . We say that B satisfies the Hausdorff distance condition if the equality dH (Fp,λ , Fp′ ,λ′ ) = 2 holds for every pair (p, λ) and (p′ , λ′ ) in P × T such that p 6= p′ . By Lemma 3.1 in [26], dH (Fp,λ , Fp′ λ′ ) = 2 provided that p 6= p′ and Fp,λ ∩Fp′ ,−λ′ = 6 ∅. We can formulate the notion of the condition of the Hausdorff distance in terms of Q. Lemma 4.7. A complex Banach space B satisfies the Hausdorff distance condition if and only if dH (Fq,λ , Fq′ ,λ′ ) = 2 for every pair q and q ′ of Q with q 6∼ q ′ . A proof is a routine argument and is omitted. Lemma 4.8 (Lemma 3.4 in [26]). Let Bj be a complex Banach space for j = 1, 2 and T : S(B1 ) → S(B2 ) a surjective isometry. Suppose that B1 satisfies the Hausdorff

The Mazur-Ulam property and point-separation property

67

distance condition. Let P1 be a set of representatives for FB1 . Then we have ϕ(p, λ) = ϕ(p, λ′ ) for every p ∈ P1 and λ, λ′ ∈ T. Put P1+ = {p ∈ P1 : τ (p, i) = iτ (p, 1)} and P1− = {p ∈ P1 : τ (p, i) = ¯iτ (p, 1)}. Then P1+ and P1− are possibly empty disjoint subsets of P1 such that P1+ ∪ P1− = P1 . Furthermore we have τ (p, λ) = λτ (p, 1), p ∈ P1+ , λ ∈ T and ¯ (p, 1), τ (p, λ) = λτ

p ∈ P1− , λ ∈ T.

Proof. See the proof of [26, Lemma 3.4]

§ 4.3.

□

The set Mp,α and the Mazur-Ulam property

We exhibit the definition of Mp,α for a real or complex Banach space. The case of a ¯ = {z ∈ K : |z| ≤ 1}, where complex Banach space is in [26, Definition 4.1]. We denote D K = R if the corresponding Banach space is a real one and K = C if the corresponding Banach space is a complex one. Definition 4.9. Let B be a real or complex Banach space and P a set of repre¯ we denote sentatives for FB . For p ∈ P and α ∈ D Mp,α = {a ∈ S(B) : d(a, Fp,α/|α| ) ≤ 1 − |α|, d(a, Fp,−α/|α| ) ≤ 1 + |α|}, where we read α/|α| = 1 if α = 0. Lemma 4.10 (cf. Lemma 4.2 in [26]). Suppose that Bj is a real or complex Banach space for j = 1, 2, and T : S(B1 ) → S(B2 ) is a surjective isometry. If Bj is a real Banach space for j = 1, 2, then we have T (Mp,α ) = τ (p, 1)Mϕ(p),α for every (p, α) ∈ P1 × T. If Bj is a complex Banach space j = 1, 2 and B1 satisfies the Hausdorff distance condition, then we have  τ (p, 1)M p ∈ P1+ ϕ(p),α , T (Mp,α ) = τ (p, 1)Mϕ(p),α , p ∈ P − 1 for every (p, α) ∈ P1 × T. Here P1+ and P1− are defined as in Lemma 4.8.

68

Osamu Hatori

Proof. According to the definition of the map Ψ we have α ) = F α α T (Fp, |α| ϕ(p, |α| ),τ (p, |α| )

and α ) = F α α T (Fp,− |α| ϕ(p,− |α| ),τ (p,− |α| )

Suppose that Bj is a real Banach space. Then by the definition T = {±1}. By (4.3) we have ϕ(p, 1) = ϕ(p, −1) for every p ∈ P1 . Hence ϕ(p, λ) does not depend on the second term for a real Banach space. We also have τ (p, −1) = −τ (p, 1) for every p ∈ P1 by (4.3). It follows that α ) = F α α T (Fp, |α| τ (p,1) = τ (p, 1)Fϕ(p), |α| ϕ(p), |α|

and α ) = F α α T (Fp,− |α| ϕ(p),− |α| τ (p,1) = τ (p, 1)Fϕ(p),− |α|

As T is a surjective isometry we have α ) = d(T (a), F α ) = d(T (a), F α α d(a, Fp, |α| ),τ (p, |α| ) τ (p,1) ) ϕ(p, |α| ϕ(p), |α| α ) = d(τ (p, 1)T (a), F α ) = d(T (a), τ (p, 1)Fϕ(p), |α| ϕ(p), |α|

and α ) = d(T (a), F α α α ) = d(T (a), F d(a, Fp,− |α| ϕ(p,− |α| ϕ(p),− |α| ),τ (p,− |α| ) τ (p,1) ) α ) = d(τ (p, 1)T (a), F α ). = d(T (a), τ (p, 1)Fϕ(p),− |α| ϕ(p),− |α|

As T is a bijection we conclude that τ (p, 1)T (Mp,α ) = Mϕ(p),α ¯ so for every p ∈ P1 and α ∈ D, T (Mp,α ) = τ (p, 1)Mϕ(p),α ¯ for every p ∈ P1 and α ∈ D A proof for the case where Bj is a complex Banach space is in [26, Proof of Lemma 4.2]. □ 4.3.1. A sufficient condition for the Mazur-Ulam property : the case of a real Banach space. Proposition 4.11. Let B be a real Banach space and P a set of representatives for FB . Suppose that (4.7)

Mp,α = {a ∈ S(B) : p(a) = α}

for every p ∈ P and −1 ≤ α ≤ 1. Then B has the Mazur-Ulam property.

The Mazur-Ulam property and point-separation property

69

Proof. Let B2 be a real Banach space and T : S(B1 ) → S(B2 ) a surjective isometry. We first prove the following equation (4.8) for every p ∈ P1 and a ∈ S(B1 ) without assuming that |p(a)| = 1; (4.8)

ϕ(p)(T (a)) = τ (p, 1)p(a)

for every p ∈ P1 and a ∈ S(B1 ) with |p(a)| ≤ 1. Let p ∈ P1 and a ∈ S(B1 ). Put α = p(a). Then by (4.7) a ∈ Mp,α . We have by Lemma 4.10 that ϕ(p)(T (a)) = ατ (p, 1) = τ (p, 1)p(a). It follows that for the homogeneous extension Te of T we have      c c e ϕ(p)(T (c)) = ϕ(p) kckT = kckτ (p, 1)p = τ (p, 1)p(c) kck kck for every 0 6= c ∈ B1 . As the equality ϕ(p)(Te(0)) = τ (p, 1)p(0) holds, we obtain for a, b ∈ B1 and a real number r that ϕ(p)(Te(a + rb) = τ (p, 1)p(a + rb) = τ (p, 1)p(a) + rτ (p, 1)p(b) and ϕ(p)(Te(a) + rTe(b)) = ϕ(p)(Te(a)) + rϕ(p)(Te(b)) = τ (p, 1)p(a) + rτ (p, 1)p(b). It follows that ϕ(p)(Te(a + rb)) = ϕ(p)(Te(a) + rTe(b)) for every p ∈ P1 , a, b ∈ B1 , and every real number r. As ϕ(P1 ) = P2 is a norming family we see that Te is real-linear on B1 . As the homogeneous extension is a norm-preserving bijection as is described in the subsection 4.1 we complete the proof. □ 4.3.2. A sufficient condition for the complex Mazur-Ulam property : the case of a complex Banach space. The case of a complex Banach space is exhibited in Proposition 4.4 in [26]. Proposition 4.12 (Proposition 4.4 in [26]). Let B be a complex Banach space and P a set of representatives for FB . Assume the following two conditions: (i) B satisfies the Hausdorff distance condition, (ii) Mp,α = {a ∈ S(B) : p(a) = α} for every p ∈ P and α ∈ D. Then B has the complex Mazur-Ulam property.

70

Osamu Hatori

§ 5.

Banach spaces which satisfy the condition (∗)

Definition 5.1. Let B be a real or complex Banach space. We say that B satisfies the condition (∗) whenever there exists a set of representative P for FB with the condition : for every p ∈ P , ε > 0, and a closed subset F of P with respect to the relative topology induced by the weak*-topology on B ∗ such that p 6∈ F , there exists a ∈ S(B) such that p(a) = 1 and |q(a)| ≤ ε for all q ∈ F . Example 5.2. Suppose that E is a uniformly closed C-regular K-linear subspace of C0 (Y, K). Put P = {τx : x ∈ Ch(E)}. By Theorem 3.11 every point in Ch(E) is a strong boundary point. Hence Fp,λ for any (p, λ) ∈ P ×T is a maximal convex set. Since p 6∼ q for p, q ∈ P with p 6= q, we have that P is a set of representatives. The condition (∗) holds with P . We proved that a closed subalgebra of C0 (Y, C) which separates the points of Y and has no common zeros is extremely C regular. Thus such an algebra satisfies the condition (∗). Throughout the section we assume that B is a real or complex Banach space and P is a set of representatives for FB for which the conditon (∗) is satisfied. Lemma 5.3. Let p1 , p2 ∈ P with p1 6= p2 . Let µ1 , µ2 ∈ T. For every ε > 0 and an open neighborhood U of {p1 , p2 } with respect to the relative topology on P induced by the weak*-topology on B ∗ , there exists h ∈ B such that khk ≤ 1 + ε, pj (h) = µj for j = 1, 2, and |q(h)| ≤ ε for every q ∈ P \ U . Proof. We may assume that ε ≤ 1/3. Let Vj be an open neighborhood of pj for j = 1, 2 in P such that V1 ∪ V2 ⊂ U and V1 ∩ V2 = ∅. Choose any positive real number δ 4δ with 0 < 1−δ 2 < ε. By the condition (∗) there exists fj ∈ B such that pj (fj ) = 1 = kfj k and |q(fj )| ≤ δ for every q ∈ P \ Vj for j = 1, 2. As V1 ∩ V2 = ∅ and p2 ∈ V2 we have p2 ∈ P \ V1 , so |p2 (f1 )| ≤ δ. We also have that |p1 (f2 )| ≤ δ. Hence we infer that 0 < 1 − |p1 (f2 )p2 (f1 )|. Put f1 − p2 (f1 )f2 h1 = . 1 − p1 (f2 )p2 (f1 ) Then we infer that p1 (h1 ) = 1 and p2 (h1 ) = 0. By a simple calculation we have kh1 k ≤

kf1 k + |p2 (f1 )|kf2 k ≤ 1/(1 − δ). 1 − |p1 (f2 )||p2 (f1 )|

For q ∈ P \ V1 we have |q(h1 )| ≤

|q(f1 )| + |p2 (f1 )||q(f2 )| ≤ 2δ/(1 − δ 2 ). 1 − |p1 (f2 )||p2 (f1 )|

The Mazur-Ulam property and point-separation property

71

In a similar way, we have p2 (h2 ) = 1 , p1 (h2 ) = 0, kh2 k ≤ 1/(1 − δ) and |q(h2 )| ≤ 2δ/(1 − δ 2 ) for every q ∈ P \ U2 , where h2 =

f2 − p1 (f2 )f1 . 1 − p1 (f2 )p2 (f1 )

Put h = µ1 h1 + µ2 h2 . Then pj (h) = µj for j = 1, 2. We prove that khk ≤ 1 + ε. Let q ∈ V1 . Then q ∈ P \ V2 . Hence |q(h)| ≤ |q(h1 )| + |q(h2 )| ≤ kh1 k + 2δ/(1 − δ 2 ) ≤ 1/(1 − δ) + 2δ/(1 − δ 2 ) < 1 + ε. Let q ∈ V2 . Then we have q ∈ P \ V1 , and |q(h)| < 1 + ε follows. For q ∈ P \ (V1 ∪ V2 ). We infer that |q(h)| ≤ |q(h1 )| + |q(h2 )| ≤ 4δ/(1 − δ 2 ) < ε. In particular, we have |q(h)| < ε for every q ∈ P \ U since V1 ∪ V2 ⊂ U . Since P is a norming family we infer that khk ≤ 1 + ε. □ Let p1 , p2 ∈ P with p1 6= p2 , and µ1 , µ2 ∈ T. Then there Proposition 5.4. exists f ∈ S(B) such that pj (f ) = µj for j = 1, 2. Proof. The idea of proof comes from the proof of the Bishop’s 14 − 34 criterion (cf. [12, Theorem 2.3.2]). We define inductively a sequence {Un } of open (with respect to the relative topology on P induced by the weak*-topology) neighborhoods of {p1 , p2 }, and a sequence {hn } in B as follows : let ε be as 0 < ε ≤ 1/3. Let U1 be any open neighborhood of {p1 , p2 }. Then by Lemma 5.3 there exists h1 ∈ B such that kh1 k ≤ 1 + ε, pl (h1 ) = µl for l = 1, 2, and |q(h1 )| ≤ ε for every q ∈ P \ U1 . Having defined U1 , · · · , Un−1 and h1 , · · · , hn−1 , set Un = {{q ∈ Un−1 : |q(hj )| < 1 + 2−n ε, 1 ≤ j ≤ n − 1}. By Lemma 5.3 there exists hn ∈ B such that khn k ≤ 1 + ε, pl (hn ) = µl for l = 1, 2, and |q(hn )| ≤ ε for every q ∈ P \ Un . Now let h=

∞ X hn . n 2 n=1

In fact the series converges and h ∈ B since khn k ≤ 1 + ε for every n. We have that ! ∞ ∞ X X hn pl (hn ) pl (h) = pl = = µl n n 2 2 n=1 n=1 for l = 1, 2.

72

Osamu Hatori

To prove khk ≤ 1, it is enough to observe |q(h)| ≤ 1 for every q ∈ P since P is a S∞ norming family. We consider three cases: i) q ∈ P \ n=1 Un ; ii) there exists n such T∞ that q ∈ Un \ Un+1 ; iii) q ∈ n=1 Un . It is possible since Un ⊃ Un+1 for every n. Suppose that i) occurs. We have |q(hn )| ≤ ε for every n, hence |q(h)| ≤ ε ≤ 1/3. Suppose that ii) occurs. If q ∈ U1 \ U2 , we have q 6∈ Um for m ≥ 2 since {Un } is decreasing. Thus |q(h1 )| ≤ kh1 k ≤ 1 + ε and |q(hm )| ≤ ε for every m ≥ 2. Therefore we have that ∞ 1+ε X ε + = 1/2 + ε < 1 |q(h)| ≤ m 2 2 m=2 since ε ≤ 1/3. If q ∈ Un \Un+1 for some n ≥ 2, then |q(hj )| < 1+2−n ε for 1 ≤ j ≤ n−1. Since q 6∈ Un+1 we have that q 6∈ Uk for k ≥ n+1. Hence |q(hk )| ≤ ε for every k ≥ n+1. Therefore we get |q(h)| ≤

n−1 X j=1

∞ X |q(hj )| |q(hn )| |q(hk )| + + j n 2 2 2k k=n+1

≤ (1 + 2−n ε)(1 − 2−(n−1) ) + (1 + ε)2−n + 2−n ε ≤ 1 since ε ≤ 1/3. Suppose that iii) occurs. We have |q(hj )| < 1+2−n ε for all n > j. Hence |q(hj )| ≤ 1 for all j, so |q(h)| ≤ 1. We conclude that |q(h)| ≤ 1 for every q ∈ P . As P is a norming family we infer that khk ≤ 1. Thus khk = 1 since 1 = |p1 (h)| ≤ khk. □ The following proposition is a version of an additive Bishop’s lemma. The proof of one we proved in [27, Lemma 5.3] requires the existence of the constants in the target algebra. Proposition 5.5 is a generalization of Lemma 5.3 in [27] which is valid for every Banach space with the condition (∗). α We read |α| = 1 provided that α = 0. Proposition 5.5. Let p ∈ P and f ∈ Ball(B). For every 0 < r < 1 there exist ′ H and H in B such that (5.1)

kH + rf k = 1, p(H + rf ) =

p(f ) , kHk ≤ 1 − r|p(f )|, |p(f )|

and (5.2)

kH ′ + rf k = 1, p(H ′ + rf ) = −

p(f ) , kH ′ k ≤ 1 + |p(f )|. |p(f )|

Proof. Put α = p(f ). The proof of of (5.1) is similar to that of Lemma 5.3 in [27], but a small revision is required since B needs not to be closed under the multiplication.

The Mazur-Ulam property and point-separation property

73

The proof of (5.2) is different from that of Lemma 5.3 in [27]. It requires substantial changes. We first prove (5.1). Let ε be any real number such that 0 < ε < 1 − r|α|. Put F0 = {q ∈ P : |rα − rq(f )| ≥ ε/4} and Fn = {q ∈ P : ε/2n+2 ≤ |rα − rq(f )| ≤ ε/2n+1 } for a positive integer n. By the condition (∗), for every positive integer n there exists un ∈ B such that p(un ) = 1 = kun k 

and |q(un )| ≤ min for every q ∈ F0 ∪ Fn . Put u=

1−r 1 , n+1 1 − r|α| 2



∞ X un . n 2 n=1

P∞ n) Then u ∈ Ball(B) since kun k = 1 for every n. Since p(u) = n=1 p(u = 1, we see 2n that kuk = 1.   α Letting H = |α| − rα u, we prove that kH + rf k ≤ 1 in the following three cases: S∞ (i) q ∈ F0 ; (ii) q ∈ Fn for some n ≥ 1; (iii) q ∈ P \ k=0 Fk . (i) Let q ∈ F0 . We have α 1−r |q(H + rf )| ≤ − rα |q(u)| + r|q(f )| ≤ (1 − r|α|) + r = 1. |α| 1 − r|α| (ii) Let q ∈ Fn for some n ≥ 1. In this case we have |q(u)| ≤

X |q(um )| |q(un )| 1 1 + ≤ 1 − n + n n+1 . m n 2 2 2 2 2

m̸=n

As 0 < ε < 1 − r|α| we have |q(H + rf )| ≤ (1 − r|α|)|q(u)| + r|α| + |rq(f ) − rα| ≤ (1 − r|α|)(1 − 1/2n + 1/(2n 2n+1 )) + r|α| + ε/2n+1 < (1 − r|α|)(1 − 1/2n + 1/(2n 2n+1 ) + 1/2n+1 ) + r|α| < 1 (iii) Let q ∈ P \

S∞ n=0

Fn . In this case we have q(f ) = α, hence

|q(H + rf )| ≤ (1 − r|α|)|q(u)| + r|α| ≤ 1 − r|α| + r|α| = 1.

74

Osamu Hatori

By (i), (ii) and (iii) we have kH + rf k ≤ 1 since P is a norming family. As we will see that p(H + rf ) = α/|α|, it will follow that kH + rf k = 1. By simple calculations we have

 

α

α



− rα u = − rα kuk = 1 − r|α| kHk = |α| |α| and

 p(H + rf ) = p

    α α α − rα u + rf = − rα p(u) + rp(f ) = . |α| |α| |α|

We have completed the proof of (5.1). Next we prove (5.2). Suppose that |α| = 1. Put H ′ = −(1 + r)f . Then we α and have kH ′ k ≤ 1 + r ≤ 1 + |α|. We also have that p(H ′ + rf ) = −p(f ) = − |α| kH ′ +rf k = k−f k ≤ 1. Since 1 = |p(H ′ +rf )| ≤ kH ′ +rf k, we infer that kH ′ +rf k = 1. Thus (5.2) holds if |α| = 1. Suppose that α = 0. As k − f k ≤ 1 and p(−f ) = 0 = α, α we have by (5.1) that there exists H ∈ B such that kH − rf k = 1, p(H − rf ) = |α| and ′ ′ ′ kHk ≤ 1 − r|α| = 1. Letting H = −H we have that kH + rf k = 1, p(H + rf ) = −1 = α − |α| and kH ′ k = kHk ≤ 1 = 1 + |α|. Thus (5.2) holds if α = 0. We assume 0 < |α| < 1. To prove (5.2) we apply induction. Define a sequence {an } by a1 = 1/3, an+1 = (an + 1)/2 for every positive integer n. Put I1 = (0, a1 ] and In = (an−1 , an ] for each positive integer n ≥ 2. For each positive integer n, put Cn = {g ∈ Ball(B) : |p(g)| ∈ In }. By induction on n we prove that for any g ∈ Cn and 0 < s < 1, there exists Hg,s ∈ B such that (5.3)

kHg,s + sgk = 1, p(Hg,s + sg) = −

p(g) , kHg,s k ≤ 1 + |p(g)|. |p(g)|

If it will be proved, then (5.3) will hold for every g ∈ Ball B such that 0 < |p(g)| < 1 since limn→∞ an = 1. Combining with the results for α = 0 or |α| = 1, it will follows that (5.2) holds for every f ∈ Ball(B) and 0 < r < 1. Suppose that g1 ∈ C1 and 0 < s1 < 1. Put α1 = p(g1 ). Then 0 < |α1 | ≤ a1 . Let ε1 √ be as 0 < ε1 < 1 − s1 and K1 = {q ∈ P : |α − q(g1 )| ≥ ε1 /2}. By the condition (∗) there exists v1 ∈ S(B) such that p(v1 ) = 1 = kv1 k and |q(v1 )| ≤ √ √ ε1 /2 for every q ∈ K1 . Look at k − 2 s1 α1 v1 + s1 g1 k. If q ∈ K1 , then |q(v1 )| ≤ ε1 /2. Therefore √ √ √ √ √ √ |q(−2 s1 α1 v1 + s1 g1 )| ≤ 2 s1 |α1 ||q(v1 )| + s1 |q(g1 )| ≤ 2 s1 |α1 |ε1 /2 + s1 < 1

The Mazur-Ulam property and point-separation property

75

since |α1 | ≤ 1/3. If q ∈ P \ K1 , then |α1 − q(g1 )| < ε1 /2. Thus √ √ √ √ √ √ |q(−2 s1 α1 v1 + s1 g1 )| ≤ s1 |α1 −2α1 q(v1 )|+ s1 |q(g1 )−α1 | ≤ 3 s1 |α1 |+ s1 ε1 /2 < 1 since since |α1 | ≤ 1/3 and ε1 < 1 −

√

s1 . We conclude that

√ √ k − 2 s1 α1 v1 + s1 g1 k ≤ 1 √ √ √ since P is a norming family. Put g1′ = −2 s1 α1 v1 + s1 g1 Then p(g1′ ) = − s1 α1 . √ Applying (5.1) with s1 and g1′ instead of r and f respectively, we find H+ ∈ B such that √ √ α1 p(g1′ ) kH+ + s1 g1′ k = 1, p(H+ + s1 g1′ ) = =− ′ |p(g1 )| |α1 | and kH+ k ≤ 1 −

√

s1 |p(g1′ )| = 1 − s1 |α1 |.

Letting Hg1 ,s1 = H+ − 2s1 α1 v1 we infer that Hg1 ,s1 + sg1 = H+ − 2s1 α1 v1 + s1 g1 = H+ +

√

√ √ √ s1 (−2 s1 α1 v1 + s1 g1 ) = H+ + s1 g1′ .

Hence kHg1 ,s1 + s1 g1 k = 1,

p(Hg1 ,s1 + s1 g1 ) = −

α1 . |α1 |

We also have kHg1 ,s1 k ≤ kH+ k + 2s1 |α1 |kv1 k ≤ 1 − s1 |α1 | + 2s1 |α1 | = 1 + s1 |α1 | ≤ 1 + |α1 |. We have proved that (5.3) holds for n = 1 Suppose that (5.3) holds for every 1 ≤ n ≤ m. Let gm+1 ∈ Cm+1 and 0 < sm+1 < 1 √ arbitrary. Put αm+1 = p(gm+1 ). Put 0 < εm+1 < 1 − sm+1 and Km+1 = {q ∈ P : |αm+1 − q(gm+1 )| ≥ εm+1 /2}. By the condition (∗) there exists vm+1 ∈ B such that p(vm+1 ) = 1 = kvm+1 k and |q(vm+1 )| ≤

εm+1 2(am+1 − am )

for every q ∈ Km+1 . Put fm+1 =

√

 sm+1

  αm+1 am · − αm+1 vm+1 + gm+1 . |αm+1 |

76

Osamu Hatori

αm+1 Note that am · |α − α = |αm+1 | − am since am < |αm+1 | ≤ am+1 . Suppose m+1 m+1 | that q ∈ Km+1 . Then |q(fm+1 )| ≤

√ √ sm+1 (|αm+1 | − am )|q(vm+1 )| + sm+1 |q(gm+1 )| √ sm+1 (|α| − am )εm+1 √ √ + sm+1 ≤ sm+1 (εm+1 /2 + 1) < 1 ≤ 2(am+1 − am )

since |αm+1 | ≤ am+1 . Suppose that q ∈ P \ Km+1 . Then   αm+1 √ |q(fm+1 )| ≤ sm+1 am · − αm+1 |q(vm+1 )| + |αm+1 | + |q(gm+1 ) − αm+1 | |αm+1 | √ √ ≤ sm+1 ((|αm+1 | − am ) + |αm+1 | + εm+1 /2) = sm+1 (2|αm+1 | − am + εm+1 /2) √ √ ≤ sm+1 (2am+1 − am + εm+1 /2) = sm+1 (1 + εm+1 /2) < 1 Therefore we have that kfm+1 k ≤ 1 since P is a norming family. By a calculation we √ √ αm+1 have p(fm+1 ) = sm+1 am · |α , hence |p(fm+1 )| = sm+1 am < am . It means that m+1 | |p(fm+1 )| ∈ Ik for some 1 ≤ k ≤ m. By the hypothesis of induction there exists Hk ∈ B such that kHk +

√ sm+1 fm+1 k = 1,

p(Hk +

√

sm+1 fm+1 ) = −

and kHk k ≤ 1 + |p(fm+1 )| = 1 + Put

Then Hgm+1 ,sm+1

p(fm+1 ) αm+1 =− |p(fm+1 )| |αm+1 |

√ sm+1 am

  α Hgm+1 ,sm+1 = Hk + sm+1 am · − α vm+1 . |α| √ + sm+1 gm+1 = Hk + sm+1 fm+1 and

kHgm+1 ,sm+1 + sm+1 gm+1 k = 1,

p(Hgm+1 ,sm+1 + sm+1 gm+1 ) = −

αm+1 . |αm+1 |

We also have

 

α m+1

kHgm+1 ,sm+1 k ≤ kHk k + s a · − α v m+1 m+1 m+1 m

|αm+1 | √ √ √ ≤ 1 + sm+1 am + sm+1 (|αm+1 | − am ) < 1 + sm+1 am + sm+1 (|αm+1 | − am ) ≤ 1 + |αm+1 |. We conclude by induction that (5.3) holds if 0 < |p(g)| < 1. □

The Mazur-Ulam property and point-separation property

§ 6.

77

Banach spaces which satisfy the condition (∗) and the Mazur-Ulam property

Theorem 6.1. Let B be a real Banach space which satisfies the condition (∗). Then B has the Mazur-Ulam property. Proof. Let P be a set of representative for FB with which the condition (∗) is satisfied. We prove that (4.7) of Proposition 4.11 holds. Let p ∈ P and −1 ≤ α ≤ 1 arbitrary. In the same way as [26, Lemma 4.3] we infer that Mp,α ⊂ {a ∈ S(B) : p(a) = α}. We prove the inverse inclusion. Suppose that f ∈ S(B) with p(f ) = α. Let 0 < r < 1 α arbitrary. By Proposition 5.5 there exists H ∈ B such that H + rf ∈ Fp, |α| and kHk ≤ 1 − r|α|. Thus kH + rf − f k ≤ 1 − r|α| + 1 − r. α ) ≤ 1 − |α|. We also have by Proposition 5.5 that there exists We infer that d(f, Fp, |α| α and kH ′ k ≤ 1 + |α|. Hence H ′ ∈ B such that H ′ + rf ∈ Fp,− |α|

kH ′ + rf − f k ≤ 1 + |α| + 1 − r. α ) ≤ 1 + |α|. Thus f ∈ M We infer that d(f, Fp,− |α| p,α . Hence {f ∈ S(B) : p(f ) = α} ⊂ Mp,α . It follows from Proposition 4.11 that B has the Mazur-Ulam property. □

Let E be a uniformly closed extremely C-regular R-linear subCorollary 6.2. space of C0 (Y, R) for a locally compact Hausdorff space Y . Then E has the Mazur-Ulam property. In particular, C0 (Y, R) itself and a uniformly closed extremely regular subspace of C0 (Y, R) has the Mazur-Ulam property. Proof. We prove that E satisfies the condition (∗). Put P = {τx : x ∈ Ch(E)}. By Theorem 3.11 every point in Ch(E) is a strong boundary point. Hence Fp,λ for any (p, λ) ∈ P × {±1} is a maximal convex set. Since p 6∼ q for p, q ∈ P with p 6= q, we have that P is a set of representatives. The condition (∗) holds with P . Then by Theorem 6.1 we see that E has the Mazur-Ulam property. □ Let B be a complex Banach space which satisfies the condition Theorem 6.3. (∗). Then B has the complex Mazur-Ulam property. Proof. Let P be a set of representative for FB with which the condition (∗) is satisfied. We prove (i) and (ii) of Proposition 4.12. Let Fp,λ , Fp′ ,λ′ ∈ FB such that

78

Osamu Hatori

p 6= p′ . By Proposition 5.4 there exists f ∈ S(B) such that p(f ) = −λ and p′ (f ) = λ′ . Then f ∈ Fp′ ,λ′ . For any g ∈ Fp,λ we infer that 2 = |p(f ) − p(g)| ≤ kf − gk ≤ 2, hence d(f, Fp,λ ) = 2. Hence dH (Fp,λ , Fp′ ,λ′ ) = 2. As a pair Fp,λ and Fp′ ,λ′ with p 6= p′ is arbitrary, we see that B satisfies the Hausdorff distance condition holds; (i) of Proposition 4.12 holds. ¯ arbitrary. Let 0 < r < 1 We prove (ii) of Proposition 4.12. Let p ∈ P and α ∈ D α be arbitrary. By Proposition 5.5 there exists H ∈ B with H + rf ∈ Fp, |α| and kHk ≤ α and kH ′ k ≤ 1 + |α|. Hence 1 − r|α|. There also exists H ′ ∈ B with H ′ + rf ∈ Fp,− |α| kH + rf − f k ≤ 1 − r|α| + 1 − r, and kH ′ + rf − f k ≤ 1 + |α| + 1 − r. α ) ≤ 1 − |α| and d(f, F α ) ≤ 1 + |α|. It As r is arbitrary we infer that d(f, Fp, |α| p,− |α| follows that f ∈ Mp,α . Thus {f ∈ S(B) : p(f ) = α} ⊂ Mp,α . The inverse inclusion

Mp,α ⊂ {f ∈ S(B) : p(f ) = α} is by [26, Lemma 4.3]. We conclude that (ii) of Proposition 4.12 holds. It follows from Proposition 4.12 that B has the complex Mazur-Ulam property.

□

Corollary 6.4. A uniformly closed extremely C-regular C-linear subspace of C0 (Y, C) for a locally compact Hausdorff space has the complex Mazur-Ulam property. In particular, C0 (Y, C) itself and a uniformly closed extremely regular C-linear subspace of C0 (Y, C) has the complex Mazur-Ulam property. Proof. The proof that E satisfies the condition (∗) is essentially the same as that for Corollary 6.2 (cf. Example 5.2). Then by Theorem 6.3 we see that E has the complex Mazur-Ulam property. □ Hatori [26, Theorem 4.5] proved that a uniform algebra has the complex MazurUlam property. In the proof of Theorem 4.5 in [26], it is crucial that a uniform algebra contains the constants. Cabezas, Cueto-Avellaneda, Hirota, Miura and Peralta [14, Corollary 3.2] proved that C0 (Y, C) satisfies the complex Mazur-Ulam property. Cueto-Avellaneda, Hirota, Miura and Peralta [17, Theorem 2.1] have proved that each surjective isometry between the unit spheres of two uniformly closed algebras on locally compact Hausdorff spaces which separates the points without common zeros admits an extension to a surjective real linear isometry between these algebras. The following generalizes a both theorems of Cueto-Avellaneda, Hirota, Miura and Peralta [17, Theorem 2.1] and Cabezas, Cueto-Avellaneda, Hirota, Miura and Peralta [14, Corollary 3.2].

The Mazur-Ulam property and point-separation property

79

Corollary 6.5. A (non-zero) closed subalgebra of C0 (Y, C) has the complex MazurUlam property. In particular, a uniform algebra has the complex Mazur-Ulam property. Proof. Let A be a non-zero closed subalgebra of C0 (Y, C). Let C = {y ∈ Y : f (y) = 0 for all f ∈ A}. As we assume A 6= {0}, there is a non-zero function in A, so C is a proper subset of Y . For any pair of points x and y in Y \ C, we denote x ∼ y if f (x) = f (y) for all f ∈ A. Then ∼ is an equivalence relation on Y \ C. Let Y0 be the quotient space induced by the relation ∼. Then Y0 is a locally compact, possibly compact, Hausdorff space induced by the quotient topology. We may suppose that A is a closed subalgebra of C0 (Y0 , C) which separates the points in Y0 and has no common zeros. Then by Proposition 3.13 A is a uniformly closed extremely C-regular C-linear subspace of C0 (Y0 , C). It follows by Corollary 6.4 that A has the complex Mazur-Ulam property. Suppose that A is a uniform algebra on a compact Hausdorff space. Then A is a closed subalgebra of C(X, C). Then by the first part we have that A has the complex Mazur-Ulam property. □

§ 7.

Final remarks

In section 5 we introduced the condition (∗) for Banach spaces and we proved that an extremely C-regular closed subspace of C0 (Y, K) satisfies the condition (∗) in section 6. On the other hand we have not enough examples of Banach spaces which satisfy the condition. Cabezas, Cueto-Avellaneda, Hirota, Miura and Peralta [14] proved that every commutative JB∗ triple satisfies the complex Mazur-Ulam property. The author does not know if a commutative JB∗ triple satisfies the condition (∗) or not. If it would satisfy the condition (∗), we would get an alternative proof of a theorem of Cabezas, Cueto-Avellaneda, Hirota, Miura and Peralta [14, Theorem 3.1]. It is interesting to exhibit enough examples of Banach spaces which satisfy the condition (∗). We have proved the complex Mazur-Ulam property especially for a closed subalgebra of C0 (Y, C) which separates the points of Y and has no common zeros, we expect it also has the Mazur-Ulam property. It is trivial that the complex Mazur-Ulam property follows the Mazur-Ulam property. The author does not know the converse statement : does the Mazur-Ulam property follow the complex Mazur-Ulam property? Acknowledgments. The author would like to express his thanks to Professor Antonio Peralta for fruitful discussions which started on the end of 2021. It is certain that the discussions and his useful comments are the starting point of the research in this paper.

80

Osamu Hatori

Special thanks are due to Professor Shiho Oi for organizing the RIMS workshop “Research on preserver problems on Banach algebras and related topics” and for editing RIMS Kˆokyˆ uroku Bessatsu. The author records his sincerest appreciation to the referee for their valuable comments and advice which have improved the presentation of this paper substantially.

References [1] T. A. Abrahamsen, O. Nygaard and M. P˜ oldvere, New applications of extremely regular function spaces, Pacifc J. Math. 301 (2019), 385–394 doi:10.2140/pjm.2019.301.385 [2] J. Araujo and J. J. Font, Linear isometries between subspaces of continuous functions, J. Araujo and J. J. Font, Trans. Amer. Math. Soc. 349 (1997), 413–428 ˇ [3] J. Araujo and J. J. Font, On Silov boundaries for subspaces of continuous functions, Proceedings of the First Ibero-American Conference on Topology and its Applications (Benicassim, 1995) Topology Appl. 77 (1997), 79–85 doi:10.1016/S0166-8641(96)00132-0 [4] T. Banakh, Every 2-dimensional Banach space has the Mazur-Ulam property, Linear Algebra Appl. 632 (2022), 268–280 doi:10.1016/j.laa.2021.09.020 [5] H. S. Bear, The Silov boundary for a linear space of continuous functions, Amer. Math. Monthly 68 (1961), 483–485 [6] J. Becerra-Guerrero, M. Cueto-Avellaneda, F. J. Fern´ andez-Polo and A. M. Peralta, On the extension of isometries between the unit spheres of a JBW*-triple and a Banach space, J. Inst. Math. Jussieu 20 (2021), 277–303 doi:10.1007/s13324-022-00448-2 [7] E. Bishop and K. de Leeuw, The representation of linear functionals by measures on sets of extreme points, Ann. Inst. Fourier, Grenoble 9 (1959), 305–331 [8] D. P. Blecher, The Shilov boundary of an operator space and the characterization theorems, J. Funct. Anal. 182 (2001), 280–343 doi:10.1006/jfan.20003734 [9] H. F. Bohnenblust and S. Karlin, Geometrical properties of the unit sphere of Banach algebras, Ann. Math., II. Ser. 62 (1955), 217–229 [10] K. Boyko, V. Kadets, M. Mart´ın and J. Mer´ı, Properties of lush spaces and applications to Banach spaces with numerical index 1, Studia Math. 190 (2009), 117–133 doi:10.4064/sm190-2-2 [11] K. Boyko, V. Kadets, M. Mart´ın and D. Werner, Numerical index of Banach spaces and duality, Math. Proc. Cambridge Philos. Soc. 142 (2007), 93–102 doi:10.1017/S0305004106009650 [12] A. Browder, Introduction to function algebras, W. A. Benjamin, Inc., New YorkAmsterdam 1969 Xii+273 pp [13] J. Cabello S´ anchez, A reflection on Tingley’s problem and some applications, J. Math. Anal. Appl. 476 (2019), 319–336 doi:10.1016/j.jmaa.2019.03.041 [14] D. Cabezas, M. Cueto-Avellaneda, D. Hirota, T. Miura and A. Peralta, Every commutative JB*-triple satisfies the complex Mazur-Ulam property, Ann. Funct. Anal. 13 (2022), Paper No. 60, 8pp [15] L. Cheng and Y. Dong, On a generalized Mazur-Ulam question: extension of isometries between unit spheres of Banach spaces, J. Math. Anal. Appl. 377 (2011), 464–470 doi:10.1016/j.jmaa.2020.11.025 [16] B. Chengiz, On extremely regular function spaces, Pac. J. Math. 49 (1973), 335–338

The Mazur-Ulam property and point-separation property

81

[17] M. Cueto-Avellaneda, D. Hirota, T. Miura and A. Peralta, Exploring new solutions to Tingley’s problem for function algebras Quaest. Math. Published online : 02 Jun 2022 [18] M. Cueto-Avellaneda and A. M. Peralta, On the Mazur-Ulam property for the space of Hilbert-space-valued continuous functions, J. Math. Anal. Appl. 479 (2019), 875–902 doi:10.1016/j.jmaa.2019.06.056 [19] M. Cueto-Avellaneda and A. M. Peralta, The Mazur-Ulam property for commutative von Neumann algebras, Linear Multilinear Algebra 68 (2020), 337–362 doi:10.1080/03081087.2018.1505823 [20] G. G. Ding, On extension of isometries between unit spheres of E and C(Ω), Acta Math. Sin. (Engl. Ser.) 19 (2003), 793–800 [21] G. G. Ding, The isometric extension of the into mapping from a L∞ (Γ)-type space to some Banach space, Illinois J. Math. 51 (2007), 445–453 [22] J. Duncan, C. McGregor, J. Pryce and A. White, The numerical index of a normed space, J. London Math. Soc. 2 (1970), 481–488 [23] X. N. Fang and J. H. Wang, Extension of isometries between the unit spheres of normed space and C(Ω), Acta Math.Sinica Engl Ser. 22 (2006), 1819–1824 [24] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129. Chapman & Hall/CRC, Boca Raton, FL, 2003. x+197 pp. ISBN: 1-58488-040-6 [25] T. W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969 [26] O. Hatori, The Mazur-Ulam property for uniform algebras, Studia Math. 265 (2022), 227–239 doi: 10.4064/sm210703-11-9 [27] O. Hatori, S. Oi and R. Shindo Togashi, Tingley’s problems on uniform algebras Jour. Math. Anal. Appl. 503 (2021), Paper No. 125346, 14 pp. doi: 10.1016/j.jmaa.2021.125346 [28] A. Jim´enez-Vargas, A. Morales-Campoy, A. M. Peralta and M. I. Ram´ırez, The MazurUlam property for the space of complex null sequences, Linear Multilinear Algebra 67 (2019), 799–816 doi:10.1080/03081087.2018.1433625 [29] V. M. Kadets and M. M. Popov, The Daugavet property for narrow operators in rich subspaces of C[0, 1] and L1 [0, 1], St. Petersburg Math. J. 8 (1997), 571–584 ˇ boundary, Math. Z. 202 [30] P. Kajetanowicz, A general approach to the notion of Silov (1989), 391–395 [31] O. F. K. Kalenda and A. M. Peralta, Extension of isometries from the unit sphere of a rank-2 Cartan factor, Anal. Math. Phys. 11, Article number:15 (2021) [32] R. Liu, On extension of isometries between unit spheres of L∞ (Γ)-type space and a Banach space E, J. Math. Anal. Appl. 333 (2007), 959–970 doi:10.1016/j.jmaa.2006.11.044 [33] T. Miura, Real-linear isometries between function algebras, Cent. Eur. J. Math. 9 (2011), 778–788 doi:10.2478/s11533-011-0044-9 [34] M. Mori, Tingley’s problem through the facial structure of operator algebras, J. Math. Anal. Appl. 466 (2018), 1281–1298 doi:10.1016/j.jmaa.2018.06.050 [35] M. Mori and N. Ozawa, Mankiewicz’s theorem and the Mazur-Ulam property for C ∗ algebras, Studia Math. 250 (2020), 265–281 doi:10.4064/sm180727-14-11 [36] W. Novinger, Linear isometries of subspaces of continuous functions, Studia Math. 53 (1975), 273–276 [37] R. R. Phelps, Lectures on Choquet’s Theorem. Second edition, Lecture Notes in Mathematics, 1757. Springer-Verlag, Berlin, 2001 [38] A. M. Peralta, Extending surjective isometries defined on the unit sphere of ℓ∞ (Γ) Rev. Mat. Complut. 32 (2019), 99–114 doi:10.1007/s13163-018-0269-2

82

Osamu Hatori

[39] A. M. Peralta, On the extension of surjective isometries whose domain is the unit sphere of a space of compact operators, Filmat 36 (2022), 3075–3090 ˇ [40] A. Peralta and R. Svarc, A strengthened Kadison’s transitivity theorem for unital JB*algebras with applications to the Mazur-Ulam property, preprint 2023, arXiv:2301.00895 [41] Ch. Pomerenke, Boundary behavior of conformal maps, Grundlehrender Mathematischen Wissenschaften, 299 Springer-Verlag, Berlin, 1992, x+300 pp [42] N. V. Rao and A. K. Rao, Multiplicatively spectrum-preserving maps of function algebras. II, Proc. Edinb. Math. Soc. 48 (2005), 219–229 ˇ [43] T. S. S. R. K. Rao and A. K. Roy, On Silov boundary for function spaces, Topology Appl. 193 (2015), 175–181 doi:10.1016/j.topol.2015.07.005 [44] Z. Semadeni, Banach spaces of continuous functions, Monografie Matematyczne, Warszawa, 1971 ˇ [45] G. Silov, On the extension of maximal ideals, Dokl. Akad. Nauk SSSR 29 (1940), 83–84 (in Russian) [46] E. L. Stout, The theory of uniform algebras, Bogden & Quigley, Inc. Publishers, Tarrytown-on-Hudson, N. Y., 1971 [47] D. N. Tan, Extension of isometries on unit spheres of L∞ , Taiwanese J. Math. 15 (2011), 819–827 [48] D. N. Tan, On extension of isometries on the unit spheres of Lp -spaces for 0 < p ≤ 1, Nonlinear Anal. 74 (2011), 6981–6987 doi:10.1016/j.na.2011.07.035 [49] D. N. Tan, Extension of isometries on the unit sphere of Lp spaces, Acta Math. Sin. (Engl. Ser.) 28 (2012), 1197–1208 doi:10.1007/s10114-011-0302-6 [50] D. Tan, X. Huang and R. Liu, Generalized-lush spaces and the Mazur-Ulam property, Studia Math. 219 (2013), 139–153 doi:10.4064/sm219-2-4 [51] R. Tanaka, A further property of spherical isometries, Bull. Aust. Math. Soc. 90 (2014), 304–310 doi:10.1017/S0004972714000185 [52] R. Tanaka, The solution of Tingley’s problem for the operator norm unit sphere of complex n × n matrices, Linear Algebra Appl. 494 (2016), 274–285 doi:10.1016/j.laa.2016.01.020 [53] D. Tingley, Isometries of the unit sphere, Geom. Dedicata 22 (1987), 371–378 [54] R. S. Wang, Isometries between the unit spheres of C0 (Ω) type spaces, Acta Math. Sci. (English Ed.) 14 (1994), 82–89 (n) [55] Risheng Wang, Isometries of C0 (X), Hokkaido Math. J. 25 (1996), 465–519 doi:10.14492/hokmj/1351516747 [56] Ruidong Wang and X. Huang, The Mazur-Ulam property for two dimensional somewhereflat spaces, Linear Algebra Appl. 562 (2019), 55–62 doi:10.1016/j.laa.2018.09.024 [57] P. Wojtaszczyk, Some remarks on the Daugavet equation, Proc. Amer. Math. Soc. 115 (1992), 1047–1052 [58] X. Yang and X. Zhao, On the extension problems of isometric and nonexpansive mappings, In:Mathematics without boundaries. Edited by Themistocles M. Rassias and Panos M. Pardalos, 725– Springer, New York, 2014

RIMS Kˆ okyˆ uroku Bessatsu B93 (2023), 83–107

Surjective isometries on an algebra of analytic functions with C n -boundary values By

Yuta Enami∗ and Takeshi Miura∗∗

Abstract Let D, D and T be the open unit disk, closed unit disk and unit circle in C. Let An (D) denote the algebra of all continuous functions f on D which are analytic in D and whose restrictions f |T to T are of class C n . For each f ∈ An (D), the k-th derivative of f |T as a function on T is denoted by Dk (f ). We characterize surjective, not necessarily linear, isometries ∑ k on An (D) with respect to the norm ∥f ∥D + n k=1 ∥D (f )∥T /k!, where ∥ · ∥D and ∥ · ∥T are the supremum norms on D and T, respectively.

§ 1.

Introduction

A mapping T : E1 → E2 between two normed spaces (E1 , k · k1 ) and (E2 , k · k2 ) is called an isometry if kT (f ) − T (g)k2 = kf − gk1 for every f, g ∈ E1 . We emphasize that we do not assume linearity for T . The characterization of isometries is a classical problem. Banach [1] characterized surjective, not necessarily linear, isometries on the Banach space CR (K) of all continuous realvalued functions on a compact metric space K with the supremum norm. After that, characterizations of surjective linear isometries were given for various Banach spaces. For the space C 1 [0, 1] of all continuously differentiable functions on [0, 1], Rao and Roy Received March 31, 2022. Revised June 23, 2022. 2020 Mathematics Subject Classification(s): 46B04, 46J15 Key Words: extreme point, function space, surjective isometry. The first author partially supported by JSPS KAKENHI Grant Number JP 21J21512. The second author partially supported by JSPS KAKENHI Grant Number JP 20K03650 ∗ Graduate School of Science and Technology, Niigata University, Niigata 950-2181 Japan. e-mail: [email protected] ∗∗ Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-2181 Japan. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

84

Y. Enami and T. Miura

[14] determined the general form of surjective complex-linear isometries on C 1 [0, 1] with respect to the norm kf k = kf k∞ + kf ′ k∞ , where k · k∞ stands for the supremum norm. Novinger and Oberlin [13] consider the space S p of all analytic functions on the open unit disk whose derivatives belong to the Hardy space H p . They gave a characterization of complex-linear isometries on S p (1 ≤ p < ∞) with respect to the norm kf k = kf k∞ + kf ′ kH p . Jarosz investigated a class of unital semisimple commutative Banach algebras with the so-called natural norm. Jarosz [7] proved that every surjective unital complex-linear isometry with respect to the natural norm is actually an isometry with respect to the supremum norm. Note that the norm kf k = kf k∞ + kf ′ k∞ becomes a natural norm on C 1 [0, 1]. One of the most interesting results on study of isometries was proved by Mazur and Ulam. The Mazur-Ulam theorem [10] states that every surjective isometry between normed spaces must be (real) affine. Applying the Mazur-Ulam theorem, surjective, not necessarily linear, isometries were studied on various normed spaces by many researchers. Hatori and the second author [6] gave the characterization of surjective isometries between function algebras. Kawamura, Koshimizu and the second author [9] introduced a unified framework to treat several norms on C 1 [0, 1], and gave the characterization of surjective isometries on C 1 [0, 1] with respect to various norms including kf k = kf k∞ + kf ′ k∞ . Concerning such a framework, Kawamura [8] also considers the algebra C 1 (T) of all continuously differentiable functions on the unit circle T, and gave the characterization of surjective isometries on C 1 (T) with respect to norms belonging to the framework. The second author and Niwa [11, 12] introduce the Novinger-Oberlin type space SA of all analytic functions whose derivatives belong to the disk algebra. The space SA admits several norms. They determined general forms of surjective isometries with respect to some norms, including kf k = kf k∞ + kf ′ k∞ .

§ 1.1.

Notations and Main results

In this paper, let N and N0 be the sets of all positive integers and non-negative 2 integers, respectively. For m1 , m2 ∈ N0 with m1 ≤ m2 , we set Nm m1 = {k ∈ N0 : m1 ≤ k ≤ m2 }. For a compact Hausdorff space K, let C(K) denote the Banach space of all complexvalued continuous functions on K, with the supremum norm kf kK = sup |f (x)|

(f ∈ C(X)).

x∈K

The constant functions on K taking the value only 0 and 1 are denoted by 0 and 1, respectively. Let T be the unit circle in the complex plane C. For n ∈ N, a function f : T → C is said to be of class C n if the function F on R defined by F (t) = f (e2πit ) is of class C n in the usual sense. We denote by C n (T) the subalgebra of C(T) consisting

Surjective isometries on An (D)

85

of all functions of class C n . Let D be the open unit disk, and let D = D ∪ T be the closed unit disk. The disk algebra A(D) is the Banach algebra of all continuous functions on D which are analytic in D, with the supremum norm k · kD . Note that, by the maximum modulus principle, kf kD = kf kT for every f ∈ A(D). Throughout this paper, we fix n ∈ N. The main object of this paper is the algebra An (D) = {f ∈ A(D) : f |T ∈ C n (T)}. For each f ∈ An (D) and k ∈ Nn1 , the k-th derivative of f |T at e2πit0 ∈ T is denoted by  k

D (f )(e

2πit0

)=

1 2π

k

dk f (e2πit ). dtk t=t0

Let D0 (f ) = f |T . Since f |T is a function of class C n , the function Dk (f ) : T → C is continuous on T for every k ∈ Nn0 . Note that Dk satisfies the Leibniz rule k   X k D (f g) = Dk−j (f )Dj (g) j j=0 k

for every f, g ∈ An (D). For each f ∈ An (D), set n n X X 1 1 k kf kΣ = kf kD + kD (f )kT = kDk (f )kT . k! k! k=0

k=1

Then (An (D), k · kΣ ) is a unital commutative Banach algebra. The following theorem is the main result of this paper. Theorem 1.1. Suppose that T : An (D) → An (D) is a surjective, not necessarily linear, isometry with respect to the norm k · kΣ . Then there exist constants c, λ ∈ T such that T (f )(z) = T (0)(z) + cf (λz)

(∀f ∈ An (D), ∀z ∈ D),

T (f )(z) = T (0)(z) + cf (λz)

(∀f ∈ An (D), ∀z ∈ D).

or

Conversely, every mapping T : An (D) → An (D) which is one of the above forms is a surjective isometry on An (D) with respect to the norm k · kΣ , where T (0) is an arbitrary function in An (D).

§ 1.2.

Some remarks

Note first that (An (D), k · kΣ ) is a unital semisimple commutative Banach algebra. Moreover, the norm k · kΣ is a natural norm in the sense of Jarosz [7]. Hence it is

86

Y. Enami and T. Miura

relatively easy to determine the general form of surjective complex-linear isometry T on An (D) with T (1) = 1 by the result of Jarosz [7, Theorem and Proposition 2]. On the other hand, our study is more complicated. In fact, we will investigate surjective isometry T on An (D), which need not be complex-linear nor unital, that is, T (1) = 1 in Theorem 1.1. The second author and Niwa [11] introduce the space SA of all analytic functions f on D whose derivative f ′ is continuously extended to D, where f ′ is the usual derivative with respect to the complex variable. It is well-known that a holomorphic function f : D → C is continuously extended to D with absolutely continuous boundary value if and only if the derivative f ′ belongs to the Hardy space H 1 (see [4, Theorem 3.11]). As a consequence of the fact, every function in SA is continuously extended to D. The continuous extension of f will be denoted by fˆ. Now, for each f ∈ SA , we set kf kΣ,SA = kfˆkD + kfb′ kD . Then the space SA becomes a unital commutative Banach algebra. The Banach algebra SA is isometrically isomorphic to A1 (D). More precisely, we have the following proposition, which can be verified by the same argument as [4, Theorem 3.11]. Proposition 1.2. A holomorphic function f : D → C is continuously extended to D and its extension fˆ belongs to A1 (D) if and only if f belongs to SA . Moreover, if f ∈ SA , then kfˆkΣ = kf kΣ,SA . In [12], a characterization of surjective, not necessarily linear, isometries on SA with respect to the norm k · kΣ,SA was given. Hence Theorem 1.1 is considered as a generalization of the result.

§ 2.

Preliminaries and embedding of An (D) into C(X)

§ 2.1.

Polynomials

First, we consider each polynomial p as a function on D. It is obvious that p ∈ A (D). Let p(z) = a0 + · · · + am z m . For k ∈ N, let p(k) denote the k-th formal derivative of p, that is, p(k) (z) = k k ak + · · · + mk am z m−k , where mk is the falling factorial m(m − 1) · · · (m − k + 1). Note that Dk (p) = p(k) |T does not hold. In fact, Dk (ιj )(z) = ik j k z j , where ιj (z) = z j . More generally, we see that Dk (p) can be represented as n

(2.1)

k

k

D (p)(z) = i

m X j=1

j k aj z j .

Surjective isometries on An (D)

87

On the other hand, the chain rule implies that D1 (p)(z) = ip(1) (z) and D2 (p)(z) = −p(2) (z)z 2 − p(1) (z)z. By induction, we see that Dk (p) can also be represented as (2.2)

k

D (p)(z) =

k X

cj p(j) (z)z j ,

j=1

where c1 , . . . , ck are constants independent of the polynomial p. For m ∈ N0 , let Mm+1,n+1 (T) denote the set of all (m + 1) × (n + 1) matrices whose entries belong to T. Let m ∈ N0 . Let W = [wj,k ]j,k ∈ Mm+1,n+1 (T), and assume Proposition 2.1. that w0,0 6∈ {w1,0 , . . . , wm,0 }. Then there exists a polynomial p such that p(w0,0 ) 6= 0 and Dk (p)(wj,k ) = 0 for every (j, k) 6= (0, 0), that is,     p(w0,0 ) D1 (p)(w0,1 ) · · · Dn (p)(w0,n ) ∗0··· 0      p(w1,0 ) D1 (p)(w1,1 ) · · · Dn (p)(w1,n )   0 0 · · · 0    = . .  (2.3) .. .. .. ..    . . . . ..  . .   . . . . . . . p(wm,0 ) D1 (p)(wm,1 ) · · · Dn (p)(wm,n ) 00··· 0 n Proof. Let I0 = {(j, k) ∈ Nm 0 × N0 : wj,k 6= w0,0 }, and let Y q(z) = (z − wj,k )k+1 . (j,k)∈I0

By definition, q(w0,0 ) 6= 0. If (j, k) ∈ I0 , then the formal derivatives q(z), q (1) (z), . . . , q (k) (z) have the factor (z − wj,k ), and thus, by equality (2.2), we have Dk (q)(wj,k ) = 0. Hence we obtain q(w0,0 ) 6= 0 = Dk (q)(wj,k ) for every (j, k) ∈ I0 . If, in addition, Dk (q)(w0,0 ) = 0 for every k ∈ Nn1 , then q satisfies the condition (2.3). In this cases, q is the desired polynomial. Now, assume that Dk (q)(w0,0 ) 6= 0 for some k ∈ Nn1 . Let k1 ∈ Nn1 be the smallest k ∈ Nn1 such that Dk (q)(w0,0 ) 6= 0. Then D1 (q)(w0,0 ) = · · · = Dk1 −1 (q)(w0,0 ) = 0 6= Dk1 (q)(w0,0 ). In particular, Dk (q)(w0,0 ) = 0 for all k ∈ Nk11 −1 . Let r(z) = q(z) − 2q(w0,0 ). Since r(w0,0 ) = −q(w0,0 ) 6= 0, we have (qr)(w0,0 ) 6= 0. Moreover, if (j, k) ∈ I0 , then the Leibniz rule shows that Dk (qr)(wj,k ) = 0. Note that Dk (r)(w0,0 ) = Dk (q)(w0,0 ) = 0 for every k ∈ Nk11 −1 , and that Dk1 (r)(w0,0 ) = Dk1 (q)(w0,0 ) 6= 0. By the Leibniz rule, Dk (qr)(w0,0 ) = 0 for every k ∈ Nk11 −1 . We also have Dk1 (qr)(w0,0 ) = q(w0,0 ) · Dk1 (r0 )(w0,0 ) + Dk1 (q)(w0,0 ) · r(w0,0 ) = q(w0,0 ) · Dk1 (q)(w0,0 ) − Dk1 (q)(w0,0 ) · q(w0,0 ) = 0.

88

Y. Enami and T. Miura

Hence we obtain Dk (qr)(w0,0 ) = 0 for all k ∈ Nk11 . This shows that the polynomial qr has not only the same properties as q, but also Dk1 (qr)(w0,0 ) = 0. Finally, applying the above argument repeatedly, at most finitely many times, we obtain a polynomial p satisfying condition (2.3) . The proof is completed. □ n Proposition 2.2. Let m ∈ N0 , and let k0 ∈ N0 . Let W = [wj,k ]j,k ∈ Mm+1,n+1 (T), and assume that w0,k0 6∈ {w1,k0 , . . . , wm,k0 }. Then there exists a polynomial p such that Dk0 (p)(w0,k0 ) 6= 0 and Dk (p)(wj,k ) = 0 for every (j, k) 6= (0, k0 ), that is,     p(w0,0 ) · · · Dk0 (p)(w0,k0 ) · · · Dn (p)(w0,n ) 0··· ∗··· 0      p(w1,0 ) · · · Dk0 (p)(w1,k0 ) · · · Dn (p)(w1,n )   0 · · · 0 · · · 0    = .  (2.4) .. .. .. .. ..    . . . .. . . ..  . . . . .   . . . . . . k0 n p(wm,0 ) · · · D (p)(wm,k0 ) · · · D (p)(wm,n ) 0··· 0··· 0 n Proof. Let I1 = {(j, k) ∈ Nm 0 × N0 : wj,k 6= w0,k0 }, and let {z1 , . . . , zm′ } be an enumeration of {wj,k : (j, k) ∈ I1 }. Applying Proposition 2.1 to the following (m′ + 1) × (n + 1) matrix   w0,k0 w0,k0 · · · w0,k0   z1 · · · z1   z1 ′ W = .. . . .   .. , . ..   . . zm ′ zm ′ · · · zm ′

we see that there exists a polynomial q such that  q(w l n 0,k0 ) 6= 0 = D (q)(w0,k0 ) (∀l ∈ N1 ), Dl (q)(wj,k ) = 0 (∀(j, k) ∈ I1 , ∀l ∈ Nn ). 0

Assume that we have constructed a polynomial r such that (2.5)

Dk0 (r)(w0,k0 ) = 1 6= 0 = Dl (r)(w0,k0 )

for every l ∈ Nn0 \ {k0 }. Set p(z) = q(z)r(z). Since q(w0,k0 ) 6= 0 = Dl (q)(w0,k0 ) for every l ∈ Nn1 , the Leibniz rule implies that Dk0 (p)(w0,k0 ) 6= 0 = Dk (p)(w0,k0 ) for every k ∈ Nn0 \ {k0 }. Moreover, if (j, k) ∈ I1 , then Dl (q)(wj,k ) = 0 for every l ∈ Nn0 , and thus the Leibniz rule implies that Dk (p)(wj,k ) = 0. Hence p(z) satisfies the condition (2.4). Now, it remains to construct a polynomial r satisfying the condition (2.5). It follows from equality (2.1) that a polynomial r(z) = a0 + a1 z + · · · + an z n satisfies the condition (2.5) if and only if the coefficients of r satisfy the system of n + 1 linear equations  n i−k0 (k = k ), X 0 k j j w0,k0 aj = 0 (otherwise). j=0

Surjective isometries on An (D)

89

The system of linear equations has a solution (a0 , . . . , an ) ∈ Cn+1 . Indeed, the determinant 2 n 1 w0,k0 2w0,k 1 1 1 . . . 1 . . . nw 0,k 0 0 0 w 0 1 2 . . . n 2 2 2 n 0,k0 2 w0,k0 . . . n w0,k0 n(n+1) 3 2 3 n 2 2 0 w0,k0 2 w0,k0 . . . n w0,k0 = w 2 0 1 2 . . . n 0,k0 .. .. .. .. .. .. .. . . .. .. . . . . . . . . . . n 2 n n 0 w 0 1 2n . . . n n 0,k0 2 w0,k0 . . . n w0,k0 is non-zero, because the right-hand side of the above equality is a determinant of a Vandermonde matrix whose columns are the geometric sequence with pairwise distinct common ratios. Hence we can find a polynomial r satisfying the condition (2.5). The proof is completed. □ Proposition 2.3. Let k0 ∈ Nn0 , let wk0 , . . . , wn ∈ T, and assume that wk0 6∈ {wk0 +1 , . . . , wn }. For each ε > 0 and each neighborhood V of wk0 in T, there exists a polynomial p such that   kDl (p)kT < ε (l ∈ Nk00 −1 ),     kDk0 (p)k = Dk0 (p)(w ) = k !, T k0 0 (2.6)  kDk0 (p)kT\V < ε,     |Dl (p)(w )| < ε (l ∈ Nnk0 +1 ), l where kDk0 (p)kT\V is the supremum of |Dk0 (p)| on T \ V . Proof. For each m ∈ N, consider the polynomial   m (−i)k0 X 1 m pm (z) = (wk0 z)m+j . m k 0 2 (m + j) j j=0 Let us show that the sequence {pm }m has the following   kDl1 (pm )kT → 0     kDk0 (p )k = Dk0 (p )(w ) = 1 m T m k0 (2.7)  kDk0 (pm )kT\V → 0     |Dl2 (p )(w )| → 0 m

l2

properties (m → ∞), (∀m ∈ N), (m → ∞), (m → ∞)

for every neighborhood V of wk0 in T, l1 ∈ Nk00 −1 and l2 ∈ Nnk0 +1 . First, by equality (2.1), we have   m (−i)k0 −l X 1 m l D (pm )(z) = (wk0 z)m+j m k −l 0 2 (m + j) j j=0

90

Y. Enami and T. Miura

for every l ∈ Nk00 . In particular,



k0

D (pm )(z) =

wk0 z + (wk0 z)2 2

m .

For each l ∈ Nk00 and w ∈ T,     m m 1 m 1 X 1 m 1 1 X |D (pm )(w)| ≤ m ≤ m = k0 −l , k −l k −l 0 0 2 j=0 (m + j) j 2 j=0 m j m l

and thus kDl (pm )kT ≤ 1/mk0 −l . This shows that kDl (pm )kT → 0 as m → ∞ for every l ∈ Nk00 −1 , and that kDk0 (pm )kT ≤ 1 for every m ∈ N. Since Dk0 (pm )(wk0 ) = 1, we obtain kDk0 (pm )kT = Dk0 (pm )(wk0 ) = 1. Let V be a neighborhood of wk0 in T. Since wk0 z + (wk0 z)2 < 1, sup 2 z∈T\V we have kDk0 (pm )kT\V → 0 as m → ∞. 0 Let us verify the rest of the property in (2.7). Let l ∈ Nn−k . By equality (2.2), 1 D

k0 +l

l

(pm )(z) = i

l X

cj (Dk0 (pm ))(j) (z)z j ,

j=1

where c1 , . . . , cl are constants independent of m. Thus, to show that Dk0 +l (pm )(z) → 0 as m → ∞, it suffices to prove that (Dk0 (pm ))(j) (wk0 +l ) → 0 as m → ∞ for every j ∈ Nl1 . Fix j0 ∈ Nl1 . It is easy to see that for each positive integer m with j0 < m, the j0 -th formal derivative of Dk0 (pm ) can be written as k0

(j0 )

(D (pm ))

(z) =

j0 X j=1

 j

m qj (z)

wk0 z + (wk0 z)2 2

m−j ,

where q1 , . . . , qj0 are polynomials independent of m. By our hypothesis on wk0 , we have |wk0 wk0 +l + (wk0 wk0 +l )2 |/2 < 1, and thus  m−j wk0 wk0 +l + (wk0 wk0 +l )2 j m qj (wk0 +l ) →0 (m → ∞) 2 for every j ∈ Nj00 , and thus (Dk0 (pm ))(j0 ) (wk0 +l ) → 0 as m → ∞, as desired. Now, let ε > 0, and let V be a neighborhood of wk0 in T. Choose m ∈ N so large that kDl1 (pm )kT , kDk0 (pm )kT\V , |Dl2 (pm )(wl2 )|
ε and |µ|(K2 ) > ε. Proof. Suppose that K1 , K2 are disjoint compact sets in X with |µ| measure greater than ϵ. Let U1 , U2 be disjoint open sets in X containing K1 , K2 , respectively. It follows from the Hahn decomposition theorem (for |µ| restricting to K1 ) (see, e.g., [15, Theorem 6.14]), we can assume that |µ|(K1 ) = µ(A) − µ(B) > ε where K1 = A ∪ B is a partition into the positive and negative subsets of K1 with respect to the real measure µ. By the regularity of µ, we can choose compact subsets A1 , B1 of A, B such that µ(A \ A1 ) − µ(B \ B1 ) < ε − µ(A) + µ(B).

128

An, Chen, Li and Liu

By the Urysohn Lemma, we can choose a function f1 in C(X) such that f1 = 1 on A1 , f1 = −1 on B1 , −1 ≤ f1 ≤ 1 and f1 = 0 outside U1 . Then ∫ ∆(f1 ) = f1 dµ > µ(A1 ) − µ(B1 ) − µ(A \ A1 ) + µ(B \ B1 ) > ε. X

In a similar manner, we have f2 ∈ C(X) such that f1 ⊥f2 and ∆(f2 ) > ε. This disjoint pair f1 , f2 verifies that ∆ is not ε-disjointness preserving. Conversely, suppose that, for every two disjoint compact sets K1 and K2 , we have that |µ|(K1 ) ≤ ε or |µ|(K2 ) ≤ ε. Let f ⊥ g with |f | ≤ 1 and |g| ≤ 1. Without loss of generality, one can assume that there is a compact subset L1 ⊂ coz(f ) such that |µ|(L1 ) > ε, this forces that, for every compact subset L2 ⊂ coz(g), |µ|(L2 ) ≤ ε. It follows from the regularity of µ that |µ|(coz(g)) ≤ ε, and then ∫ ∫ |∆(g)| = | gdµ| ≤ |g|d|u| X ∫ X = |g|d|u| ≤ |µ|(coz(g)) ≤ ε. coz(g) This implies that |∆(f )| ∧ |∆(g)| ≤ ε, and then ∆ is a bounded ε-disjointness preserving linear functional. Recall that R ⊕1 C(X) is the Banach lattice with the canonical order and norm, where X is a compact Hausdorff space and C(X) is the spaces of continuous functions on X. Suppose that ψ is a linear functional on R ⊕1 C(X). Let ψl : r ∈ R 7→ ψ(r, 0) ∈ R and ψr : f ∈ C(X) 7→ ψ(0, f ) ∈ R, we have that ψ(r, f ) = ψl (r) + ψr (f ) for all (r, f ) ∈ R ⊕1 C(X). Theorem 2.2. is of the form

Let ψ be a continuous linear functional on R ⊕1 C(X). Then ψ ∫ ψ(r, f ) = kr +

f dµ, X

where k is a real number and µ is a regular Borel finite measure. Proof. Since ψ is continuous, ψ1 and ψr both are continuous, then there exists a regular Borel finite measure µ such that ∫ ψr (f ) = f dµ, ∀ f ∈ C(X), X

□

Almost disjointness preserving functionals

and there exists a real number k such that ψl (r) = kr for all r ∈ R.

129

□

It is easy to show that Theorem 2.3. Let ψ be a bounded ε-disjointness preserving linear functional on R ⊕1 C(X). Then ψl and ψr are both bounded ε-disjointness preserving. Theorem 2.4. Suppose that ψ is a continuous linear functional on R ⊕1 C(X), then ψ is ε-disjointness preserving if and only if following two conditions holds. (i) kψl k ≤ ε or kψr k ≤ ε. (ii) ψl and ψr are bounded ε-disjointness preserving linear functional. Proof. Since ψ is a bounded ε-disjointness preserving linear functional, it follows from Theorem 2.3 that ψl and ψr are bounded ε-disjointness preserving. Suppose on the contrary that kψl k > ε and kψr k > ε, then there exists a f ∈ C(X) with kf k = 1 such that |ψr (f )| > ε. Let ξ1 = (0, f ) and ξ2 = (1, 0), we can derive that ξ1 ⊥ ξ2 , |ψ(ξ1 )| = |ψr (f )| > ε and |ψ(ξ2 )| = |ψl (1)| = kψl k > ε, which implies ψ is not εdisjointness preserving. Conversely, suppose that η1 = (r1 , f1 ) and η2 = (r2 , f2 ) be in R ⊕1 C(X) with η1 ⊥ η2 and kη1 k ≤ 1 and kη2 k ≤ 1, then one can derive that r1 r2 = 0 and f1 f2 = 0. Without loss of generality, we can assume that r1 =0 and r2 6= 0. It follows from Theorem 2.3 that ψr is ε-disjointness preserving, which implies that |ψr (f1 )| ∧ |ψr (f2 )| ≤ ε. Since kη2 k = |r2 | + kf2 k∞ ≤ 1, then we have that kf2 k∞ ≤ 1. 1 − |r2 | On the one hand, if |ψ(η1 )| = |ψr (f1 )| ≤ ε, then |ψ(η1 )| ∧ |ψ(η2 )| ≤ ε. On the other hand, if |ψ(η1 )| = |ψr (f1 )| > ε, since ψr is ε-disjointness preserving, then one can derive that |ψr (

(2.2)

f2 )| ≤ ε. 1 − |r2 |

In case of kψl k ≤ ε, we have that |ψ(η2 )| = |ψl (r2 ) + ψr (f2 )| f2 ) 1 − |r2 | ≤ ε|r2 | + (1 − |r2 |)ε = ε,

≤ ε|r2 | + (1 − |r2 |)ψr (

130

An, Chen, Li and Liu

and then |ψ(η1 )| ∧ |ψ(η2 )| ≤ ε. In case of kψr k ≤ ϵ, by the similar argument, one can derive that |ψ(η1 )| = |ψr (f1 )| ≤ kψr k ≤ ε, and then |ψ(η1 )| ∧ |ψ(η2 )| ≤ ε.

□

Theorem 2.5. Suppose that ψ is a bounded linear functional on C 1 [0, 1], then ψ is ε-disjointness preserving if and only if ∫ ′ ψf = kf (0) + f dµ, [0,1]

where k is a real number and µ is a regular Borel finite measure such that |k| ≤ ε or |µ|([0, 1]) ≤ ε, and µ satisfies the conditions of Theorem 2.1. Proof. Since ψ is ε-disjointness preserving if and only if ψπ −1 is a ε-disjointness preserving on R ⊕1 C[0, 1], we can complete the proof using Theorem 2.4. □ Acknowledgement This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. The authors would like to thank the referees for the careful reading, detailed helpful remarks and valuable comments.

References [1] Y. A. Abramovich and A. K. Kitover, Inverses of disjointness preserving operators, Mem. Amer. Math. Soc., 143(2000), no. 679. [2] J. Araujo, Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity, Adv. Math., 187 (2004), 488-520. [3] K. Boulabiar, Recent trends on order bounded disjointness preserving operators, Irish Math. Soc. Bull., 62(2008), 43-69. [4] L. G. Brown and N.-C. Wong, Unbounded disjointness preserving linear functionals, Monatsh. Math., 141(2004), 21-32. [5] D. H. Leung, L. Li and Y.-S. Wang, Inverses of disjointness preserving operators, Studia Math., 234(3)(2016), 217-240. [6] G. Dolinar, Stability of disjointness preserving mappings, Proc. Amer. Math. Soc., 130(1)(2002), 129-138. [7] K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull., 33(1990), 139-144. [8] R. Kantrowitz and M. Neumann, Disjointness preserving and local operators on algebras of differentiable functions, Glag. Math. J., 43(2001), 295-309.

Almost disjointness preserving functionals

131

[9] H. Koshimizu, Finite codimensional linear isomortries on spaces of differentiable and Lipschitz functions, Cent. Eur. J. Math., 9(2011), 139-146. [10] D. H. Leung, L. Li and Y.-S. Wang, Inverses of disjointness preserving operators, Studia Math., 234(3)(2016), 217-240. [11] P. Meyer-Nieberg, Banach lattices, Springer, Berlin, 1991. [12] T. Oikhberg, On the stability of some preservers, Linear Algebra Appl., 563(2019), 494526, [13] T. Oikhberg, A. M. Peralta and D. Puglisi, Automatic continuity of orthognality or disjointness preserving bijections, Rev. Mat. Complut., 26(2013), 57-88. [14] T. Oikhberg and P. Tradacete, Almost band preservers, Canad. J. Math., 69(2017), 650686. [15] W. Rudin, Real and Complex Analysis(3rd Edition), McGraw-Hill Book Co., Singapore, 1986.

RIMS Kˆ okyˆ uroku Bessatsu B93 (2023), 133–141

Isometries, Jordan ∗-isomorphisms and order isomorphisms on spaces of a unital C ∗ -algebra-valued continuous maps By

Shiho Oi∗

Abstract This article presents a survey of the paper [14] with applications. In this paper, we study Jordan ∗-isomorphisms, surjective linear isometries and order isomorphisms on the spaces of continuous maps taking values in unital C ∗ -algebras.

§ 1.

Introduction

Let X be a compact Hausdorff space and (A, k·kA ) a unital C ∗ -algebra. We denote the unit of A by 1A . In this paper, every C ∗ -algebra is assumed to be unital. An element a in a C ∗ -algebra is positive if a is self-adjoint and the spectrum σ(a) ⊂ {r ∈ R |r ≥ 0}. We denote by A+ the collection of positive elements in A. We denote the set of all pure states on A by PS(A). For any ρ ∈ PS(A), there exists an irreducible representation πρ : A → B(Hρ ), where B(Hρ ) is the Banach space of all bounded linear operators on a Hilbert space Hρ . We denote C(X, A) by the space of all A-valued continuous maps on X with the supremum norm k · k∞ , that is, kF k∞ = sup{kF (x)kA : x ∈ X}. When A = C, we denote C(X, C) by C(X). For any F ∈ C(X, A), we define F ∗ ∈ C(X, A) by F ∗ (x) = [F (x)]∗ for any x ∈ X. Then C(X, A) is a unital C ∗ -algebra. The unit of C(X, A) is a constant map 1 : X → A which satisfies 1(x) = 1A for any x ∈ X. Throughout the paper let X1 , X2 be compact Hausdorff spaces and A1 , A2 be unital C ∗ -algebras. Received March 10, 2022. Revised November 8, 2022. 2020 Mathematics Subject Classification(s): 46E40, 47C10, 46B04 Key Words: Banach-Stone property, C ∗ -algebra, vector-valued continuous map. Supported by JSPS KAKENHI Grant Numbers JP21K13804 ∗ Niigata University, Niigata 950-2181, Japan. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

134

Shiho Oi

Definition 1.1.

Let T : A1 → A2 be a bijective linear map.

ˆ If the map T satisfies that kT a − T bkA2 = ka − bkA1 for any a, b ∈ A1 , T is called an isometry. + ˆ If both T and T −1 preserve the order structure, i.e., a ∈ A+ 1 if and only if T a ∈ A2 , then T is called an order isomorphism.

ˆ If the map T holds T (a∗ ) = (T a)∗ and T (a2 ) = (T a)2 for any a ∈ A1 , then T is called a Jordan ∗-isomorphism.

In this paper we study Jordan ∗-isomorphisms, surjective linear isometries and order isomorphisms from C(X1 , A1 ) onto C(X2 , A2 ). In the classical period three results stand out, namely the theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky. Theorem 1.2 (Banach and Stone [1, 16]). Let X1 and X2 be compact Hausdorff spaces. Let T : C(X1 ) → C(X2 ) be a map. Then T is a surjective linear isometry if and only if there is a homeomorphism φ : X2 → X1 and a continuous function u ∈ C(X2 ) with |u| = 1 such that T f (y) = u(y)(f (φ(y))),

f ∈ C(X1 ),

y ∈ X2 .

Remark. Banach proved in [1] that if T is a surjective linear isometry from CR (X1 ) onto CR (X2 ), where CR (Xj ) is the space of all real-valued continuous functions on a compact metric space Xj , then T is a weighted composition operator. In the case when Xj are compact Hausdorff spaces, Stone proved that every surjective linear isometry between CR (Xj ) is also a weighted composition operator in [16]. The statement of Theorem 1.2 is considered as the modern version of Banach’s and Stone’s contributions and is called the Banach-Stone theorem. Theorem 1.3 (Kaplansky [10]). Let X1 and X2 be compact Hausdorff spaces. Let T : C(X1 ) → C(X2 ) be a map. Then T is an order isomorphism if and only if there is a homeomorphism φ : X2 → X1 and a positive invertible element u ∈ C(X2 ) such that T f (y) = u(y)(f (φ(y))), f ∈ C(X1 ), y ∈ X2 . Theorem 1.4 (Gelfand and Kolmogorov [4]). Let X1 and X2 be compact Hausdorff spaces. Let T : C(X1 ) → C(X2 ) be a map. Then T is an algebra isomorphism if and only if there is a homeomorphism φ : X2 → X1 such that T f (y) = f (φ(y)),

f ∈ C(X1 ),

y ∈ X2 .

Isometries, Jordan ∗-isomorphisms and order isomorphisms on C(X, A)

135

These imply that a compact Hausdorff space X is determined by the metric structure, the order structure and the algebraic structure respectively. How about the case of the vector-valued maps? Jerison in [7] obtained the first vector-valued version of the Banach-Stone Theorem: Suppose that E is a strictly convex Banach space. Let T be a surjective linear isometry from C(X1 , E) onto C(X2 , E). Then there is a homeomorphism φ : X2 → X1 and a continuous map U with the strong operator topology from X2 into the space of surjective linear isometries on E, where Uy : E → E is a surjective linear isometry for any y ∈ X2 , such that T F (y) = Uy (F (φ(y))),

F ∈ C(X1 , E),

y ∈ X2 .

Cambern, in [2], defined the Banach-Stone property. A Banach space E is said to have the Banach-Stone property if every surjective linear isometry T : C(X1 , E) → C(X2 , E) admits a homeomorphism φ : X2 → X1 and a strongly continuous family {Vy }y∈X2 of surjective linear isometries on E such that T F (y) = Vy (F (φ(y))),

F ∈ C(X1 , E),

y ∈ X2 .

Later many results of surjective linear isometries are exhibited including in [3, 5, 6, 11]. They studied Banach spaces which have the Banach-Stone property. In particular, in [5], the authors studied unital surjective linear isometries between the injective tensor product of a uniform algebra and a unital factor C ∗ -algebra. A unital C ∗ -algebra A is called a factor if its center is trivial. As a corollary of their main theorems, they show that unital factor C ∗ -algebras have the Banach-Stone property as the following: Theorem 1.5 (Corollary 5 in [5]). Let Ai be a unital factor C ∗ -algebra for i = 1, 2. Then a bijective linear map T : C(X1 , A1 ) → C(X2 , A2 ) is a surjective linear isometry if and only if there exists a homeomorphism φ : X2 → X1 , a strongly continuous family {Vy }y∈X2 of Jordan ∗-isomorphisms from A1 onto A2 and a unitary element U ∈ C(X2 , A2 ) such that T F (y) = U (y)Vy (F (φ(y))) for any F ∈ C(X1 , A1 ) and y ∈ X2 . The authors studied hermitian operators on the injective tensor product of a uniform algebra and a unital factor C ∗ -algebra. They applied the notion of hermitian operators and the technique, which was introduced by Lumer in [12, 13], to characterize surjective linear isometries on the spaces. In this paper, we study surjective linear isometries by applying studies of Jordan ∗-isomorphisms. In Theorem 3.3 we prove that a primitive C ∗ -algebra A has the Banach-Stone property. Since a primitive C ∗ -algebra is factor, Theorem 3.3 follows

136

Shiho Oi

from Theorem 1.5. However the proof is quite different and simple. We do not need studying hermitian operators on C(X, A).

§ 2.

Jordan ∗-isomorphisms

We introduce the studies of Jordan ∗-isomorphisms on C(X, A) in [14]. In addition, we will proceed to sketch the main ideas of the proof for completeness. We refer the reader to the paper [14] for more details. In the proof, we consider the algebraic tensor product space. The algebraic tensor product space of C(X) and A over C is denoted by C(X) ⊗ A. Theorem 2.1 (Theorem 2.4. in [14]). Let J : C(X1 , A1 ) → C(X2 , A2 ) be a Jordan ∗-isomorphism. Then there exist a continuous map φρ : X2 → X1 for every ρ ∈ PS(A2 ) and a Jordan ∗-homomorphism Vy : A1 → A2 for each y ∈ X2 such that πρ (JF (y)) = πρ (Vy (F (φρ (y)))) for all F ∈ C(X1 , A1 ) and all y ∈ X2 . Outline of the proof of Theorem 2.1. We sketch the outline of the proof of Theorem 2.1. We have PS(C(X2 , A2 )) = {ρ ◦ δy | ρ ∈ PS(A2 ), y ∈ X2 }, where δy is a complex linear operator from C(X2 , A2 ) into A2 such that δy (F ) = F (y) for any F ∈ C(X2 , A2 ). We denote the commutant of πρ◦δy (C(X2 , A2 )) by πρ◦δy (C(X2 , A2 ))′ . We denote the identity operator on Hρ◦δy by IHρ◦δy . Since a space πρ◦δy (C(X2 , A2 )) acts irreducibly on Hρ◦δy , we get πρ◦δy (C(X2 , A2 ))′ = CIHρ◦δy . Corollary 3.4 in [15] implies that πρ◦δy ◦ J : C(X1 , A1 ) → B(Hρ◦δy ) is either a ∗homomorphism or an anti ∗-homomorphism. Thus we get for any f ∈ C(X1 ), πρ◦δy ◦ J(f ⊗ 1A1 ) ∈ CIHρ◦δy . We define λf ∈ C by (2.1)

πρ◦δy ◦ J(f ⊗ 1A1 ) = λf · IHρ◦δy .

Since J ∗ (ρ ◦ δy ) ∈ PS(C(X1 , A1 )), there exist ϕρ,y ∈ PS(A1 ) and x ∈ X1 such that J ∗ (ρ ◦ δy ) = ϕρ,y ◦ δx . We define φρ : X2 → X1 by J ∗ (ρ ◦ δy ) = ϕρ,y ◦ δφρ (y) . Since J(1) = 1, we get f (φρ (y)) = λf for any f ∈ C(X1 ). Moreover (2.1) shows that πρ (J(f ⊗ 1A1 )(y)) = f (φρ (y))IHρ . Fix y ∈ Y2 . We define a Jordan ∗-homomorphism Vy : A1 → A2 by Vy (a) = J(1 ⊗ a)(y)

Isometries, Jordan ∗-isomorphisms and order isomorphisms on C(X, A)

137

for any a ∈ A1 . As ρ ∈ PS(A2 ), πρ is an irreducible representation of A2 . It follows that πρ (J(f ⊗ a)(y)) = πρ (J(f ⊗ 1)(y))πρ (J(1 ⊗ a)(y)) = πρ (Vy (f (φρ (y))a)) = πρ (Vy (f ⊗ a(φρ (y)))). Since C(X1 ) ⊗ A1 is dense in C(X1 , A1 ) and J is a bounded operator with k · k∞ , we have πρ (JF (y)) = πρ (Vy (F (φρ (y)))) for all F ∈ C(X1 , A1 ) and all y ∈ X2 . □ Since there is a Jordan ∗-isomorphism from C({a}, C2 ) onto C({x, y}, C), Remark. C({a}, C2 ) and C({x, y}, C) is isometric ∗-isomorphic. On the other hand, {a} is not homeomorphic to {x, y} and C2 is not ∗-isomorphic to C. We can not expect to get that Vy is an isomorphism and φρ is a homeomorphism. Theorem 2.2 (Theorem 2.7. in [14]). Assume that A1 and A2 are primitive. Then J : C(X1 , A1 ) → C(X2 , A2 ) is a Jordan ∗-isomorphism if and only if there exist a homeomorphism φ from X2 onto X1 and a Jordan ∗-isomorphism Vy : A1 → A2 for each y ∈ X2 so that the map y 7→ Vy is continuous with respect to the strong operator topology such that (2.2)

JF (y) = Vy (F (φ(y)))

for all F ∈ C(X1 , A1 ) and y ∈ X2 . Outline of the proof of Theorem 2.2. We omit the proof of the statement that if J is of the form described as (2.2) in the statement then J is a Jordan ∗-isomorphism from C(X1 , A1 ) onto C(X2 , A2 ). We only mention that if J : C(X1 , A1 ) → C(X2 , A2 ) is a Jordan ∗- isomorphism, then J has the form as (2.2) with the desired properties for φ and Vy . Since A2 is a primitive C ∗ -algebra, there is a ρ ∈ PS(A2 ) such that πρ is faithful. We define φ : X2 → X1 by φ = φρ . By Theorem 2.1, we have JF (y) = Vy (F (φ(y))). We prove that φ is a homeomorphism. Let y1 , y2 ∈ X2 such that φ(y1 ) = φ(y2 ). Since A1 is a primitive C ∗ - algebra, there exist a continuous map ψ : X1 → X2 and a Jordan ∗-homomorphism Sx : A2 → A1 for each x ∈ X1 such that J −1 F (x) = Sx (F (ψ(x))), for all F ∈ C(X2 , A2 ). This implies that for all F ∈ C(X1 , A1 ), we have F (x) = Sx (Vψ(x) F (φ(ψ(x)))). As C(X1 , A1 ) separates the points of X1 , we get φ(ψ(x)) = x for any x ∈ X1 . Applying a similar argument, we get F (y) = Vy (Sφ(y) (F (ψ(φ(y))))) for all F ∈ C(X2 , A2 ) and ψ(φ(y)) = y for any y ∈ X2 . As X1 and X2 are compact

138

Shiho Oi

Hausdorff spaces and φ is a continuous map, φ is a homeomorphism. By φ−1 = ψ, we get Sφ(y) ◦ Vy = IA1 and Vy ◦ Sφ(y) = IA2 , where IAi is the identity operator on Ai for i = 1, 2. Thus Vy is a bijection and Vy is a Jordan ∗-isomorphism such that Vy−1 = Sφ(y) . Let {yλ } ⊂ X2 be a net with yλ → y0 . For any a ∈ A1 , kVyλ (a) − Vy0 (a)kA2 = kJ(1 ⊗ a)(yλ ) − J(1 ⊗ a)(y0 )kA2 → 0 as J(1 ⊗ a) ∈ C(X2 , A2 ). This implies that the map y 7→ Vy is continuous with respect to the strong operator topology.

□

§ 3.

Applications of Theorems 2.1 and 2.2

§ 3.1.

Surjective linear isometries

Kadison in 1951 obtained the following characterization for surjective complex linear isometries between unital C ∗ -algebras. Theorem 3.1 (Kadison [8]). Let Ai be a unital C ∗ -algebra for i = 1, 2. Then T : A1 → A2 is a surjective linear isometry if and only if there is a unitary element u ∈ A2 and a Jordan ∗-isomorphism J : A1 → A2 such that T (a) = uJ(a),

a ∈ A1 .

Applying Theorem 3.1, we obtain the following theorems. Theorem 3.2. Let T : C(X1 , A1 ) → C(X2 , A2 ) be a surjective linear isometry. Then there exist a unitary element U ∈ C(X2 , A2 ), a continuous map φρ : X2 → X1 for every ρ ∈ PS(A2 ) and a Jordan ∗-homomorphism Vy : A1 → A2 for each y ∈ X2 such that πρ (T F (y)) = πρ (U (y)Vy (F (φρ (y)))) for all ρ ∈ PS(A2 ), F ∈ C(X1 , A1 ) and y ∈ X2 . Proof. By theorem 3.1, there is a unitary element U ∈ C(X2 , A2 ) and a Jordan ∗-isomorphism J : C(X1 , A1 ) → C(X2 , A2 ) such that T (F ) = U J(F ) for any F ∈ C(X1 , A1 ). By Theorem 2.1, there exist a continuous map φρ : X2 → X1 for every ρ ∈ PS(A2 ) and a Jordan ∗-homomorphism Vy : A1 → A2 for each y ∈ X2 such that πρ (JF (y)) = πρ (Vy (F (φρ (y)))) for all F ∈ C(X1 , A1 ) and all y ∈ X2 . Since πρ : A2 → B(Hρ ) is an irreducible representation for any ρ ∈ PS(A2 ), we have πρ (T F (y)) = πρ (U (y)J(F )(y)) = πρ (U (y))πρ (J(F )(y)) = πρ (U (y))πρ (Vy (F (φρ (y)))) = πρ (U (y)Vy (F (φρ (y))))

Isometries, Jordan ∗-isomorphisms and order isomorphisms on C(X, A)

for any F ∈ C(X1 , A1 ), y ∈ X2 .

139

□

In particular when A1 and A2 are primitive C ∗ -algebras, we have the following characterization. Theorem 3.3. Assume that A1 and A2 are primitive. Then T : C(X1 , A1 ) → C(X2 , A2 ) is a surjective linear isometry if and only if there exist a unitary element U ∈ C(X2 , A2 ), a homeomorphism φ from X2 onto X1 and a Jordan ∗-isomorphism Vy : A1 → A2 for each y ∈ X2 so that the map y 7→ Vy is continuous with respect to the strong operator topology such that (3.1)

T F (y) = U (y)Vy (F (φ(y)))

for all F ∈ C(X1 , A1 ) and y ∈ X2 . Proof. Firstly suppose that T is of the form described as (3.1). Then by Theorem 2.2, the map F 7→ V· (F (φ(·))) from C(X1 , A1 ) onto C(X2 , A2 ) is a Jordan ∗ isomorphism. Thus Theorem 3.1 implies that T is a surjective linear isometry. Now the converse statement is clear by Theorem 3.1 and Theorem 2.2. □ Remark 3.4. Let A be a unital C ∗ -algebra. If A is primitive then it is factor. But the converse is not true. Theorem 3.3 is a corollary of Theorem 1.5. Thus, this result was already obtained but our proof is much simpler than that presented in [5].

§ 3.2.

Order isomorphisms

In [9], Kadison proved that every order isomorphism which carries the identity into identity between unital C ∗ -algebras is a Jordan ∗-isomorphism. We obtain the following characterization of order isomorphisms between unital C ∗ -algebras as a corollary of the theorem of Kadison. Although this is a well-known fact, we give a proof by applying [9, Corollary 5] for completeness. Theorem 3.5 (Kadison [9]). Let Ai be a unital C ∗ -algebra for i = 1, 2. Then T : A1 → A2 is an order isomorphism if and only if there is a positive invertible element u ∈ A2 and a Jordan ∗-isomorphism J : A1 → A2 such that T (a) = uJ(a)u,

a ∈ A.

Proof. Let T : A1 → A2 be an order isomorphism. Then there is a positive element a ∈ A1 with a 6= 0 such that T (a) = 1A2 . Since a ≤ kak1A1 , we get 1A2 = T (a) ≤ kakT (1A1 ).

140

Shiho Oi 1

A2 As kak 6= 0, we get ∥a∥ ≤ T (1A1 ). By the Gelfand representation applied to the C ∗ algebra generated by T (1A1 ) and 1A2 , we get 0 ∈ / σ(T 1A1 ) and T 1A1 is an invertible 1 element. Hence we put u = (T 1A1 ) 2 . Since u is also invertible and positive in A2 , we define a map J : A1 → A2 by J(a) = u−1 T (a)u−1 for any a ∈ A1 . It is clear that J is an order isomorphism which satisfies that J(1A1 ) = 1A2 . By [9, Corollary 5], J is a Jordan ∗-isomorphism. We conclude that T (a) = uJ(a)u for all a ∈ A1 . Conversely we assume that T (a) = uJ(a)u for any a ∈ A1 , where J is a Jordan ∗-isomorphism. Then T is a linear bijective map. Since J is an order isomorphism, for any a ∈ A1 there is b ∈ A2 such that J(a∗ a) = b∗ b. Hence we obtain that

T (a∗ a) = uJ(a∗ a)u = ub∗ bu = (bu)∗ (bu) ≥ 0. In addition, T −1 (b) = J −1 (u−1 bu−1 ) for any b ∈ A2 . We have T −1 (b∗ b) = J −1 (u−1 b∗ bu−1 ) = J −1 ((bu−1 )∗ bu−1 ) ≥ 0 for all b ∈ A2 . Thus T is an order isomorphism.

□

By applying Theorem 3.5 and similar arguments as in the case of surjective linear isometries, we obtain the following theorems. Theorem 3.6. Let T : C(X1 , A1 ) → C(X2 , A2 ) be an order isomorphism. Then there exist a positive invertible element U ∈ C(X2 , A2 ), a continuous map φρ : X2 → X1 for any ρ ∈ PS(A2 ) and a Jordan ∗-homomorphism Vy : A1 → A2 for each y ∈ X2 such that πρ (T F (y)) = πρ (U (y)Vy (F (φρ (y)))U (y)) for all ρ ∈ PS(A2 ), F ∈ C(X1 , A1 ) and y ∈ X2 . Theorem 3.7. Assume that A1 and A2 are primitive. Then T : C(X1 , A1 ) → C(X2 , A2 ) is an order isomorphism if and only if there exist a positive invertible element U ∈ C(X2 , A2 ), a homeomorphism φ from X2 onto X1 and a Jordan ∗-isomorphism Vy : A1 → A2 for each y ∈ X2 where the map y 7→ Vy is continuous with respect to the strong operator topology such that (3.2)

T F (y) = U (y)Vy (F (φ(y)))U (y)

for all F ∈ C(X1 , A1 ) and y ∈ X2 . Remark 3.8. We obtain that for any unital C ∗ -algebra A, Jordan ∗-isomorphisms, surjective linear isomorphisms and order isomorphisms on C(X, A) are represented by weighted composition operators by using the irreducible representations on A. Moreover

Isometries, Jordan ∗-isomorphisms and order isomorphisms on C(X, A)

141

when A is a primitive C ∗ -algebra, we obtain complete representations of these operators. These are one of the vector-valued versions of the classical theorems by Banach and Stone, Gelfand and Kolmogorov, and Kaplansky. It is not clear to the author whether complete representations of Jordan ∗-isomorphisms, surjective linear isomorphisms and order isomorphisms on C(X, A) are obtained for other classes of C ∗ -algebras A. Acknowledgement The author expresses her gratitude to the reviewer for his/her constructive comments. This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. This work was supported by JSPS KAKENHI Grant Numbers JP21K13804.

References [1] S. Banach, Th´eorie des Op´erations Lin´eaires, Monograf. Mat. 1, Warszawa, 1932; reprint, Chelsea, New York, 1963. [2] M. Cambern, Reflexive spaces with the Banach-Stone property, Rev. Roumaine Math. Pures Appl., 23 (1978), no. 7, 1005–1010. [3] R. J. Fleming and J. E. Jamison, Hermitian operators on C(X, E) and the Banach-Stone Theorem, Math. Z., 170 (1980) 77–84. [4] I. Gelfand and A. Kolmogoroff, On rings of continuous functions on topological spaces, Dokl. Akad. Nauk. SSSR (C. R. Acad. Sci. USSR), 22 (1939) 11–15. [5] O. Hatori, K. Kawamura and S. Oi, Hermitian operators and isometries on injective tensor products of uniform algebras and C ∗ -algebras., J. Math. Anal. Appl., 472 (2019), no. 1, 827–841. [6] J.-S. Jeang and N.-C. Wong On the Banach-Stone problem, Studia Math., 155 (2003), 95–105. [7] M. Jerison, The space of bounded maps into a Banach space, Ann. of Math, 52(1950), 309–327. [8] R. V. Kadison, Isometries of operator algebras, Ann. of Math., 54 (1951), 325–338. [9] R. V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2), 56 (1952), 494–503. [10] I. Kaplansky, Lattices of continuous functions, Bull. Amer. Math. Soc., 53 (1947) 617–623. [11] K. S. Lau, A representation theorem for isometries of C(X, E), Pacific J. Math, 60 (1975), 229–233. [12] G. Lumer, Isometries of Orlicz spaces, Bull. Amer. Math. Soc., 68 (1962) 28–30. [13] G. Lumer, On the isometries of reflexive Orlicz spaces, Ann. Inst. Fourier (Grenoble), 13 (1963) 99–109. [14] S. Oi, Jordan ∗-homomorphisms on the spaces of continuous maps taking values in C ∗ algebras, Studia Math., 269 (2023), no.1, 107–119. [15] E. Størmer, On the Jordan structure of C ∗ -algebras, Trans. Amer. Math. Soc., 120 (1965), 438–447. [16] M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375–481.

RIMS Kˆ okyˆ uroku Bessatsu B93 (2023), 143–155

Topological reflexivity of isometries on algebras of matrix-valued Lipschitz maps By

´nez-Vargas∗∗ M. G. Cabrera-Padilla∗ and A. Jime

Abstract Let X and Y be compact metric spaces and let Mn (C) be the Banach algebra of all n × n complex matrices. We prove that the set of all unital surjective linear isometries from Lip(X, Mn (C)) to Lip(Y, Mn (C)), whenever both spaces are endowed with the sum norm, is topologically reflexive.

§ 1.

Introduction and statement of the result

Let E and F be Banach algebras, B(E, F ) be the space of all linear continuous operators from E to F and S be a nonempty subset of B(E, F ). We define the algebraic reflexive closure of S by refalg (S) = {T ∈ B(E, F ) : ∀e ∈ E, ∃Te ∈ S | Te (e) = T (e)} and the topological reflexive closure of S by n o reftop (S) = T ∈ B(E, F ) : ∀e ∈ E, ∃{Te,n }n∈N ⊂ S | lim Te,n (e) = T (e) . n→∞

The set S is said to be algebraically reflexive (respectively, topologically reflexive) if refalg (S) = S (respectively, reftop (S) = S). It is straightforward to verify that the Received October 28, 2021. Revised March 10, 2022. 2020 Mathematics Subject Classification(s): 46B04, 46B07, 46E40. Key Words: algebraic reflexivity, topological reflexivity, surjective linear isometry, vector-valued Lipschitz map. This work was supported by the Research Institute for Mathematical Sciences (an International Joint Usage/Research Center located in Kyoto University) and by Junta de Andaluc´ıa grant FQM194, project UAL-FEDER grant UAL2020-FQM-B1858 and project P 20 00255. ∗ Departamento de Matem´ aticas, Universidad de Almer´ıa, 04120 Almer´ıa, Spain. e-mail: m [email protected] ∗∗ Departamento de Matem´ aticas, Universidad de Almer´ıa, 04120 Almer´ıa, Spain. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

144

´nez-Vargas M. G. Cabrera-Padilla and A. Jime

topological reflexivity of S implies its algebraic reflexivity. The elements of refalg (S) and reftop (S) are known as S-local maps and approximate S-local maps, respectively. The study of these S-local maps was addressed when S is the set of surjective linear isometries, the set of algebra automorphisms or the set of derivations of E to F (see the monograph [8] by Moln´ar). In a recent paper [9], Oi proved that both the set of surjective linear isometries between spaces of complex-valued Lipschitz functions Lip(X) with the sum norm (see Theorem 3.1 in [9]) and the set of unital surjective linear isometries between spaces of matrix-valued Lipschitz maps Lip(X, Mn (C)) with the sum norm (see Theorem 4.1 in [9]) are algebraically reflexive. In [5], we proved that the former set is topologically reflexive. In this note, we prove that the latter set is also topologically reflexive. Two proofs of this result are presented: one –which is more direct– is based on Oi’s second result, and the other on Oi’s first result. We hope that these proofs with different viewpoints would give us some new ideas to deal with the local reflexivity problem of isometries on spaces of Lipschitz maps which take values in a non-commutative Banach algebra. In [2, 4], some results were stated on the algebraic reflexivity of *-isomorphisms on Lip(X, B(H)), whenever B(H) is the C*-algebra of all bounded linear operators on a complex infinite-dimensional separable Hilbert space H, but, apparently, nothing is known on reflexivity of the isometry groups of Lip(X, B(H)). Given n ∈ N, the matrix algebra Mn (C) is the unital Banach algebra of all complex matrices of order n, with the operator norm: kAkMn (C) = sup {kAxk2 : x ∈ Cn , kxk2 ≤ 1}

(A ∈ Mn (C))

where k·k2 is the Euclidean norm of Cn . The identity of Mn (C) is the unit matrix In of order n, and the group of invertible elements of Mn (C) is Inv(Mn (C)) = {A ∈ Mn (C) : det(A) 6= 0} . A result due to Schur [11] establishes that a map Φ : Mn (C) → Mn (C) is a unital surjective linear isometry if and only if there exists a unitary matrix V such that either Φ(A) = V AV −1 for all A ∈ Mn (C), or Φ(A) = V At V −1 for all A ∈ Mn (C). Given a compact metric space (X, dX ), the linear space ( ) kF (x) − F (y)kMn (C) sup Lip(X, Mn (C)) = F : X → Mn (C) | 0 on Z, we conclude σ(Z) = 0, and thus σ({η0 } × {z0 }) = 1. This proves that any representing measure for δx0 is the Dirac measure concentrated at x0 . □

162

D. Hirota and T. Miura

Lemma 2.3. Ch(B) = M × T.

For each x0 = (η0 , z0 ) ∈ M × T, we have x0 ∈ Ch(B), that is,

Proof. We shall prove that δx0 belongs to ext(B1∗ ). Suppose that δx0 = (ξ1 + ξ2 )/2 for ξ1 , ξ2 ∈ B1∗ . For j = 1, 2, there exists a representing measure σj for ξj by the Hahn– Banach theorem and the Riesz representation theorem (see, for example, [25, Theorems fI ) + ξ2 (1 fI ) = 2δx0 (1 fI ) = 2 with |ξj (1 fI )| ≤ 1, we have 5.16 and 2.14]). Since ξ1 (1 fI ) = 1 = ∥ξj ∥ for j = 1, 2. Applying the same argument in [2, p.81] to σj , we see ξj ( 1 that σj is a positive measure. We put σ = (σ1 + σ2 )/2, and then σ is a positive measure. First, we prove that σ is a representing measure for δx0 . Because σj is a representing measure for ξj , we get Z M×T

fedσ =

Z

σ1 + σ2 ξ1 (fe) + ξ2 (fe) )= = δx0 (fe) (fe ∈ B). fed( 2 2 M×T

R fI into the above equality, we have σ(M × T) = fdσ = 1, which Entering fe = 1 1 M×T I shows that ∥σ∥ = 1 = ∥δx0 ∥. Therefore, σ is a representing measure for δx0 . By Lemma 2.2, σ = (σ1 + σ2 )/2 is the Dirac measure, τx0 , concentrated at x0 . We note that σj is a positive measure with j = 1, 2. For each Borel set D with x0 ∈ / D, we obtain (σ1 (D) + σ2 (D))/2 = σ(D) = 0, and thus, σj (D) = 0. Having in mind that ∥σj ∥ = ∥ξj ∥ = 1, we conclude that σj = τx0 for j = 1, 2. Hence, R ξj (fe) = M×T fedσj = fe(x0 ) = δx0 (fe) for any fe ∈ B, which implies that ξ1 = δx0 = ξ2 . This proves δx0 ∈ ext(B1∗ ), which yields x0 ∈ Ch(B). □ We now characterize the set of all maximal convex subsets FB of SB . The following result is proved by Hatori, Oi and Shindo Togashi in [15] for uniform algebras. The proof below of the next proposition is quite similar to that of [15, Lemma 3.2]. Proposition 2.4. Let F be a subset of SB . Then F ∈ FB if and only if there exist λ ∈ T and x ∈ M × T such that F = λVx . Proof. Suppose that F is a maximal convex subset of SB . By [15, Lemma 3.1], F = ξ −1 (1) ∩ SB for some ξ ∈ ext(B1∗ ) = {λδx ∈ B1∗ : λ ∈ T, x ∈ M × T}, where we have used Lemma 2.3. There exist λ ∈ T and x ∈ M × T such that ξ = λδx . Now we can write F = (λδx )−1 (1) ∩ SB = {fe ∈ SB : λfe(x) = 1} = λVx . We thus obtain F = λVx with λ ∈ T and x ∈ M × T. Conversely, suppose that F = λVx for some λ ∈ T and x ∈ M × T. It is routine to check that F is a convex subset of SB . Using Zorn’s lemma, we can prove that there exists a maximal convex subset K of SB with F ⊂ K. By the above paragraph, we see

Tingley’s problem for Lip(I)

163

that K = µVy for some µ ∈ T and y ∈ M × T. Then λVx = F ⊂ K = µVy . Lemma 2.1 shows that (λ, x) = (µ, y), which implies that F = K. Consequently, F is a maximal convex subset of SB . □ Tanaka [28, Lemma 3.5] proved that every surjective isometry between the unit spheres of two Banach spaces preserves maximal convex subsets of the spheres (see also [3, Lemma 5.1]). By these results, we can prove the following lemma. Lemma 2.5. such that

There exist maps α : T×(M×T) → T and ϕ : T×(M×T) → M×T

(2.5)

T (λVx ) = α(λ, x)Vϕ(λ,x)

for all (λ, x) ∈ T × (M × T). Proof. For each (λ, x) ∈ T × (M × T), λVx is a maximal convex subset of SB by Proposition 2.4. By [28, Lemma 3.5], surjective isometry T : SB → SB preserves maximal convex subsets of SB , that is, there exists (µ, y) ∈ T × (M × T) such that T (λVx ) = µVy . If, in addition, T (λVx ) = µ′ Vy′ for some (µ′ , y ′ ) ∈ T × (M × T), then we obtain (µ, y) = (µ′ , y ′ ) by Lemma 2.1. Therefore, if we define α(λ, x) = µ and ϕ(λ, x) = y, then α : T × (M × T) → T and ϕ : T × (M × T) → M × T are well defined maps with T (λVx ) = α(λ, x)Vϕ(λ,x) . □ Lemma 2.6. isfying

The maps α and ϕ from Lemma 2.5 are both surjective maps satα(−λ, x) = −α(λ, x)

and

ϕ(−λ, x) = ϕ(λ, x)

for all (λ, x) ∈ T × (M × T). Proof. Take any (λ, x) ∈ T × (M × T), and then λVx is a maximal convex subset of SB by Proposition 2.4. We get T (−λVx ) = −T (λVx ), which was proved by Mori [20, Proposition 2.3] in a general setting. Lemma 2.5 shows that α(−λ, x)Vϕ(−λ,x) = T (−λVx ) = −T (λVx ) = −α(λ, x)Vϕ(λ,x) . Applying Lemma 2.1, we obtain α(−λ, x) = −α(λ, x) and ϕ(−λ, x) = ϕ(λ, x). There exist well defined maps β : T × (M × T) → T and ψ : T × (M × T) → M × T such that T −1 (µVy ) = β(µ, y)Vψ(µ,y) ((µ, y) ∈ T × (M × T)), since T −1 has the same property as T . For each (µ, y) ∈ T × (M × T), we have, by (2.5), µVy = T (T −1 (µVy )) = T (β(µ, y)Vψ(µ,y) ) = α(β(µ, y), ψ(µ, y))Vϕ(β(µ,y),ψ(µ,y)) .

164

D. Hirota and T. Miura

We derive from Lemma 2.1 that µ = α(β(µ, y), ψ(µ, y)) and y = ϕ(β(µ, y), ψ(µ, y)). These prove that both α and ϕ are surjective. □ By definition, ϕ(λ, x) ∈ M × T for each (λ, x) ∈ T × (M × T). There exist ϕ1 (λ, x) ∈ M and ϕ2 (λ, x) ∈ T such that ϕ(λ, x) = (ϕ1 (λ, x), ϕ2 (λ, x)). We shall regard ϕ1 and ϕ2 as maps defined on T × (M × T) to M and T, respectively. By Lemma 2.6, both ϕ1 and ϕ2 are surjective maps with (2.6)

ϕj (−λ, x) = ϕj (λ, x)

((λ, x) ∈ T × (M × T), j = 1, 2).

Lemma 2.7. Let λj ∈ T and (ηj , zj ) ∈ M × T for j = 1, 2. If η1 ̸= η2 , then there exist fej ∈ SB such that fej ∈ λj V(ηj ,zj ) for j = 1, 2 and ∥fe1 − fe2 ∥∞ = 1. Proof. Take j ∈ {1, 2} and open sets Oj in M with ηj ∈ Oj and O1 ∩ O2 = ∅. By Urysohn’s lemma, there exists uj ∈ SC(M) such that uj (ηj ) = 1 and uj = 0 on M \ Oj . Let fj = I(λj zj uj ), and then we see that fej (η, z) = λj zj uj (η)z for all (η, z) ∈ M × T by (2.1) and (2.3). It follows from fej ∈ λj V(ηj ,zj ) for j = 1, 2 that 1 = |fe1 (η1 , z1 ) − fe2 (η1 , z1 )| ≤ ∥fe1 − fe2 ∥∞ . Hence, it is enough to prove that ∥fe1 − fe2 ∥∞ ≤ 1. We shall prove |fe1 (η, z) − fe2 (η, z)| ≤ 1 for all (η, z) ∈ M × T. Fix an arbitrary (η, z) ∈ M × T. If η ∈ O1 , then u2 (η) = 0, since O1 ∩ O2 = ∅, and thus |fe1 (η, z) − fe2 (η, z)| = |λ1 z1 u1 (η) − λ2 z2 u2 (η)| ≤ |u1 (η)| + |u2 (η)| ≤ 1. If η ∈ M \ O1 , then |fe1 (η, z) − fe2 (η, z)| ≤ 1 by the choice of u1 . We conclude that |fe1 (η, z) − fe2 (η, z)| ≤ 1 for all (η, z) ∈ M × T, which yields ∥fe1 − fe2 ∥∞ ≤ 1. □ If λ ∈ T and x ∈ M × T, then ϕ1 (λ, x) = ϕ1 (1, x); we shall write Lemma 2.8. ϕ1 (λ, x) = ϕ1 (x) for simplicity. Proof. Take any λ ∈ T and x ∈ M × T. Then T (Vx ) = α(1, x)Vϕ(1,x) and T (λVx ) = α(λ, x)Vϕ(λ,x) by (2.5). Suppose, on the contrary, that ϕ1 (λ, x) ̸= ϕ1 (1, x). There exist fe1 ∈ α(1, x)Vϕ(1,x) = T (Vx ) and fe2 ∈ α(λ, x)Vϕ(λ,x) = T (λVx ) such that ∥fe1 − fe2 ∥∞ = 1 by Lemma 2.7. We infer from the choice of fe1 and fe2 that T −1 (fe1 ) ∈ Vx and T −1 (fe2 ) ∈ λVx , which implies that T −1 (fe1 )(x) = 1 and T −1 (fe2 )(x) = λ. If Re λ ≤ √ 0, then |1 − λ| ≥ 2, and thus √ 2 ≤ |1 − λ| = |T −1 (fe1 )(x) − T −1 (fe2 )(x)| ≤ ∥T −1 (fe1 ) − T −1 (fe2 )∥∞ = ∥fe1 − fe2 ∥∞ = 1,

Tingley’s problem for Lip(I)

165

where we have used that T is an isometry on SB . We arrive at a contradiction, which shows ϕ1 (λ, x) = ϕ1 (1, x), provided that Re λ ≤ 0. Now we consider the case when Re λ > 0. Then ϕ1 (−λ, x) = ϕ1 (1, x), since Re(−λ) < 0. By (2.6), ϕ1 (λ, x) = ϕ1 (−λ, x) = ϕ1 (1, x), even if Re λ > 0. □ Lemma 2.9. (2.7)

For each λ1 , λ2 ∈ T and x ∈ M×T, the following inequality holds: |λ1 − λ2 | ≤ |1 − α(λ1 , x)α(λ2 , x)|.

Proof. Fix λ1 , λ2 ∈ T and x ∈ M × T. We set fj = α(λj , x)1I ∈ SLip(I) for each j ∈ {1, 2}. We see that fej ∈ α(λj , x)Vϕ(λj ,x) = T (λj Vx ) by (2.5). Then T −1 (fej ) ∈ λj Vx , and hence T −1 (fej )(x) = λj . We obtain |λ1 − λ2 | = |T −1 (fe1 )(x) − T −1 (fe2 )(x)| ≤ ∥T −1 (fe1 ) − T −1 (fe2 )∥∞ = ∥fe1 − fe2 ∥∞ fI ∥∞ = |1 − α(λ1 , x)α(λ2 , x)|. = |α(λ1 , x) − α(λ2 , x)| ∥1 Thus, |λ1 − λ2 | ≤ |1 − α(λ1 , x)α(λ2 , x)| holds for all λ1 , λ2 ∈ T and x ∈ M × T.

□

For each x ∈ M × T, there exists ε0 (x) ∈ {±1} such that Lemma 2.10. ε0 (x) α(λ, x) = λ α(1, x) for all λ ∈ T; for simplicity, we shall write α(1, x) = α(x). Proof. Let λ ∈ T \ {±1} and x ∈ M × T. Taking λ1 = 1 and λ2 = ±λ in (2.7), we obtain |1 − λ| ≤ |1 − α(1, x)α(λ, x)|

and |1 + λ| ≤ |1 + α(1, x)α(λ, x)|,

where we have used Lemma 2.6. Since α(1, x)α(λ, x) ∈ T, we conclude that α(1, x)α(λ, x) ∈ {λ, λ}. If we consider the case when λ = i, then we have α(1, x)α(i, x) ∈ {±i}. This implies that α(i, x) = iε0 (x)α(1, x) for some ε0 (x) ∈ {±1}. Entering λ1 = i and λ2 = λ into (2.7) to get |i − λ| ≤ |1 − α(i, x)α(λ, x)| = |1 + iε0 (x)α(1, x)α(λ, x)| = |i − ε0 (x)α(1, x)α(λ, x)|, and thus |i−λ| ≤ |i−ε0 (x)α(1, x)α(λ, x)|. Because α(−λ, x) = −α(λ, x) by Lemma 2.6, we get |i + λ| ≤ |i + ε0 (x)α(1, x)α(λ, x)|. These inequalities imply ε0 (x)α(1, x)α(λ, x) ∈ {λ, −λ}, since ε0 (x)α(1, x)α(λ, x) ∈ T. Then α(1, x)α(λ, x) ∈ {λ, λ} ∩ {ε0 (x)λ, −ε0 (x)λ}. We have two possible cases to consider. If ε0 (x) = 1, then we obtain α(1, x)α(λ, x) ∈ {λ, λ} ∩ {λ, −λ}. Since λ ̸= ±1, we conclude that α(1, x)α(λ, x) = λ, and hence

166

D. Hirota and T. Miura

α(λ, x) = λε0 (x) α(1, x). If ε0 (x) = −1, then α(1, x)α(λ, x) ∈ {λ, λ} ∩ {−λ, λ}, which yields α(1, x)α(λ, x) = λ. Thus, α(λ, x) = λε0 (x) α(1, x). These identities are valid even for λ = ±1. By the liberty of the choice of λ ∈ T, we conclude that α(λ, x) = λε0 (x) α(1, x) for all λ ∈ T and x ∈ M × T. □ By Lemmas 2.8 and 2.10, we can rewrite (2.5) as T (λVx ) = λε0 (x) α(x)V(ϕ1 (x),ϕ2 (λ,x))

(2.8)

for all λ ∈ T and x ∈ M × T. Definition 1. Let λVx and µVy be maximal convex subsets of SB , where λ, µ ∈ T and x, y ∈ M × T. We denote by dH (λVx , µVy ) the Hausdorff distance of λVx and µVy , that is, ( ) (2.9) dH (λVx , µVy ) = max sup d(fe, µVy ), sup d(λVx , ge) , fe∈λVx

g e∈µVy

h − ge∥∞ . h∥∞ and d(λVx , ge) = inf eh∈λVx ∥e where d(fe, µVy ) = inf eh∈µVy ∥fe − e Since T is a surjective isometry on SB , we obtain d(T (fe), T (µVy )) =

inf

e h∈T (µVy )

∥T (fe) − e h∥∞ =

inf

T −1 (e h)∈µVy

∥fe − T −1 (e h)∥∞ = d(fe, µVy )

for every fe ∈ λVx . Hence, supT (fe)∈T (λVx ) d(T (fe), T (µVy )) = supfe∈λVx d(fe, µVy ). By the same reasoning, we get supT (eg)∈T (µVy ) d(T (λVx ), T (e g )) = supge∈µVy d(λVx , ge), and thus (2.10)

dH (T (λVx ), T (µVy )) = dH (λVx , µVy )

Remark 2.

(λ, µ ∈ T, x, y ∈ M × T).

Let λ ∈ T and (η, z) ∈ M × T. For each fe ∈ λV(η,z) , we observe that λf (0) ∈ [0, 1]

and fb′ (η)λz = ∥fb′ ∥∞ .

In fact, f (0) + fb′ (η)z = λ by the definition of λV(η,z) . Then 1 = λ{f (0) + fb′ (η)z} = |λ{f (0) + fb′ (η)z}| ≤ |λf (0)| + |fb′ (η)λz| ≤ ∥f ∥σ = 1, and thus, |λf (0)+ fb′ (η)λz| = |λf (0)|+|fb′ (η)λz|. This implies that λf (0) = tfb′ (η)λz for some t ≥ 0, provided fb′ (η) ̸= 0. Since λ{f (0)+ fb′ (η)z} = 1, we have fb′ (η)λz = 1/(1+t) and λf (0) = t/(1 + t) ∈ [0, 1]. If fb′ (η) = 0, then λf (0) = 1, and hence λf (0) ∈ [0, 1] as well. In particular, λf (0) = |f (0)|. We infer from fb′ (η)λz = 1 − λf (0) and ∥fb′ ∥∞ = 1 − |f (0)| that fb′ (η)λz = ∥fb′ ∥∞ .

Tingley’s problem for Lip(I)

For each η ∈ M, z ∈ T and k ∈ {±1}, the following equalities

Lemma 2.11. hold: (2.11)

167

sup fe∈kV(η,k)

d(fe, kV(η,z) ) =

sup g e∈kV(η,z)

d(kV(η,k) , ge) = |1 − kz|.

In particular, dH (kV(η,k) , kV(η,z) ) = |1 − kz| for all η ∈ M, z ∈ T and k = ±1. Proof. Fix an arbitrary fe ∈ kV(η,k) and ge ∈ kV(η,z) , and then f (0) + fb′ (η)k = k

(2.12)

and g(0) + gb′ (η)z = k.

We notice that kf (0), kg(0) ∈ [0, 1], fb′ (η) = ∥fb′ ∥∞ and gb′ (η)kz = ∥gb′ ∥∞ by Remark 2. We deduce from the choice of fe and ge that |(1 − kz)(kf (0) − 1)| ≤ |kf (0) − kg(0)| + |kg(0) − 1 − kz(kf (0) − 1)| = |f (0) − g(0)| + |z(g(0) − k) − (kf (0) − 1)| = |f (0) − g(0)| + |gb′ (η) − fb′ (η)|

by (2.12)

≤ |f (0) − g(0)| + ∥fb′ − gb′ ∥∞ = ∥f − g∥σ = ∥fe − ge∥∞ . That is, |1 − kz|(1 − kf (0)) ≤ ∥fe− ge∥∞ . We also have |(1 − kz)(kg(0) − 1)| ≤ ∥fe− ge∥∞ by a similar calculation, and thus, |1 − kz|(1 − kg(0)) ≤ ∥fe − ge∥∞ . By the liberty of the choice of fe ∈ kV(η,k) and ge ∈ kV(η,z) , we obtain |1 − kz|(1 − kf (0)) ≤ d(fe, kV(η,z) )

and |1 − kz|(1 − kg(0)) ≤ d(kV(η,k) , ge).

Setting f1 = f (0) + I(kz fb′ ) and g1 = g(0) + I(kz gb′ ), we see that fe1 (η, z) = f (0) + k fb′ (η) = k and ge1 (η, k) = g(0) + z gb′ (η) = k by (2.12), where we have used that I(u)(0) = 0 for u ∈ C(M). Consequently, fe1 ∈ kV(η,z) and ge1 ∈ kV(η,k) . By the choice of f1 , we have ∥fe − fe1 ∥∞ =

sup

|fe(ζ, ν) − fe1 (ζ, ν)| =

(ζ,ν)∈M×T

sup

|(1 − kz)fb′ (ζ)ν|

(ζ,ν)∈M×T

= |1 − kz| ∥fb′ ∥∞ = |1 − kz| fb′ (η) = |1 − kz|(1 − kf (0)) by (2.12). In the same way, we get ∥ge1 − ge∥∞ =

sup

|(kz − 1)gb′ (ζ)ν| = |kz − 1| ∥gb′ ∥∞ = |1 − kz|(1 − kg(0)),

(ζ,ν)∈M×T

which yields d(fe, kV(η,z) ) = |1 − kz|(1 − kf (0)) and d(kV(η,k) , ge) = |1 − kz|(1 − kg(0)). Having in mind that kf (0), kg(0) ∈ [0, 1], we conclude that supfe∈kV(η,k) d(fe, kV(η,z) ) = |1 − kz| = supge∈kV(η,z) d(kV(η,k) , ge). □

168

D. Hirota and T. Miura

Lemma 2.12. The identity ϕ1 (η, z) = ϕ1 (η, 1) holds for all η ∈ M and z ∈ T; we shall write ϕ1 (η, z) = ϕ1 (η) for the sake of simplicity of notation. Proof. Fix arbitrary k ∈ {±1}, η ∈ M and z ∈ T \ {±1}. We assume that ϕ1 (η, z) ̸= ϕ1 (η, k). There exists uk ∈ SC(M) such that uk (ϕ1 (η, z)) = kα(η, z)ϕ2 (k, (η, z))

and uk (ϕ1 (η, k)) = −kα(η, k)ϕ2 (k, (η, k)).

Setting gk = I(uk ), we see that gek ∈ kα(η, z)Vϕ(k,(η,z)) ∩ (−kα(η, k))Vϕ(k,(η,k)) , where we have used ϕ1 (λ, x) = ϕ1 (x) by Lemma 2.8. For any fe ∈ kα(η, k)Vϕ(k,(η,k)) , we obtain 2 = |kα(η, k) + kα(η, k)| = |fe(ϕ(k, (η, k))) − gek (ϕ(k, (η, k)))| ≤ ∥fe − gek ∥∞ ≤ 2, which shows d(kα(η, k)Vϕ(k,(η,k)) , gek ) = 2. Combining (2.8), (2.9), (2.10) and (2.11), we get 2≤

sup g e∈kα(η,z)Vϕ(k,(η,z))

d(kα(η, k)Vϕ(k,(η,k)) , ge)

≤ dH (kα(η, k)Vϕ(k,(η,k)) , kα(η, z)Vϕ(k,(η,z)) ) = dH (T (kV(η,k) ), T (kV(η,z) )) = dH (kV(η,k) , kV(η,z) ) = |1 − kz|, which implies z = −k. This contradicts z ̸= ±1, and thus ϕ1 (η, z) = ϕ1 (η, k) for z ̸= ±1. Entering z = i and k = ±1 into the last equality, we get ϕ1 (η, 1) = ϕ1 (η, i) = ϕ1 (η, −1). Therefore, we conclude ϕ1 (η, z) = ϕ1 (η, 1) for all η ∈ M and z ∈ T. □ Lemma 2.13.

The following inequalities hold for all λ, µ ∈ T and x ∈ M × T;

(2.13) |λε0 (x) ϕ2 (λ, x)ϕ2 (µ, x) − µε0 (x) | ≤ |λ − µ|, and

|λε0 (x) ϕ2 (λ, x)ϕ2 (µ, x) + µε0 (x) | ≤ |λ + µ|.

Proof. Take any λ, µ ∈ T and x ∈ M × T. For each fe ∈ λVx and ge ∈ µVx , we obtain |λ − µ| = |fe(x) − ge(x)| ≤ ∥fe − ge∥∞ , which yields |λ − µ| ≤ d(fe, µVx ). Set f0 = λµf , and then we see that fe0 ∈ µVx with ∥fe − fe0 ∥∞ = ∥(1 − λµ)fe∥∞ = |λ − µ|. This implies d(fe, µVx ) = |λ−µ|. By the same argument, we see that d(λVx , ge) = |λ−µ|. Consequently, dH (λVx , µVx ) = |λ − µ| by (2.9). Let us define f1 = α(λ, x)ϕ2 (λ, x)I(1M ), and then we see that fe1 ∈ α(λ, x)Vϕ(λ,x) = T (λVx ) by (2.3) and (2.5). Set ge1 = T (e g ) for each ge ∈ µVx . Then ge1 ∈ T (µVx ) = α(µ, x)Vϕ(µ,x) . By the definition of the set νVy , we have fb1′ (ϕ1 (x))ϕ2 (λ, x) = λε0 (x) α(x) and g1 (0) + gb1′ (ϕ1 (x))ϕ2 (µ, x) = µε0 (x) α(x), where we have used (2.8). We deduce from α(x), ϕ2 (λ, x), ϕ2 (µ, x) ∈ T that |λε0 (x) ϕ2 (λ, x) − µε0 (x) ϕ2 (µ, x)| ≤ |fb1′ (ϕ1 (x)) − gb1′ (ϕ1 (x))| + |g1 (0)| ≤ |f1 (0) − g1 (0)| + ∥fb1′ − gb1′ ∥∞ = ∥f1 − g1 ∥σ = ∥fe1 − ge1 ∥∞ ,

Tingley’s problem for Lip(I)

169

which shows |λε0 (x) ϕ2 (λ, x) − µε0 (x) ϕ2 (µ, x)| ≤ d(fe1 , T (µVx )). We infer from (2.9) and (2.10) that |λε0 (x) ϕ2 (λ, x) − µε0 (x) ϕ2 (µ, x)| ≤

sup T (fe)∈T (λVx )

d(T (fe), T (µVx ))

≤ dH (T (λVx ), T (µVx )) = dH (λVx , µVx ) = |λ − µ|. Thus, |λε0 (x) ϕ2 (λ, x)ϕ2 (µ, x) − µε0 (x) | ≤ |λ − µ|. Noting that ϕ2 (−µ, x) = ϕ2 (µ, x) by (2.6), we obtain |λε0 (x) ϕ2 (λ, x)ϕ2 (µ, x) + µε0 (x) | ≤ |λ + µ|. □ For each x ∈ M × T, there exists ε1 (x) ∈ {±1} such that Lemma 2.14. ε0 (x)−ε1 (x) ϕ2 (λ, x) = λ ϕ2 (1, x) for all λ ∈ T. Proof. Fix arbitrary x ∈ M × T and λ ∈ T \ {±1}. We obtain |λε0 (x) ϕ2 (λ, x)ϕ2 (1, x) ± 1| ≤ |λ ± 1| by (2.13) with µ = 1, which implies λε0 (x) ϕ2 (λ, x)ϕ2 (1, x) ∈ {λ, λ}. Hence, ϕ2 (λ, x)ϕ2 (1, x) ∈ {λ1−ε0 (x) , λ−1−ε0 (x) }. In particular, ϕ2 (i, x)ϕ2 (1, x) ∈ {±ε0 (x)}, and thus ϕ2 (i, x) = ε1 (x)ε0 (x)ϕ2 (1, x) for some ε1 (x) ∈ {±1}. Entering µ = i into (2.13) to get |λ − i| ≥ |λε0 (x) ϕ2 (λ, x)ϕ2 (i, x) − ε0 (x)i| = |λε0 (x) ϕ2 (λ, x)ε1 (x)ϕ2 (1, x) − i|. By the same reasoning, we have |λ + i| ≥ |λε0 (x) ϕ2 (λ, x)ε1 (x)ϕ2 (1, x) + i|. Then we derive from these two inequalities that λε0 (x) ϕ2 (λ, x)ε1 (x)ϕ2 (1, x) ∈ {λ, −λ}. Thus, ε1 (x)ϕ2 (λ, x)ϕ2 (1, x) ∈ {λ1−ε0 (x) , −λ−1−ε0 (x) }. Now we obtain ϕ2 (λ, x)ϕ2 (1, x) ∈ {λ1−ε0 (x) , λ−1−ε0 (x) } ∩ {ε1 (x)λ1−ε0 (x) , −ε1 (x)λ−1−ε0 (x) }. Note that λ ̸= ±1. If ε1 (x) = 1, then we get ϕ2 (λ, x)ϕ2 (1, x) = λ1−ε0 (x) , and if ε1 (x) = −1, then ϕ2 (λ, x)ϕ2 (1, x) = λ−1−ε0 (x) . These imply that ϕ2 (λ, x)ϕ2 (1, x) = λε1 (x)−ε0 (x) for λ ∈ T \ {±1}. The last identity is valid even for λ ∈ {±1} by (2.6). Therefore, we conclude that ϕ2 (λ, x) = λε0 (x)−ε1 (x) ϕ2 (1, x) for all λ ∈ T. □ We shall write ϕ2 (1, x) = ϕ2 (x) for x ∈ M × T. Let λ ∈ T and x ∈ M × T. By (2.8), T (fe)(ϕ1 (x), ϕ2 (λ, x)) = λε0 (x) α(x) = α(λ, x) for f ∈ SLip(I) with fe ∈ λVx . Noting ]) by (2.4), we infer from Lemma 2.12 that that T (fe) = ∆(f (2.14)

\ ∆(f )(0) + ∆(f )′ (ϕ1 (η))ϕ2 (λ, x) = α(λ, x)

170

D. Hirota and T. Miura

for all λ ∈ T, x = (η, z) ∈ M×T and f ∈ SLip(I) with fe ∈ λVx . If we apply Lemma 2.14, then we can rewrite the last equality as \ ∆(f )(0) + ∆(f )′ (ϕ1 (η))λε0 (x)−ε1 (x) ϕ2 (x) = λε0 (x) α(x)

(2.15)

for λ ∈ T, x = (η, z) ∈ M × T and f ∈ SLip(I) satisfying fe ∈ λVx . ′ \ Lemma 2.15. Suppose that ∆(λ0 1I )(0) = 0 for some λ0 ∈ T. Then ∆(λ 0 id) = 0 on M for the identity function id on I.

fI ∈ Proof. Fix arbitrary η ∈ M and z ∈ T, and we set x = (η, z). We note λ0 1 −ε (x) 1 ′ \ λ0 Vx , and then equality (2.15) shows that ∆(λ ϕ2 (x) = α(x). We set 0 1I ) (ϕ1 (η))λ0 ′ \ e(η) = ∆(λ0 1I ) (ϕ1 (η)) for the sake of simplicity of notation. Then we can rewrite the above equality as −ε1 (x)

(2.16)

e(η)λ0

ϕ2 (x) = α(x).

Since λ0 id ∈ λ0 zV(η,z) , we get, by (2.15), ε0 (x)−ε1 (x) ′ \ ∆(λ0 id)(0) + ∆(λ ϕ2 (x) = (λ0 z)ε0 (x) α(x). 0 id) (ϕ1 (η))(λ0 z)

Combining (2.16) with the last equality, we obtain −ε (x) ε0 (x)−ε1 (x) ′ \ ∆(λ0 id)(0) + ∆(λ ϕ2 (x) = (λ0 z)ε0 (x) e(η)λ0 1 ϕ2 (x), 0 id) (ϕ1 (η))(λ0 z)

which leads to ε0 (x)

∆(λ0 id)(0) = (λ0 z)

n o ε1 (x) ′ \ e(η)z − ∆(λ0 id) (ϕ1 (η)) (λ0 z)−ε1 (x) ϕ2 (x).

Note that |e(η)| = 1 by (2.16). Taking the modulus of the above equality, we get ′ \ |∆(λ0 id)(0)| = |z ε1 (x) −e(η) ∆(λ 0 id) (ϕ1 (η))|. Since z ∈ T is arbitrary, the last equality ′ \ holds for z = ±1, i. Then we have ∆(λ 0 id) (ϕ1 (η)) = 0. Having in mind that η ∈ M is ′ \ arbitrarily fixed, we obtain ∆(λ 0 id) = 0 on M, where we have used ϕ1 (M) = M by Lemmas 2.6, 2.8 and 2.12. □ Lemma 2.16.

For each λ ∈ T, the value ∆(λ1I )(0) is nonzero.

Proof. Suppose, on the contrary, that ∆(λ0 1I )(0) = 0 for some λ0 ∈ T. Then ′ \ ∆(λ 0 id) = 0 on M by Lemma 2.15. We define a function f0 ∈ SLip(I) by f0 = λ0 (2 id + id2 )/4. We shall prove that fb0′ (η0 ) = λ0 for some η0 ∈ M. Let R(id) be the essential range of id ∈ Lip(I), that is, R(id) is the set of all ζ ∈ C for which {t ∈ I : | id(t) − ζ| < ϵ} has positive measure for all ϵ > 0. By definition, we see that b R(id) = id(I) = I. For the spectrum σ(id) of id, we observe that R(id) = σ(id) = id(M)

Tingley’s problem for Lip(I)

171

b 0 ) = 1, (see, for example, [6, Lemma 2.63]). Thus, there exists η0 ∈ M such that id(η b 0 ))/4 = λ0 as is claimed. Fix an arbitrary z ∈ T, which yields fb0′ (η0 ) = λ0 (2 + 2id(η ′ \ e ∈ λ0 zV(η ,z) with ∆(λ and then we see that λ0 id 0 id) = 0 on M. Applying (2.14) to 0 f = λ0 id, we have ∆(λ0 id)(0) = α(λ0 z, (η0 , z)). Having in mind that z ∈ T is arbitrary, we may enter z = ±1 into the last equality. Then we get (2.17)

α(λ0 , (η0 , 1)) = α(−λ0 , (η0 , −1)).

Note also that fe0 ∈ λ0 zV(η0 ,z) , and thus ′ \ ∆(f0 )(0) + ∆(f 0 ) (ϕ1 (η0 ))ϕ2 (λ0 z, (η0 , z)) = α(λ0 z, (η0 , z))

by (2.14). Since ∆(λ0 id)(0) = α(λ0 z, (η0 , z)), we can rewrite the above equality as (2.18)

′ \ ∆(f0 )(0) + ∆(f 0 ) (ϕ1 (η0 ))ϕ2 (λ0 z, (η0 , z)) = ∆(λ0 id)(0),

′ \ \′ which yields |∆(λ0 id)(0) − ∆(f0 )(0)| = |∆(f 0 ) (ϕ1 (η0 ))| ≤ ∥∆(f0 ) ∥∞ . We thus obtain ′ \ \′ 2∥∆(f 0 ) ∥∞ ≥ |∆(λ0 id)(0) − ∆(f0 )(0)| + ∥∆(f0 ) ∥∞ ′ \ \′ = |∆(λ0 id)(0) − ∆(f0 )(0)| + ∥∆(λ 0 id) − ∆(f0 ) ∥∞ 1 1 c b = ∥∆(λ0 id) − ∆(f0 )∥σ = ∥λ0 id −f0 ∥σ = ∥1 I − id∥∞ = . 2 2 ′ \ Hence, we have ∥∆(f 0 ) ∥∞ ≥ 1/4, which implies |∆(f0 )(0)| ≤ 3/4, since ∥∆(f0 )∥σ = 1. It follows from (2.18) that ′ \ 1 = |α(λ0 z, (η0 , z))| = |∆(λ0 id)(0)| = |∆(f0 )(0) + ∆(f 0 ) (ϕ1 (η0 ))ϕ2 (λ0 z, (η0 , z))|. ′ \ Since |∆(f0 )(0)| ≤ 3/4, we see that ∆(f 0 ) (ϕ1 (η0 )) ̸= 0. By the liberty of the choice of z ∈ T, we deduce from (2.18) that ϕ2 (λ0 z, (η0 , z)) is invariant with respect to z ∈ T. Entering z = ±1 into ϕ2 (λ0 z, (η0 , z)), we get

(2.19)

ϕ2 (λ0 , (η0 , 1)) = ϕ2 (−λ0 , (η0 , −1)).

b 0 ) = 1. Set f1 = λ0 (2 + id2 )/4 ∈ SLip(I) , and then we have fe1 ∈ λ0 V(η0 ,1) , because id(η We deduce from (2.14) that (2.20)

′ \ ∆(f1 )(0) + ∆(f 1 ) (ϕ1 (η0 ))ϕ2 (λ0 , (η0 , 1)) = α(λ0 , (η0 , 1)).

Combining (2.17) and (2.19) with (2.20), we have ′ \ ∆(f1 )(0) + ∆(f 1 ) (ϕ1 (η0 ))ϕ2 (−λ0 , (η0 , −1)) = α(−λ0 , (η0 , −1)).

^ Here, we recall that T (fe1 ) = ∆(f 1 ) by (2.4). Then the above equality with (2.5) and (2.14) implies that T (fe1 ) ∈ α(−λ0 , (η0 , −1))Vϕ(−λ0 ,(η0 ,−1)) = T (−λ0 V(η0 ,−1) ), which

172

D. Hirota and T. Miura

shows fe1 ∈ (−λ0 )V(η0 ,−1) . Consequently, fe1 ∈ (−λ0 )V(η0 ,−1) ∩ λ0 V(η0 ,1) , and therefore, we obtain f1 (0) − fb1′ (η0 ) = −λ0 = −{f1 (0) + fb1′ (η0 )}. This leads to f1 (0) = −f1 (0), which yields f1 (0) = 0. On the other hand, f1 (0) = λ0 (2 + id2 (0))/4 = λ0 /2 ̸= 0. This is a contradiction. We conclude that ∆(λ1I )(0) ̸= 0 for all λ ∈ T. □ Lemma 2.17. The values α(x) and ε0 (x) are both independent from the variable x ∈ M × T; we shall write α(x) = α and ε0 (x) = ε0 . Proof. Take any λ ∈ T and x = (η, z) ∈ M × T. According to (2.14), applied to f = λ1I , we have ′ \ 1 = |λε0 (x) α(x)| = |∆(λ1I )(0) + ∆(λ1 I ) (ϕ1 (η))ϕ2 (λ, x)| ′ \ ≤ |∆(λ1I )(0)| + |∆(λ1 I ) (ϕ1 (η))| ≤ ∥∆(λ1I )∥σ = 1.

The above inequalities show that ′ \ \′ |∆(λ1I )(0) + ∆(λ1 I ) (ϕ1 (η))ϕ2 (λ, x)| = 1 = |∆(λ1I )(0)| + |∆(λ1I ) (ϕ1 (η))|.

Note that ∆(λ1I )(0) ̸= 0 by Lemma 2.16. By the above equality, there exists t ≥ 0 ′ \ such that ∆(λ1 I ) (ϕ1 (η))ϕ2 (λ, x) = t∆(λ1I )(0). We thus obtain ′ \ |t∆(λ1I )(0)| = |∆(λ1 I ) (ϕ1 (η))| = 1 − |∆(λ1I )(0)|,

which yields (1 + t)|∆(λ1I )(0)| = 1. Consequently, ∆(λ1I )(0) ′ \ λε0 (x) α(x) = ∆(λ1I )(0) + ∆(λ1 I ) (ϕ1 (η))ϕ2 (λ, x) = (1 + t)∆(λ1I )(0) = |∆(λ1I )(0)| by (2.14). Then α(x) = ∆(1I )(0)/|∆(1I )(0)| is independent from x ∈ M × T. Letting λ = i in the above equality, we get iε0 (x)α(x) = ∆(i1I )(0)/|∆(i1I )(0)|. Thus, ε0 is constant on M × T. □ By Lemma 2.17, we can rewrite (2.15) as (2.21)

\ ∆(f )(0) + ∆(f )′ (ϕ1 (η))λε0 −ε1 (x) ϕ2 (x) = λε0 α

for all λ ∈ T, x = (η, z) ∈ M × T and f ∈ SLip(I) with fe ∈ λVx . Lemma 2.18. Let η ∈ M, λ ∈ T and f ∈ SLip(I) be such that fb′ (η) = λ. Then ∆(f ) satisfies ∆(f )(0) = 0 and (2.22) for all z ∈ T.

\ ∆(f )′ (ϕ1 (η))ϕ2 (λz, (η, z)) = (λz)ε0 α

Tingley’s problem for Lip(I)

173

Proof. Fix an arbitrary z ∈ T. By the choice of f , we have fe ∈ λzV(η,z) . By (2.21) with ϕ2 (λz, (η, z)) = (λz)ε0 −ε1 (η,z) ϕ2 (η, z), we obtain \ ∆(f )(0) + ∆(f )′ (ϕ1 (η))ϕ2 (λz, (η, z)) = (λz)ε0 α.

(2.23)

\ \ We observe that ∥∆(f )′ ∥∞ ̸= 0; for if ∥∆(f )′ ∥∞ = 0, then we would have ∆(f )(0) = (λz)ε0 α for all z ∈ T, which is impossible. Equality (2.23) shows that \ 1 = |∆(f )(0) + ∆(f )′ (ϕ1 (η))ϕ2 (λz, (η, z))| \ ≤ |∆(f )(0)| + |∆(f )′ (ϕ1 (η))| ≤ ∥∆(f )∥σ = 1, \ \ and hence, |∆(f )′ (ϕ1 (η))| = ∥∆(f )′ ∥∞ ̸= 0. Then there exists s ≥ 0 such that \ ∆(f )(0) = s∆(f )′ (ϕ1 (η))ϕ2 (λz, (η, z)).

(2.24)

It follows from (2.23) that \ (1 + s)∆(f )′ (ϕ1 (η))ϕ2 (λz, (η, z)) = (λz)ε0 α, \ \ \ which yields (1 + s)∥∆(f )′ ∥∞ = 1, or equivalently, s∥∆(f )′ ∥∞ = 1 − ∥∆(f )′ ∥∞ . These equalities show that \ \ ∆(f )′ (ϕ1 (η))ϕ2 (λz, (η, z)) = ∥∆(f )′ ∥∞ (λz)ε0 α. \ We deduce from the last equality with (2.24) that ∆(f )(0) = s∥∆(f )′ ∥∞ (λz)ε0 α = \ (1 − ∥∆(f )′ ∥∞ )(λz)ε0 α, that is, \ ∆(f )(0) = (1 − ∥∆(f )′ ∥∞ )(λz)ε0 α. \ By the liberty of the choice of z ∈ T, we get 1 − ∥∆(f )′ ∥∞ = 0 = ∆(f )(0). Thus, by \ (2.23), ∆(f )′ (ϕ1 (η))ϕ2 (λz, (η, z)) = (λz)ε0 α for all z ∈ T. □ Lemma 2.19.

For each λ, z ∈ T and η ∈ M, ϕ2 (λ, (η, z)) = λε0 −ε1 (η) ϕ2 (1, (η, 1))z ε1 (η) ,

where ε1 (η) = ε1 (η, 1). Proof. Fix arbitrary λ, z ∈ T and η ∈ M. Setting µ = λz and v = µ1M ∈ SC(M) , [′ (η) = µ by (2.3). We may apply (2.22) to we see that I(v) ∈ SLip(I) satisfies I(v) \ ′ (ϕ1 (η))ϕ2 (µz, (η, z)) = (µz)ε0 α. Therefore, we obtain f = I(v), and we get ∆(I(v)) \ ′ (ϕ1 (η))ϕ2 (µ, (η, 1))z ε0 . \ ′ (ϕ1 (η))ϕ2 (µz, (η, z)) = µε0 α · z ε0 = ∆(I(v) ∆(I(v))

174

D. Hirota and T. Miura

\ ′ (ϕ1 (η)) ̸= 0, and hence ϕ2 (µz, (η, z)) = ϕ2 (µ, (η, 1))z ε0 . This implies Then ∆(I(v)) ϕ2 (λ, (η, z)) = ϕ2 (λz, (η, 1))z ε0 . Applying Lemmas 2.14 and 2.17 to the last equality, we now get ϕ2 (λ, (η, z)) = ϕ2 (λz, (η, 1))z ε0 = (λz)ε0 −ε1 (η) ϕ2 (1, (η, 1))z ε0 = λε0 −ε1 (η) ϕ2 (1, (η, 1))z ε1 (η) . Consequently, ϕ2 (λ, (η, z)) = λε0 −ε1 (η) ϕ2 (1, (η, 1))z ε1 (η) .

□

We shall write ϕ2 (1, (η, 1)) = ϕ2 (η) for simplicity. According to Lemma 2.19, we can write ϕ2 (λ, (η, z)) = λε0 −ε1 (η) ϕ2 (η)z ε1 (η)

(2.25)

for all λ ∈ T and (η, z) ∈ M × T. Combining (2.21) and (2.25), with ϕ2 (λ, x) = λε0 −ε1 (x) ϕ2 (x), we obtain (2.26)

\ ∆(f )(0) + ∆(f )′ (ϕ1 (η))λε0 −ε1 (η) ϕ2 (η)z ε1 (η) = λε0 α

for all λ ∈ T, (η, z) ∈ M × T and f ∈ SLip(I) with fe ∈ λV(η,z) . Lemma 2.20. λV(η,z) . Then

Let λ ∈ T, (η, z) ∈ M × T and f ∈ SLip(I) be such that fe ∈

∆(f )(0) = |∆(f )(0)|λε0 α

and

\ \ ∆(f )′ (ϕ1 (η)) = ∥∆(f )′ ∥∞ λε1 (η) αϕ2 (η)z −ε1 (η) .

In particular, (2.27)

\ |∆(f )(0)| + |∆(f )′ (ϕ1 (η))| = |f (0)| + |fb′ (η)|

for all f ∈ SLip(I) with fe ∈ λV(η,z) . Proof. By assumption, (2.26) holds. Taking the modulus of (2.26) to get (2.28)

\ 1 ≤ |∆(f )(0)| + |∆(f )′ (ϕ1 (η))λε0 −ε1 (η) ϕ2 (η)z ε1 (η) | \ ≤ |∆(f )(0)| + ∥∆(f )′ ∥∞ = ∥∆(f )∥σ = 1.

\ \ We derive from the last inequalities that |∆(f )′ (ϕ1 (η))| = ∥∆(f )′ ∥∞ . If ∆(f )(0) = 0, then the identity ∆(f )(0) = |∆(f )(0)|λε0 α is obvious; in addition, \ \ \ ∥∆(f )′ ∥∞ = ∥∆(f )∥σ = 1, and hence ∆(f )′ (ϕ1 (η)) = ∥∆(f )′ ∥∞ λε1 (η) αϕ2 (η)z −ε1 (η) by (2.26). We next consider the case when ∆(f )(0) ̸= 0. There exists s ≥ 0 such

Tingley’s problem for Lip(I)

175

\ that ∆(f )′ (ϕ1 (η))λε0 −ε1 (η) ϕ2 (η)z ε1 (η) = s∆(f )(0) by (2.28). Entering the last equality into (2.26) to get (1 + s)∆(f )(0) = λε0 α. We thus obtain (1 + s)|∆(f )(0)| = 1, and consequently, ∆(f )(0) = |∆(f )(0)|λε0 α holds even if ∆(f )(0) ̸= 0. Having in mind that \ |∆(f )(0)| + ∥∆(f )′ ∥∞ = 1, we infer from (2.26) that \ ∥∆(f )′ ∥∞ λε0 α = (1 − |∆(f )(0)|)λε0 α = λε0 α − ∆(f )(0) \ = ∆(f )′ (ϕ1 (η))λε0 −ε1 (η) ϕ2 (η)z ε1 (η) . \ \ This shows that ∆(f )′ (ϕ1 (η)) = ∥∆(f )′ ∥∞ λε1 (η) αϕ2 (η)z −ε1 (η) . Since fe ∈ λV(η,z) , we get 1 = |λ| = |f (0) + fb′ (η)z| ≤ |f (0)| + |fb′ (η)| ≤ ∥f ∥σ = 1, \ and hence |∆(f )(0)| + |∆(f )′ (ϕ1 (η))| = 1 = |f (0)| + |fb′ (η)|.

□

For each λ ∈ T and η ∈ M, we define λPη by λPη = {u ∈ SC(M) : u(η) = λ}. Lemma 2.21. Let η0 ∈ M and f ∈ SLip(I) . We set λ = fb′ (η0 )/|fb′ (η0 )| if fb′ (η0 ) ̸= 0, and λ = 1 if fb′ (η0 ) = 0. For each t ∈ R with 0 < t < 1, there exists ut ∈ Pη0 such that n o |tf (0)|λ + tfb′ + 1 − |tf (0)| − |tfb′ (η0 )| λut ∈ λPη0 . Proof. Note first that 1 − |tf (0)| − |tfb′ (η0 )| > 0, since |tf (0)| + |tfb′ (η0 )| ≤ ∥tf ∥σ < 1. We set r = 1 − |tf (0)| − |tfb′ (η0 )|, n ro , G0 = η ∈ M : |tfb′ (η) − tfb′ (η0 )| ≥ 4n r r o ′ ′ b b and Gm = η ∈ M : m+2 ≤ |tf (η) − tf (η0 )| ≤ m+1 2 2 for each m ∈ N. We see that Gn is a closed subset of M with η0 ̸∈ Gn for all n ∈ N∪{0}. For each n ∈ N ∪ {0}, there exists vn ∈ Pη0 such that (2.29)

vn = 0 on Gn P∞ by Urysohn’s lemma. Setting ut = v0 n=1 vn /2n , we see that ut converges in C(M), since ∥vn ∥∞ = 1 for all n ∈ N. We observe that 1 = ut (η0 ) ≤ ∥ut ∥∞ ≤ ∥v0 ∥∞

∞ X ∥vn ∥∞ = 1, n 2 n=1

and hence ut ∈ Pη0 . Here, we define wt = |tf (0)|λ + tfb′ + rλut ∈ C(M).

176

D. Hirota and T. Miura

We shall prove that wt ∈ λPη0 . Since ut (η0 ) = 1 and tfb′ (η0 ) = |tfb′ (η0 )|λ, we have n o wt (η0 ) = |tf (0)|λ + tfb′ (η0 ) + 1 − |tf (0)| − |tfb′ (η0 )| λ = λ. Fix an arbitrary η ∈ M. To prove that |wt (η)| ≤ 1, we shall consider three cases. First, we consider the case when η ∈ G0 . Then v0 (η) = 0 by (2.29), and hence ut (η) = 0 by definition. We thus obtain |wt (η)| ≤ ||tf (0)|λ + tfb′ (η)| ≤ ∥tf ∥σ < 1, and consequently, |wt (η)| < 1 if η ∈ G0 . We next consider the case when η ∈ ∪∞ n=1 Gn , and then η ∈ Gm for some m ∈ N. By ′ ′ the choice of Gm , we get |tfb (η)−tfb (η0 )| ≤ r/2m+1 . Thus, |tfb′ (η)| ≤ |tfb′ (η0 )|+r/2m+1 . P We derive from (2.29) that |rλut (η)| ≤ r|v0 (η)| n̸=m |vn (η)|/2n ≤ r(1 − 2−m ). Since c′ (η0 )| = 1 − r, we obtain |tf (0)| + |tf |wt (η)| ≤ |tf (0)| + |tfb′ (η)| + |rλut (η)| ≤ |tf (0)| + |tfb′ (η0 )| + = (1 − r) −

r 2m+1

+r =1−

r 2m+1

r 2m+1

  1 +r 1− m 2

< 1.

Hence, |wt (η)| < 1 for η ∈ ∪∞ n=1 Gn . b′ b′ Finally we consider the case when η ̸∈ ∪∞ n=0 Gn . Then f (η) = f (η0 ), and hence |wt (η)| ≤ |tf (0)| + |tfb′ (η0 )| + r = 1. We thus conclude that |wt (η)| ≤ 1 for all η ∈ M, and consequently, wt ∈ λPη0 . □

§ 3.

Proof of Main results

Proof of Theorem 1.1. Fix arbitrary f ∈ SLip(I) and η ∈ M. Set ζ = ϕ1 (η) and λ = fb′ (η)/|fb′ (η)| if fb′ (η) ̸= 0, and λ = 1 if fb′ (η) = 0. Thus, fb′ (η) = |fb′ (η)|λ. For each t ∈ R with 0 < t < 1, we define r = 1 − |tf (0)| − |tfb′ (η)|, and then r > 0. By Lemma 2.21, there exists ut ∈ Pη such that wt = |tf (0)|λ + tfb′ + rλut ∈ λPη . We obtain ∥wt − fb′ ∥∞ = ∥|tf (0)|λ + (t − 1)fb′ + rλut ∥∞ ≤ |tf (0)| + (1 − t)∥fb′ ∥∞ + 1 − |tf (0)| − |tfb′ (η)| = (1 − t)∥fb′ ∥∞ + 1 − |tfb′ (η)|. ′ \ ^ Since wt ∈ λPη , we see that I(w t ) (η) = wt (η) = λ, that is, I(wt ) ∈ λV(η,1) . Then \t ))′ (ζ) = ∆(I(w \t ))′ (ϕ1 (η)) = λε1 (η) αϕ2 (η) by Lemma 2.20. ∆(I(wt ))(0) = 0 and ∆(I(w

Tingley’s problem for Lip(I)

177

We get \ \ \ 1 − |∆(f )′ (ζ)| = |λε1 (η) αϕ2 (η)| − |∆(f )′ (ζ)| ≤ |λε1 (η) αϕ2 (η) − ∆(f )′ (ζ)| \t ))′ (ζ) − ∆(f \ \t ))′ − ∆(f \ = |∆(I(w )′ (ζ)| ≤ ∥∆(I(w )′ ∥∞ = ∥∆(I(wt )) − ∆(f )∥σ − |∆(f )(0)| = ∥I(wt ) − f ∥σ − |∆(f )(0)| = |f (0)| + ∥wt − fb′ ∥∞ − |∆(f )(0)| ≤ |f (0)| + (1 − t)∥fb′ ∥∞ + 1 − |tfb′ (η)| − |∆(f )(0)|, where we have used that ∆(I(wt ))(0) = 0 = I(wt )(0) and ∆ is an isometry. Letting t ↗ 1 in the above inequalities, we have (3.1)

\ \ 1 − |∆(f )′ (ζ)| ≤ |λε1 (η) αϕ2 (η) − ∆(f )′ (ζ)| ≤ |f (0)| + 1 − |fb′ (η)| − |∆(f )(0)|.

\ In particular, we obtain |∆(f )(0)| − |∆(f )′ (ζ)| ≤ |f (0)| − |fb′ (η)|, that is, (3.2)

\ |∆(f )(0)| − |∆(f )′ (ϕ1 (η))| ≤ |f (0)| − |fb′ (η)|.

Let η0 ∈ M be such that |fb′ (η0 )| = ∥fb′ ∥∞ . There exist µ, z ∈ T such that f (0) = |f (0)|µ and fb′ (η0 ) = |fb′ (η0 )|z = ∥fb′ ∥∞ z. Thus, f (0) + fb′ (η0 )zµ = (|f (0)| + ∥fb′ ∥∞ )µ = ∥f ∥σ µ = µ, and hence fe ∈ µV(η0 ,zµ) . Equality (2.27) shows that (3.3)

\ |∆(f )(0)| + |∆(f )′ (ϕ1 (η0 ))| = |f (0)| + |fb′ (η0 )|.

\ Note that |∆(f )(0)| − |∆(f )′ (ϕ1 (η0 ))| ≤ |f (0)| − |fb′ (η0 )| holds by (3.2). If we add the last inequality to (3.3), we get |∆(f )(0)| ≤ |f (0)|. We may apply the above arguments to ∆−1 , then we obtain |∆−1 (g)(0)| ≤ |g(0)| for all g ∈ SLip(I) . Entering g = ∆(f ) into the last inequality to get |f (0)| ≤ |∆(f )(0)|, and thus |∆(f )(0)| = |f (0)|. \ It follows from (3.2) that |fb′ (η)| ≤ |∆(f )′ (ϕ1 (η))|. Having in mind that fe ∈ µV(η0 ,zµ) and f (0) = |f (0)|µ, we derive from Lemma 2.20 that (3.4)

∆(f )(0) = |∆(f )(0)|µε0 α = |f (0)|µε0 α = [f (0)]ε0 α,

where [ν]ε0 = ν if ε0 = 1 and [ν]ε0 = ν if ε0 = −1 for ν ∈ C. Now we shall prove that ϕ1 is injective. Suppose that ϕ1 (η1 ) = ϕ1 (η2 ) for η1 , η2 ∈ M. Set f1 = I(1M ), and thus fb1′ (ηj ) = 1 for j = 1, 2 by (2.3). Equalities (2.22) and ′ \ (2.25) show that ∆(f 1 ) (ϕ1 (ηj ))ϕ2 (ηj ) = α for j = 1, 2. Since ϕ1 (η1 ) = ϕ1 (η2 ), we have

178

D. Hirota and T. Miura

ϕ2 (η1 ) = ϕ2 (η2 ). Applying Lemmas 2.12, 2.17 and 2.19 to (2.8) with λ = 1, we obtain T (V(1,(η,1)) ) = αV(ϕ1 (η),ϕ2 (η)) . Therefore, we get T (V(1,(η1 ,1)) ) = T (V(1,(η2 ,1)) ), and consequently, V(1,(η1 ,1)) = V(1,(η2 ,1)) . Lemma 2.1 shows that η1 = η2 , which proves that ϕ1 is injective. Now, we may apply the arguments in the last paragraph to ∆−1 and ϕ−1 1 , −1 ′ −1 ′ ′ \ \ \ and then we obtain |∆(f ) (ζ)| ≤ |(∆ (∆(f ))) (ϕ1 (ζ))|, which shows |∆(f ) (ϕ1 (η))| ≤ \ \ |fb′ (η)|. We thus conclude that |∆(f )′ (ζ)| = |∆(f )′ (ϕ1 (η))| = |fb′ (η)|. By inequalities (3.1) and |∆(f )(0)| = |f (0)|, we obtain \ \ |λε1 (η) αϕ2 (η) − ∆(f )′ (ζ)| + |∆(f )′ (ζ)| = 1. \ The above equality implies that ∆(f )′ (ζ) = sλε1 (η) αϕ2 (η) for some s ≥ 0. Then s = \ \ |sλε1 (z) αϕ2 (η)| = |∆(f )′ (ζ)| = |fb′ (η)|, which shows ∆(f )′ (ζ) = |fb′ (η)|λε1 (η) αϕ2 (η) = [fb′ (η)]ε1 (η) αϕ2 (η), where we have used fb′ (η) = |fb′ (η)|λ. Thus, (3.5)

\ ∆(f )′ (ϕ1 (η)) = αϕ2 (η) [fb′ (η)]ε1 (η)

for all f ∈ SLip(I) and η ∈ M. We now define ∆0 : Lip(I) → Lip(I) by    g  ∥g∥σ ∆ if g ∈ Lip(I) \ {0}, ∥g∥σ ∆0 (g) =  0 if g = 0. By the definition of ∆0 with (3.4) and (3.5), we observe that (3.6)

∆0 (g)(0) = α[g(0)]ε0

ε1 (η) ′ \ b′ and ∆ 0 (g) (ϕ1 (η)) = αϕ2 (η)[g (η)]

for all g ∈ Lip(I) and η ∈ M. We thus obtain ′ \′ ∥∆0 (g1 ) − ∆0 (g2 )∥σ = |∆0 (g1 )(0) − ∆0 (g2 )(0)| + sup |∆\ 0 (g1 ) (ϕ1 (η)) − ∆0 (g2 ) (ϕ1 (η))| η∈M

= |g1 (0) − g2 (0)| + sup |gb1′ (η) − gb2′ (η)| = ∥g1 − g2 ∥σ η∈M

for all g1 , g2 ∈ Lip(I), where we have used ϕ1 (M) = M. Hence ∆0 is an isometry on Lip(I). We infer from (3.6) that ∆0 is real linear. We deduce that ∆0 is surjective, since so is ∆. Therefore, ∆0 is a surjective, real linear isometry on Lip(I) that extends ∆ to Lip(I). □ Proof of Corollary 1.2. Let ∆1 be a surjective isometry on Lip(I). By the Mazur–Ulam theorem [19], ∆1 − ∆1 (0) is a surjective, real linear isometry. Without loss of generality, we may and do assume that ∆1 is a surjective real linear isometry.

Tingley’s problem for Lip(I)

179

Since ∆−1 1 has the same property as ∆1 , we see that ∆1 maps SLip(I) onto itself. Now we may apply (3.4) and (3.5) to ∆1 , and then we obtain ∆1 (f )(0) = α[f (0)]ε0

ε1 (η) ′ \ b′ and ∆ 1 (f ) (ϕ1 (η)) = αϕ2 (η)[f (η)]

for all f ∈ Lip(I) and η ∈ M, where α ∈ T, ε0 ∈ {±1}, ϕ1 : M → M, ϕ2 : M → T and ε1 : M → {±1} are from proof of Theorem 1.1. As we proved in the second paragraph of Proof of Theorem 1.1, we know that ϕ1 is injective. By Lemma 2.6, ψ1 = ϕ−1 1 is well defined, and then we have (3.7)

ε1 (ψ1 (η)) ′ \ b′ ∆ 1 (f ) (η) = αϕ2 (ψ1 (η))[f (ψ1 (η))]

for f ∈ Lip(I) and η ∈ M. We shall prove that ψ1 and ϕ2 are both continuous. Let ′ \ {ηa } be a net in M converging to η ∈ M. By the continuity of ∆ 1 (f ) , we see that ′ \ \′ b′ |∆ 1 (f ) (ηa )| converges to |∆1 (f ) (η)| for each f ∈ Lip(I). This implies that |f (ψ1 (ηa ))| converges to |fb′ (ψ1 (η))| for every f ∈ Lip(I) by (3.7). Since the weak topology of M induced by the family {|fb′ | : f ∈ Lip(I)} is Hausdorff, we observe that the identity map from M with the original topology onto M with the weak topology is a homeomorphism. Hence, ψ1 (ηa ) converges to ψ1 (η) with respect to the original topology of M, and thus ψ1 is continuous on M. Since ψ1 is a bijective continuous map on the compact Hausdorff space M, it must be a homeomorphism. Let id be the identity function on I. Then we ′ have ∆\ 1 (id) = αϕ2 ◦ ψ1 by (3.7), which implies the continuity of ϕ2 on M. Moreover, ′ the identity ∆\ 1 (i id) = αϕ2 ◦ ψ1 i(ε1 ◦ ψ1 ) shows that ε1 ◦ ψ1 is continuous on M. Since ψ1 is a homeomorphism, we have ε1 = (ε1 ◦ ψ1 ) ◦ ψ1−1 is continuous on M as well. Then M1 = {η ∈ M : ε1 (ψ1 (η)) = 1} is a closed and open subset of M with ε1 (ψ1 (η)) = −1 for all η ∈ M \ M1 . We define a map Φ : C(M) → C(M) by Φ(u)(η) = [u(ψ1 (η))]ε1 (ψ1 (η)) for u ∈ C(M) and η ∈ M. We see that Φ is a well defined real linear map on C(M). For each v0 ∈ C(M), we set u0 (η) = [v0 (ψ1−1 (η))]ε1 (η) for η ∈ M. Then we have Φ(u0 )(η) = [u0 (ψ1 (η))]ε1 (ψ1 (η)) = [v0 (η)]ε1 (ψ1 (η))ε1 (ψ1 (η)) = v0 (η), which shows that Φ is surjective. It is routine to check that Φ is an injective homomorphism, and consequently, Φ is a real algebra automorphism on C(M). Let Γ be the Gelfand transformation from L∞ (I) onto C(M), that is, Γ(h) = b h for h ∈ L∞ (I). We define a real algebra automorphism Ψ = Γ−1 ◦ Φ ◦ Γ on L∞ (I). For each f ∈ Lip(I) and η ∈ M, we obtain [fb′ (ψ1 (η))]ε1 (ψ1 (η)) = Φ(fb′ )(η) = (Φ ◦ Γ)(f ′ )(η) = (Γ ◦ Ψ)(f ′ )(η) = Γ(Ψ(f ′ ))(η). By the continuity of ϕ2 and ψ1 , we may set h0 = Γ−1 (αϕ2 ◦ ψ1 ) ∈ L∞ (I). We derive from (3.7) that ′ ′ ′ \ \′ ∆ 1 (f ) (η) = Γ(h0 )(η)Γ(Ψ(f ))(η) = Γ(h0 Ψ(f ))(η) = h0 Ψ(f )(η)

180

D. Hirota and T. Miura

for all η ∈ M. Therefore, we conclude ∆1 (f )′ = h0 Ψ(f ′ ) for every f ∈ Lip(I). According to (2.2), we have Z t Z t ′ ε0 ∆1 (f )(t) = ∆1 (f )(0) + ∆1 (f ) dm = α[f (0)] + h0 Ψ(f ′ ) dm 0

0

for every t ∈ I and f ∈ Lip(I).

□

Acknowledgement The authors would like to express our gratitude to the referee for his/her valuable suggestions and comments which have improved the original manuscript. The second author is supported by JSPS KAKENHI (Japan) Grant Number JP 20K03650.

References [1] T. Banakh, Every 2-dimensional Banach space has the Mazur-Ulam property, Linear Algebra Appl., 632 (2022), 268–280. [2] A. Browder, Introduction to function algebras, W.A. Benjamin, Inc., New YorkAmsterdam, 1969. [3] L. Cheng and Y. Dong, On a generalized Mazur-Ulam question: extension of isometries between unit spheres of Banach spaces, J. Math. Anal. Appl., 377 (2011), 464–470. [4] M. Cueto-Avellaneda, D. Hirota, T. Miura and A.M. Peralta, Exploring new solutions to Tingley’s problem for function algebras, Quaest. Math., DOI: 10.2989/16073606.2022.2072787. [5] M. Cueto-Avellaneda, A.M. Peralta, On the Mazur-Ulam property for the space of Hilbertspace-valued continuous functions, J. Math. Anal. Appl., 479 (2019), 875–902. [6] R.G. Douglas, Banach algebra techniques in operator theory. Second edition, Graduate Texts in Mathematics 179, Springer-Verlag, New York, 1998. [7] F.J. Fern´ andez-Polo, J.J. Garc´es, A.M. Peralta, I. Villanueva, Tingley’s problem for spaces of trace class operators, Linear Algebra Appl., 529 (2017), 294–323. [8] F.J. Fern´ andez-Polo, E. Jord´ a, A.M. Peralta, Tingley’s problem for p-Schatten von Neumann classes, J. Spectr. Theory, 10 (2020), 809–841. [9] F.J. Fern´ andez-Polo, A.M. Peralta, On the extension of isometries between the unit spheres of a C ∗ -algebra and B(H), Trans. Amer. Math. Soc. Ser. B, 5 (2018), 63–80. [10] F.J. Fern´ andez-Polo, A.M. Peralta, On the extension of isometries between the unit spheres of von Neumann algebras, J. Math. Anal. Appl., 466 (2018), 127–143. [11] F.J. Fern´ andez-Polo, A.M. Peralta, Low rank compact operators and Tingley’s problem, Adv. Math., 338 (2018), 1–40. [12] F.J. Fern´ andez-Polo, A.M. Peralta, Tingley’s problem through the facial structure of an atomic JBW ∗ -triple, J. Math. Anal. Appl., 455 (2017), 750–760. [13] R. Fleming and J. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 129, Boca Raton, 2003.

Tingley’s problem for Lip(I)

181

[14] O. Hatori, The Mazur-Ulam property for uniform algebras, Studia Math., 265 (2022), 227–239. [15] O. Hatori, S. Oi and R. Shindo Togashi, Tingley’s problems on uniform algebras, J. Math. Anal. Appl., 503 (2021), 125346. [16] H. Koshimizu, Linear isometries on spaces of continuously differentiable and Lipschitz continuous functions, Nihonkai Math. J., 22 (2011), 73–90. [17] C.W. Leung, C.K. Ng, N.C. Wong, Metric preserving bijections between positive spherical shells of non-commutative Lp -spaces, J. Operator Theory, 80 (2018), 429–452. [18] C.W. Leung, C.K. Ng, N.C. Wong, On a variant of Tingley’s problem for some function spaces, J. Math. Anal. Appl., 496 (2021), 124800. [19] S. Mazur and S. Ulam, Sur les transformationes isom´etriques d’espaces vectoriels norm´es, C. R. Acad. Sci. Paris 194 (1932), 946–948. [20] M. Mori, Tingley’s problem through the facial structure of operator algebras, J. Math. Anal. Appl., 466 (2018), 1281–1298. [21] M. Mori, N. Ozawa, Mankiewicz’s theorem and the Mazur-Ulam property for C ∗ -algebras, Studia. Math., 250 (2020), 265–281. [22] A.M. Peralta, Extending surjective isometries defined on the unit sphere of ℓ∞ (Γ), Rev. Mat. Complut., 32 (2019), 99–114. [23] A.M. Peralta, On the unit sphere of positive operators, Banach J. Math. Anal., 13 (2019), 91–112. [24] A.M. Peralta, R. Tanaka, A solution to Tingley’s problem for isometries between the unit spheres of compact C ∗ -algebras and JB ∗ -triples, Sci. China Math., 62 (2019), 553–568. [25] W. Rudin, Real and complex analysis, Third Edition, McGraw-Hill Book. Co., New York, 1987. [26] D. Tan, X. Huang, R. Liu, Generalized-lush spaces and the Mazur-Ulam property, Studia. Math., 219 (2013), 139–153. [27] D. Tan, R. Liu, A note on the Mazur-Ulam property of almost-CL-spaces, J. Math. Anal. Appl., 405 (2013), 336–341. [28] R. Tanaka, A further property of spherical isometries, Bull. Aust. Math. Soc., 90 (2014), 304–310. [29] R. Tanaka, The solution of Tingley’s problem for the operator norm unit sphere of complex n × n matrices, Linear Algebra Appl., 494 (2016), 274–285. [30] R. Tanaka, Spherical isometries of finite dimensional C ∗ -algebras, J. Math. Anal. Appl., 445 (2017), 337–341. [31] R. Tanaka, Tingley’s problem on finite von Neumann algebras, J. Math. Anal. Appl., 451 (2017), 319–326. [32] D. Tingley, Isometries of the unit sphere, Geom. Dedicata, 22 (1987), 371–378. (n) [33] R. Wang, Isometries of C0 (X), Hokkaido Math. J., 25 (1996), 465–519. [34] R. Wang and A. Orihara, Isometries on the ℓ1 -sum of C0 (Ω, E) type spaces, J. Math. Sci. Univ. Tokyo, 2 (1995), 131–154.

RIMS Kˆ okyˆ uroku Bessatsu B93 (2023), 183–222

On the norm of normal matrices By

Ludovick Bouthat∗, Javad Mashreghi∗∗ and Fr´ed´eric ´rin∗∗∗ Morneau-Gue

Abstract In this article, we present some recent results related to the calculation of the induced p-norm of n × n circulant matrices A(n, a, b) with diagonal entries equal to a ∈ R and offdiagonal entries equal to b ∈ R. For circulant matrices with nonnegative entries, an explicit formula for the induced p-norm (1 ≤ p ≤ ∞) is given, whereas for A(n, −a, b), a > 0 the situation is no longer so simple and calls for a more subtle analysis. As a matter of fact, while the 2-norm of A(n, −a, b) is precisely determined, the exact value of the induced p-norm for 1 < p < ∞, p ̸= 2, still remains elusive. Nevertheless, we provide a lower bound as well as two different categories of upper bounds. As an indication of not being far from the exact values, our estimates coincide at both ends points (i.e., p = 1 and p = ∞) as well as at p = 2 with the precise values. As an abstract approach, we also introduce the ∗-algebra generated by a normal matrix A accompanied by an axis-oriented norm, and obtain some estimations of the norm of elements of the ∗-algebra. We then exhibit the connection between the new generalized estimates and the previously obtained estimates in the special case where A is a circulant matrix. Finally, using an optimization-oriented approach, we provide insight on the ∥Ax∥ nature of the maximizing vectors for ∥x∥pp . This leads us to formulate a conjecture that, if proven valid, would make it possible to derive an exact formula for the induced p-norm of A(n, a, b) whenever a = 1−n and b = n1 . n

Received March 15, 2022. Revised May 16, 2022. 2020 Mathematics Subject Classification(s): 15A60, 15B05, 47A30, 47A60 Key Words: Circulant matrices, doubly stochastic matrices, p-norm, ∗-algebra, axis-oriented norms. This work was partially supported by research grants from NSERC (Canada) and grants or scholarships from FRQNT (Quebec). ∗ D´ epartement de math´ ematiques et de statistique, Universit´ e Laval, Qu´ ebec, QC, Canada G1K 7P4. e-mail: [email protected] ∗∗ D´ epartement de math´ ematiques et de statistique, Universit´ e Laval, Qu´ ebec, QC, Canada G1K 7P4. e-mail: [email protected] ∗∗∗ D´ ´ ´ epartement Education, Universit´ e TELUQ, Qu´ ebec, QC, Canada G1K 9H6. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

184

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

§ 1.

Introduction

Given positive integers m, n ≥ 2, let Cm×n denote the set of with entries in the complex field C. For α0 , α1 , . . . , αn−1 ∈ C, the circ(α0 , α1 , . . . , αn−1 ) is defined as  α0 α1 α2 · · · α  n−1 α0 α1 · · ·  α α α ··· circ(α0 , α1 , . . . , αn−1 ) = [αi−j ]ni,j=1 =   n−2 n−1 0 . . .. . .  . ..  . . . α1

α2

m × n matrices circulant matrix  αn−1 αn−2    αn−3  .  ..  . 

α3 · · · α0

The index k in αk is always calculated mod n, and thus runs between 0 and n − 1. The circulant matrices are a very special type of square Toeplitz matrices [20, Chapter 4]. According to the mathematician and historian of early linear algebra Thomas Muir [39, Vol 2., Ch. 14], circulant matrices were first introduced, although somewhat implicitly, by Eug`ene Catalan in [9] in 1846. More than half a century elapsed, however, before the systematic study of this important class of matrices began to gain significant momentum. The first comprehensive monograph dedicated to the study of the various properties of circulant matrices, dating from 1979, was penned by Davis [16]. In recent years, circulant and block-circulant matrices have become omnipresent in many sub-disciplines of mathematics. For reasons that will become apparent later in our presentation, these matrices naturally prove themselves to be of fundamental importance in areas of mathematics where the roots of unity come into play. Additionally, they also have a wide range of applications in various parts of modern and classical mathematics. For instance, they have been used in the study of functional equations with several complex variables [46], in solving polynomial equations [30, 5], in solving various ordinary and partial differential equations [13, 17, 45, 34], in image processing [47], in signal processing [1, 25], in numerical analysis [12, 31, 48], in Wiener-Hopf equations [10, 7, 36, 6, 42], in information theory [21], and in various branches of operator theory (see [2, 27, 26, 22, 11, 18, 24, 29, 14, 28, 38, 32] and references therein). It goes without saying that, due to the abundance of contributions, our treatment is by no means exhaustive. In order to provide a more refined definition of circulant matrices, we first need to introduce a particular operator, the right circular shift, that is acting on vectors of length n by operating a rearrangement of their entries consisting in moving the final entry to the first position while shifting all other entries to the next position. Formally, a right circular shift is an operator S : Cn → Cn defined by S(α0 , α1 , . . . , αn−1 ) := (αn−1 , α0 , . . . , αn−2 ).

185

On the norm of normal matrices

One can view A := circ(α0 , α1 , . . . , αn−1 ) as the n × n matrix whose rows are given by iterations of the right circular shift acting on the vector (α0 , α1 , . . . , αn−1 ), i.e., for k = 1, . . . , n, the k-th row of A is S k−1 (α0 , α1 , . . . , αn−1 ). By identifying a circulant matrix with its first row, one can see that circn (C), the set of all complex-valued n × n circulant matrices, forms an n-dimensional vector space with respect to the usual operations of matrix addition and multiplication of matrices by scalars. This space can be interpreted as the space of complex-valued functions on Z/nZ, the cyclic group of order n. For more information concerning the circular shift operator and its connections with circulant matrices, we point the reader to the very detailed treatment given by Fuhrmann in [19, § 5.2 and § 5.3].

§ 2.

Organization of the paper and a sketch of the main results

Recall that a function k · k : Cm×n → [0, ∞) is a matrix norm or a ring norm if, for all A, B ∈ Cm×n , it satisfies the following five axioms: (i) kAk ≥ 0, (ii) kAk = 0 if and only if A = 0,

Nonnegativity, Positivity,

(iii) kλAk = |λ|kAk for all scalar λ,

Homogeneity,

(iv) kA + Bk ≤ kAk + kBk,

Subadditivity,

(v) kABk ≤ kAkkBk,

Submultiplicativity.

The first four properties of a matrix norm are identical to the axioms for a norm. Since the set of m × n matrices with complex entries, can be equated to with the set of vectors of length m × n with complex entries, it does makes sense to endow Cm×n with a norm that does not satisfy property (v). In order to avoid any confusion, such norms shall be called vector norms. Suppose that a vector norm k · kα on Cn and a vector norm k · kβ on Cm are given. Then any m × n matrix A represents, with respect to the canonical basis, a linear operator from Cn to Cm . We define the corresponding induced matrix norm or operator norm or lub norm (this acronym stands for the least upper bound norm) on the space Cm×n of all m × n matrices as   kAxkβ n : x ∈ C , x 6= 0 . kAkα→β := sup kxkα The notation kAkα→β is a shorter replacement for kAk(Cn ,∥·∥α )→(Cm ,∥·∥β ) . Throughout this paper, the only operator norms that we shall consider are those induced by p-norms

186

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

for vectors. Recall that, for p ≥ 1, the p-norm of vector x = (x1 , . . . , xn ) is defined by kxkp =

n X

!1/p |xk |p

,

k=1

and the ∞-norm is kxk∞ = max{|x1 |, . . . , |xn |}. It is customary, in particular in operator theory texts, to denote the vector space Cn endowed by the vector p-norm by ℓnp (C). In this note, our main objective is a comprehensive study of the induced p-norm of a special class of circulant matrices, acting as operators from ℓnp (C) to ℓnp (C). The circulant matrices under consideration are those with the diagonal entries equal to a ∈ R and the off-diagonal entries equal to b ∈ R, i.e., 

(2.1)

a b   b A(n, a, b) =    ..  . b

b a b .. . b

b b a .. . b

 b b   b ,  ..  . ··· a ··· ··· ··· .. .

where a, b ∈ R. However, via a normalization process, it suffices to consider the following two cases: A(n, a, b) and A(n, −a, b), where a, b ≥ 0. Since the A(n, a, b)’s are matrices with real entries, it may appear that there is some ambiguity as to whether they are acting on Rn or on Cn , and thus possibly ends with different p-norms for the real and complex cases. As a matter of fact, this is not an issue as it was shown by Taylor in [44] (1958) that for any m × n matrix B with real entries and all q ≥ p ≥ 1, the induced norm   kBxkq kBkp→q := sup : x 6= 0 kxkp is the same whether x runs through all non-zero vectors with complex entries or only through all non-zero vectors with real entries. Note that the author points out that a proof of this result had been sketched in an earlier paper (1927) by M. Riesz [40]. Also, we refer to Crouzeix [15] since this is, to the best of our knowledge, the most simple and direct proof of this result in the case that is considered here. Our interest for estimating the induced p-norms of A(n, ±a, b) stems from other studies of the geometry of the set of doubly stochastic matrices. Indeed, one can verify that the Chebyshev radius of the n-dimensional Birkhoff polytope, i.e., the greatest lower bound of the radii of all balls containing Dn , with respect to the induced p-norm

187

On the norm of normal matrices

(1 ≤ p ≤ ∞) is given by



1

, Rp (Dn ) =

I − n K p→p

where I is the n×n identity matrix and K is the n×n all-ones matrix. This corresponds
1 to the induced p-norm of −A n, 1−n n , n . In Section 3 of the present paper, we discuss some recent results (see [8, 41]) related to the calculation of kA(n, ±a, b)kp→p for a, b ≥ 0 and 1 ≤ p ≤ ∞. We shall see that the negative sign plays a crucial role, as the p-norms of A(n, a, b) and A(n, −a, b) differ significantly. Indeed, kA(n, a, b)kp→p = a + (n − 1)b,

(1 ≤ p ≤ ∞),

whereas kA(n, −a, b)kp→p is a non-constant function of p that is monotonically nonincreasing for 1 ≤ p ≤ 2 and monotonically non-decreasing for 2 ≤ p ≤ ∞ (see [41]), with  a + b, if (n − 2)b ≤ 2a, kA(n, −a, b)k2→2 = −a + (n − 1)b, if (n − 2)b ≥ 2a. But the exact value of the induced p-norm of A(n, −a, b) for 1 < p < ∞, p 6= 2, remains unknown. However, the following lower and upper bounds  1 1  a + b ≤ kA(n, −a, b)kp→p ≤ n| p − 2 | (a + b) if (n − 2)b ≤ 2a,   

1 1 (n − 1)b − a ≤ kA(n, −a, b)kp→p ≤ n| p − 2 | ((n − 1)b − a) if (n − 2)b ≥ 2a,

were obtained in [8]. Here, as a novel contribution, we provide a proof of the following refined estimates    2 −1| (n−1)b+a | p   a + b ≤ kA(n, −a, b)kp→p ≤ (a + b) if (n − 2)b ≤ 2a,  a+b      (n − 1)b − a ≤ kA(n, −a, b)k

p→p

≤ ((n − 1) b − a)



(n−1)b+a (n−1)b−a

| p2 −1|

if (n − 2)b ≥ 2a.

In Section 4, we take a wider perspective and investigate the induced p-norm of elements of the ∗-algebra generated by a normal matrix A in the case where k · k is an axis-oriented matrix norm. We then discuss how these estimates are related to those mentioned above in the case where A is a circulant matrix. Finally, in Section 5 we open some new fronts in dealing with the question that was initially attracted our interest in estimating the induced p-norm of circulant matrices of the form given by (2.1). By leveraging the power of an optimization-oriented approach, ∥Ax∥ we gain insights on the nature of the maximizing vectors for ∥x∥pp . This leads us to

188

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

formulate a conjecture that, if proven valid, would make it possible to derive an exact formula for kI − n1 Kkp→p .

§ 3.

The induced p-norm of circulant matrices

As versatile and pervasive as circulant matrices are, a great deal of their key properties can be established using only elementary linear algebra. In this section, we will present those properties that shall prove useful later on.

§ 3.1.

The DFT matrix

The following cyclic permutation matrix of order n 

Cn

0 0  . := circ(0, 1, 0, . . . , 0) =   ..  0 1

1 0 .. . 0 0

0 1 .. . 0 0

··· ··· .. .

0 0 .. . ··· 0 ··· 0

 0 0  ..   .  1 0

is regarded as the basic circulant matrix. Indeed, Cn has the fundamental representation property (3.1)

A = α0 I + α1 Cn + α2 Cn2 + · · · + αn−1 Cnn−1 = PA (Cn ),

where A = circ(α0 , α1 , . . . , αn−1 ) and PA is the polynomial PA (z) = α0 + α1 z + α2 z 2 + · · · + αn−1 z n−1 . This polynomial is called the associated polynomial, or representer (see [16, 33]), of the circulant matrix A. The characteristic polynomial of Cn is pCn (λ) = λn − 1, and thus its eigenvalues are λj := ωnj , 0 ≤ j ≤ n − 1, where ωn = exp(2πi/n) is a primitive n-th root of unity. The corresponding normalized eigenvectors are 1 xj := √ (1, ωnj , ωn2j , . . . , ωn(n−1)j )⊺ , n

0 ≤ j ≤ n − 1.

Since these vectors are pairwise orthogonal, we deduce the diagonalization (3.2)

Cn = Wn diag(1, ωn , ωn2 , . . . , ωnn−1 ) Wn∗ ,

189

On the norm of normal matrices

where Wn is the unitary matrix of order n whose columns are x0 , x1 , . . . , xn−1 . More explicitly, 

(3.3)

Wn

1 1 1 1 ω ωn2 n   1 ωn2 ωn4 1   = √  .. .. .. n . . .   (n−2)2 n−2 ωn  1 ωn (n−1)2

1 ωnn−1 ωn

··· ··· ··· .. .

ωnn−2 2(n−2) ωn .. .

···

ωn

1

(n−2)2

(n−1)(n−2)

· · · ωn

1



     .   (n−2)(n−1)  ωn  ωnn−1 2(n−1) ωn .. .

(n−1)2

ωn

The sequence ωnk , k ≥ 0, is periodic, and thus there are only n distinct elements in Wn . For example, 

W4

1  11 =  21 1

 1 1 1 i −1 −i   . −1 1 −1 −i −1 i

As a matter of fact, Wn is a most remarkable matrix: it is easily established that it is ⊺ symmetric, i.e. Wn⊺ = Wn , and unitary, i.e., Wn−1 = Wn∗ = W n . The combination of these properties means that Wn−1 = W n . Moreover, it is the Vandermonde matrix for the roots of unity up to a normalization factor. What is more essential is that Wn is closely related to Fn , the unitary Discrete Fourier Transform (DFT) matrix [16, § 2.5] of order n which was introduced by Sylvester in 1867 [43]. Indeed, Wn = Fn∗ = Fn . Finally, we deduce from (3.1), (3.2), and Cnn = I the following essential representation. If A ∈ circn (C) then (3.4)


A = Wn diag PA (1), PA (ωn ), PA (ωn2 ), . . . , PA (ωnn−1 ) Wn∗ .

This means that the unitary matrix Wn simultaneously diagonalizes all n × n circulant matrices. For further details see [16, § 3.2]. It follows from the foregoing that, PA (z), the associated polynomial of a given A ∈ circn (C) is actually closely related to pA (z), the characteristic polynomial of A. Indeed, PA (z) is the unique polynomial of degree strictly less than n whose values at ωnj , j = 0, 1, . . . , n − 1, spans over each and every one of eigenvalues of A (counted with multiplicity), whereas pA (z) is the unique monic polynomial of degree n that vanishes precisely at the eigenvalues of A. To conclude this introductory commentary, remark that, in the light of above observations, one can easily check that the product of two circulant matrices is again circulant and that, for this set of matrices, multiplication is commutative. Thus, circn (C) forms

190

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

a commutative ∗-algebra over C, with involution given by the conjugate transpose. Furthermore, circn (C) is canonically isomorphic to diagn (C), the set of all complex-valued diagonal matrices of order n.

§ 3.2.

The induced p-norm of A(n, ±a, b), Cases p = 1 and p = ∞

As previously mentioned, we seek to find the induced p-norm of the circulant matrices A(n, ±a, b) as given by (2.1). For p = ∞, the situation is straightforward. Indeed, given any matrix A ∈ Cn×n , we have kAxk∞

X n X n = max ai,j xj ≤ kxk∞ max |ai,j |. 1≤i≤n 1≤i≤n j=1

This shows that kAk∞→∞ ≤ max

n P

1≤i≤n j=1

j=1

|ai,j |. For the reverse inequality, we choose an

index 1 ≤ i0 ≤ n that realizes the maximum in the previous expression and we consider x ∈ Cn such that xj := sgn(ai0 ,j ), where sgn(z) denotes the complex signum function defined as   -z , z 6= 0, |z| sgn(z) =  0, z = 0. Hence, kxk∞ = 1 and kAxk∞

n n n n X X X X |ai,j |. ai0 ,j xj = |ai0 ,j | = max ai,j xj ≥ = max 1≤i≤n 1≤i≤n j=1

j=1

j=1

j=1

Therefore, kAk∞→∞ =

max

1≤i≤n

n X

|ai,j |,

j=1

which is simply the maximum absolute row sum of the matrix. In particular, for a, b ≥ 0 and for A(n, ±a, b) as given by (2.1), we have kA(n, ±a, b)k∞→∞ = |a| + (n − 1)|b|. From the above calculation, we can directly also infer that kA(n, ±a, b)k1→1 = |a| + (n − 1)|b|. Indeed kAkp→p = kAkq→q for H¨older conjugate indices p and q, since A is self-adjoint [23, § 5.4 and Theorem 5.3.35]. Nevertheless, we provide a different detailed proof, which is interesting in its own right. Given any x ∈ Cn , consider a partition of a

191

On the norm of normal matrices

matrix A ∈ Cn×n according to its columns as A = [a1 |a2 | · · · |an ] ∈ Cn×n . The triangle inequality gives kAxkp

X

n

= xj a j

≤

p

j=1

n X

|xj | · kaj kp = kxk1 max kaj kp 1≤j≤n

j=1

This shows that kAk1→p ≤ max kaj kp . For the reverse inequality, we choose an index 1≤j≤n

1 ≤ j0 ≤ n that realizes the maximum in the previous expression and we consider the canonical basis vector ej0 , i.e., the vector whose components are all zero, except the j0 -th that equals one. Then kAej0 kp = kaj0 kp = max kaj kp . 1≤j≤n

Hence, kAk1→p =

max kaj kp .

1≤j≤n

For p = 1, this is simply the maximum absolute column sum of the matrix. In particular, for a, b ≥ 0 and for A(n, ±a, b) as given by (2.1), we have kA(n, ±a, b)k1→1 = |a| + (n − 1)|b|. As a by-product, we also deduce kA(n, ±a, b)k1→p =

§ 3.3.

|a|p + (n − 1)|b|p )1/p .

The induced p-norm of A(n, ±a, b), Case p = 2

For 1 < p < ∞, the situation is no longer so simple. One cannot easily calculate the induced p-norm of A(n, ±a, b) by inspection. As a matter of fact, we shall see that the sign of the real number a plays a crucial role as the p-norms of A(n, a, b) and A(n, −a, b) are entirely different in general. Before discussing the general case, we start by addressing the case p = 2. To do this, we first remark that, for a diagonal matrix, all the induced p-norms are equal to the maximum of the absolute value of the entries. In [41], Sahasranand calculated this value for diagonal matrices involved in the diagonalization of A(n, a, b) and A(n, −a, b) as in (3.4). Lemma 3.1. Let a, b ≥ 0, and let A = A(n, ±a, b) be given by (2.1). Let ∗ Wn ΛWn be the diagonalization of A as in (3.4) and assume that 1 ≤ p ≤ ∞. (i) If A = A(n, a, b) = Wn ΛWn∗ , then we have kΛkp→p = a + (n − 1)b.

192

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

(ii) If A = A(n, −a, b) = Wn ΛWn∗ , then we have  a + b, kΛkp→p = −a + (n − 1)b,

if (n − 2)b ≤ 2a, if (n − 2)b ≥ 2a.

Proof. By (3.4), the matrix A can be expressed as A = Wn ΛWn∗ , where Λ is a diagonal matrix whose entries on the main diagonal are given by Λ1,1 = PA (1) = a + (n − 1)b and, for k = 2, . . . , n, Λk,k = PA (ωnk−1 ) = a + bωnk−1 + bωn2(k−1) + · · · + bωn(n−1)(k−1) n−1 X = (a − b) + b wnj(k−1) j=0

= a − b.

The last equality results from the application of the following well-known identity (see [35, § 3, Ex. 3]),  n−1 n, X if s = 0 mod n, js ωn = 0, otherwise. j=0 The result follows by calculating 

kΛkp→p

 kΛxkp : x 6= 0 = sup kxkp  n  1/p P p     |Λ x | k,k k   k=1 = sup P 1/p : x 6= 0 n      |xk |p k=1

=

max |Λk,k |

1≤k≤n

for 1 ≤ p ≤ ∞, first for A(n, a, b) and then for A(n, −a, b).

□

As in [41], we now use the previous lemma to derive an exact expression for kA(n, a, b)k2→2 and kA(n, −a, b)k2→2 for a, b ≥ 0. In doing so, we obtain an alternative proof of the result presented in [8].

On the norm of normal matrices

Theorem 3.2.

193

Let a, b ≥ 0.

(i) For A = A(n, a, b) defined as in (2.1), we have kAk2→2 = a + (n − 1)b. (ii) For A = A(n, −a, b) defined as in (2.1), we have  a + b, if (n − 2)b ≤ 2a, kAk2→2 = −a + (n − 1)b, if (n − 2)b ≥ 2a. Proof. Let Wn ΛWn∗ be the diagonalization of A(n, ±a, b) as in (3.4). We have kAk2→2 = kWn ΛWn∗ k2→2 ≤ kWn k2→2 · kΛk2→2 · kWn∗ k2→2 . Observe that kW ∗ xk22 = (W ∗ x)∗ W ∗ x = x∗ W W ∗ x = x∗ x = kxk22 . From this relation – which bears the name of Parseval’s identity – we deduce that kW ∗ k2→2 = 1, and the same goes for W . The first part of Lemma 3.1 implies that (3.5)

kA(n, a, b)k2→2 ≤ a + (n − 1)b,

while the second part entails that (3.6)

kA(n, −a, b)k2→2 ≤ a + b

if (n − 2)b ≤ 2a, and (3.7)

kA(n, −a, b)k2→2 ≤ −a + (n − 1)b

if (n − 2)b ≥ 2a. One can easily check that x = (1, 1, . . . , 1)⊺ is actually a maximizing vector for (3.5) and (3.7), i.e., that we can replace ≥ by the equal sign in these equations. As for the inequality in (3.6), it suffices to consider the vector y = (1, −1, 0, . . . , 0)⊺ to see that it too is saturated. □ We shall now present a generalization of the case A(3, −a, b). Proposition 3.3. If A = circ(α1 , α2 , α3 ) for arbitrary α1 , α2 , α3 ∈ R, then p  α12 + α22 + α32 − (α1 α2 + α2 α3 + α3 α1 ), if α1 α2 + α2 α3 + α3 α1 ≤ 0, kAk2→2 =  |α + α + α |, if α1 α2 + α2 α3 + α3 α1 ≥ 0. 1 2 3

194

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

Note that the special case b = α2 = α3 ≥ 0 and −a = α1 < 0 yields the case A(3, −a, b). The proof of Proposition 3.3 is omitted since we shall now state and prove a more general theorem, presented in [8], that subsumes both previous results and that conceptually explains why there are two cases that arise in calculations. Theorem 3.4. Suppose A = [c1 |c2 | · · · |cn ] ∈ Rn×n is an arbitrary matrix whose columns c1 , . . . , cn ∈ Rn satisfy c⊺1 c1 = . . . = c⊺n cn = ρ,

c⊺i cj = β,

(1 ≤ i < j ≤ n),

for some scalars ρ, β ∈ R. Then kAk2→2

√  ρ−β, =  pρ + (n − 1)β,

if β ≤ 0, if β ≥ 0.

Proof. It is well-known that the square of the induced 2-norm (also called the spectral norm) of any real n × n matrix A is precisely the largest eigenvalue of the symmetric matrix A⊺ A. Then, under the above-stated hypotheses,   ρ β β ··· β β ρ β · · · β      A⊺ A =  β β ρ · · · β  = (ρ − β)I + βK, . . . . .  .. .. .. . . ..  β β β ··· ρ where I is the n × n identity matrix and K is the n × n all-ones matrix (i.e., the matrix where every element is equal to 1). The eigenvalues of the positive semidefinite matrix A⊺ A = (ρ − β)I + βK are ρ − β (with multiplicity of n − 1) and ρ + (n − 1)β (with multiplicity 1). Taking their maximum and then its square root, the result follows. □ Note that Theorem 3.2.(ii) corresponds to the case ρ = a2 + (n − 1)b2

and β = − 2ab + (n − 2)b2 ,

whereas Proposition 3.3 corresponds to the case n = 3, and ρ = α12 + α22 + α32

and β = α1 α2 + α2 α3 + α3 α1 ,

of Theorem 3.4.

§ 3.4.

The p-norm of A(n, a, b)

In [8, Theorem 3.2], it was shown that if a, b ≥ 0, then for all 1 ≤ p ≤ ∞, kAkp→p = a + (n − 1)b. Here, we present a short proof taken from [41].

On the norm of normal matrices

Theorem 3.5. 1 ≤ p ≤ ∞. Then

195

Let a, b ≥ 0, and let A = A(n, a, b) be given by (2.1). Assume kAkp→p = a + (n − 1)b.

Proof. Using the vector x = (1, 1, . . . , 1)⊺ , we see that kA(n, a, b)kp→p ≥

kA(n, a, b)xkp = a + (n − 1)b. kxkp

This implies that, in order to prove the result, it suffices to show that x = (1, 1, . . . , 1)⊺ is actually a maximizing vector. We have seen that kAk∞→∞ = a + (n − 1)b = kAk2→2 . Now, by applying the Riesz–Thorin interpolation theorem [37, Theorem 1.1.1] with p0 = q0 = 2, p1 = q1 = ∞ and pθ = qθ , we obtain, (3.8)

θ kAkpθ →pθ ≤ kAk1−θ 2→2 kAk∞→∞ ,

for (with a mild abuse of notation) 1 1−θ θ 1−θ = + = , pθ 2 ∞ 2

(0 ≤ θ ≤ 1).

Hence, kAkpθ →pθ ≤ kAk∞→∞ = a + (n − 1)b, for 2 ≤ pθ ≤ ∞. Since A is self-adjoint, the result also follows for 1 ≤ p ≤ 2.

□

Using similar arguments as in the proof of Theorem 3.5, one can show that k circ(α0 , α1 , . . . , αn−1 )kp→p ≤ α0 + α1 + · · · + αn−1 , where αj ≥ 0. Then, considering again the same maximizing vector x = (1, 1, . . . , 1)⊺ , one concludes that k circ(α0 , α1 , . . . , αn−1 )kp→p = α0 + α1 + · · · + αn−1 .

§ 3.5.

The p-norm of A(n, −a, b)

It turns out that the calculations of the induced p-norms of A(n, −a, b) with a, b ≥ 0 are more involved than those of A(n, a, b). In fact, to the best of our knowledge, the exact value of kA(n, −a, b)kp→p for 1 < p < ∞, p 6= 2 is not known yet. Note however that lower bounds were provided in [8] as well as the two different set of upper bounds that we shall now present.

196

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

Theorem 3.6. 1 ≤ p ≤ ∞. Then

Let a, b ≥ 0, and let A = A(n, −a, b) be given by (2.1). Assume kAkp→p ≤ n| p − 2 | kAk2→2 . 1

1

Proof. We follow [41]. Because A is self-adjoint, we will assume without any loss of generality that 2 ≤ p ≤ ∞. Recall that for vectors in Cn , we have kxk22 = x⊺ x = x∗ x. Moreover, for 1 ≤ r ≤ p, kxkp ≤ kxkr ≤ n r − p kxkp . 1

(3.9)

1

Both inequalities will prove useful in the special case 2 = r ≤ p. For x 6= 0, the essential representation (3.4) implies that kAxkp ≤ kAxk2 = kWn ΛWn∗ xk2 ≤ kWn k2→2 · kΛk2→2 · kWn∗ xk2 . By the Parseval’s identity and that kWn k2→2 = 1, we deduce that kAxkp ≤ kΛk2→2 · kxk2 ≤ kΛk2→2 · n 2 − p kxkp 1

1

whereby kAkp→p ≤ n 2 − p kΛk2→2 . 1

1

The desired result now follows from Lemma 3.1.

□

Using once again the Riesz-Thorin interpolation theorem, it is possible to obtain another set of upper bounds for kA(n, −a, b)kp→p . Theorem 3.7. 1 ≤ p ≤ ∞. Then

Let a, b ≥ 0, and let A = A(n, −a, b) be given by (2.1). Assume  kAkp→p ≤ kAk2→2

 1− 2 kAk∞→∞ | p | . kAk2→2

Proof. For 2 ≤ p ≤ ∞, it suffices to interpolate between between kAk2→2 and kAk∞→∞ . As in the proof of Theorem 3.5, fix p0 = q0 = 2, p1 = q1 = ∞, and qθ = pθ to obtain (3.10)

θ kAkpθ →pθ ≤ kAk1−θ 2→2 kAk∞→∞

197

On the norm of normal matrices

for (with a mild abuse of notation) 1 1−θ θ 1−θ = + = , pθ 2 ∞ 2 Remark that θ = 1 −

2 pθ .

(0 ≤ θ ≤ 1).

Hence (3.10) can be restated as 1− p2

2 p

θ θ kAkpθ →pθ ≤ kAk2→2 kAk∞→∞ ,

(2 ≤ pθ ≤ ∞).

Since A is self-adjoint, the result follows for 1 ≤ p ≤ 2 with a bit of simplification.

- Upper bound from Theorem 3.6 - Upper bound from Theorem 3.7

-Upper bound from Theorem 3.6 - Upper bound from Theorem 3.7

4

- IIAIIH

- IIAIIH

p

(a) n = 5, a =

2 3

□

and b = 13 .

p

(b) n = 23, a = 0.1 and b = 0.05.

Figure 1: Upper and lower bounds of kAkp→p for different values of n, a, b. Even though the estimations provided in Theorem 3.7 are more complex than those given in Theorem 3.6, they are more accurate. In fact, when n is small, the difference between the two estimations is marginal. However, as n grows, the first estimation becomes significantly better than the latter (see Figure 1, where the parameters a, b have been normalized so that kAk2→2 = 1). Note, however, that neither of them provide the precise value of kA(n, −a, b)kp→p . Open question 3.8.

Evaluate kA(n, −a, b)kp→p .

Using similar arguments as in the proof of Theorem 3.7, Sahasranand [41, Theorem 6] showed the following. Theorem 3.9. Let a, b ≥ 0, and let A = A(n, −a, b) be given by (2.1). For 1 ≤ p ≤ 2, kAkp→p is monotonically non-increasing in p, whereas for 2 ≤ p ≤ ∞, kAkp→p is monotonically non-decreasing in p. 1 1 + p+β = 1. This time, Proof. Fix p ≥ 2 and β > 0. Let α ≥ 0 be such that p−α we apply the Riesz–Thorin interpolation theorem with p0 = q0 = p − α, p1 = q1 = p + β

198

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

and pθ = qθ to obtain (3.11) for

θ kAkpθ →pθ ≤ kAk1−θ (p−α)→(p−α) kAk(p+β)→(p+β) ,

1−θ θ 1 = + , pθ p−α p+β

(0 ≤ θ ≤ 1).

Since p − α and p + β are H¨older conjugates and since A is self-adjoint, we have kAk(p−α)→(p−α) = kAk(p+β)→(p+β) and (3.11) can be restated as kAkp→p ≤ kAk(p+β)→(p+β) , which implies that kAkp→p is monotonically non-decreasing in p for 2 ≤ p ≤ ∞. The case 1 ≤ p ≤ 2 is dealt with similarly. □ Finally, as Sahasranand [41] pointed out, conditional upon knowing the exact value of kA(n, −a, b)kr→r for some r > p > 2, yet another use of Riesz–Thorin interpolation theorem allows us to derive an upper bound kAkp→p that is even more precise than the one given by Theorem 3.7. Corollary 3.10. Let a, b ≥ 0, and let A = A(n, −a, b) be given by (2.1). Assume 2 ≤ p ≤ ∞. Then, for β ≥ 0, 1− 1 p p+β 1− 1 2 p+β

kAkp→p ≤ kAk2→2

1−1 2 p 1− 1 2 p+β

kAk(p+β)→(p+β) .

Obviously, an analogous result holds for 1 ≤ p ≤ 2.

§ 4.

An operator theoretic approach

In this section, we adopt an abstract approach to study the norm. Our setting is general enough and in the special case of circulants, leads us to some of our previous results as well as some new estimations.

§ 4.1.

Axis-oriented norms

Suppose we endow Cn×n , the ring of complex n × n matrices, with a norm k · k which satisfies (4.1)

kdiag(z1 , . . . , zn )k = max{|z1 |, . . . , |zn |}.

Recall that the spectral radius of a given matrix A ∈ Cn×n whose eigenvalues are λ1 , . . . , λn is defined as ρ(A) := max{|λ1 |, . . . , |λn |}.

On the norm of normal matrices

199

The spectral radius, though not itself a matrix norm, is some sort of infimum over all matrix norms, in that ρ(A) ≤ kAk for all A ∈ Cn×n and all matrix norm k·k. Therefore, a norm satisfying (4.1) is optimal for all diagonal matrices. Matrix norms induced by a vector norm verifying (4.1) were said to be axis-oriented by Bauer and Fike in [3]. As it turns out, using the following equivalence relations (see [4, Theorems 2 and 3]), one can easily verify that axis-orientedness is not an uncommon property of induced matrix norms. Theorem 4.1. are equivalent.

Let n ≥ 2, and let k · k be a vector norm on Cn . The following

(i) The matrix norm induced by k · k satisfies (4.1).

(ii) The vector norm k · k is absolute, i.e., kxk = |x| for all x ∈ Cn , where |x| denote the vector the components of which are the moduli of the components of x. (iii) The vector norm k · k is monotonic, i.e., kxk ≤ kyk whenever |xi | ≤ |yi | for all 1 ≤ i ≤ n. One can easily verify from their definition that the p-norms (1 ≤ p ≤ ∞) are absolute. Therefore, their respective induced matrix norms satisfy (4.1). The extension of the concept of axis-orientedness to all matrix norms (that is, even to the matrix norms that are not induced by any vector norm) and to vector norms on Cn×n is a natural step forward since there are vector norms and non-induced matrix norms satisfying (4.1). Consider for instance the element-wise (vector) norm kAkmax :=

max |aij |,

1≤i,j≤n

which is not sub-multiplicative and thus not a matrix norm, or the sub-multiplicatives matrix norms kAkmax{p,q} := max{kAkp→p , kAkq→q },

(1 ≤ p, q ≤ ∞, p 6= q),

which are not minimal, hence non-induced (see [23, Theorem 5.6.32]).

§ 4.2.

The ∗-algebra TA

Let P denote the set of all analytic polynomials in two independent variables, i.e., p(z, w) :=

N X

amn z m wn .

m,n=0

Note that, being in the world of commutative polynomial, there is no difference between zw and wz. Hence, for an arbitrary polynomial p, the combination p(A, B) will be

200

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

meaningful if the matrices A and B commute. In particular, p(A, A∗ ) makes sense whenever A is normal, i.e., AA∗ = A∗ A. Given a normal matrix A, let TA be the ∗-algebra generated by A, i.e., TA = {p(A, A∗ ) : p ∈ P}. We may even generalize the above definition by considering the functions f (z, w) which are defined at least on the set σ(A) × σ(A). However, we stick to polynomials to make the presentation free of some technical difficulties. By the Spectral Theorem for normal matrices [23, Theorem 2.5.3], there exists a unitary matrix U such that A = U ΛU ∗ , where Λ = diag(λ1 , . . . , λn ) is the diagonal matrix whose entries are the eigenvalues of A. Therefore, an easy algebraic computation shows (4.2)


p(A, A∗ ) = U p(Λ, Λ∗ )U ∗ = U diag p(λ1 , λ1 ), . . . , p(λn , λn ) U ∗ .

In subsequent subsections, we seek to calculate (or, failing that, estimate) the value of kp(A, A∗ )k, where k · k is some axis-oriented matrix norm.

§ 4.3.

Unitarily invariant norms

A norm k · k on Cn×n is called unitarily invariant if kU Xk = kXU k = kXk for all X ∈ Cn×n and all unitary matrices U ∈ Cn×n . Important examples of unitarily p invariant norms include the spectral norm kAk = ρ(A∗ A), the Frobenius norm kAkF :=

X n n X

1/2 |ai,j |

2

,

i=1 j=1

and the Schatten p-norms kAkS∞ :=

max σi (A)

1≤i≤n

&

kAkSp :=

X n

1/p σip (A)

,

(1 ≤ p < ∞),

i=1

where σi ’s designate the singular values of A ∈ Cn×n . One needs only look to the Frobenius norm to see that not all unitarily invariant norms on Cn×n satisfy the condition (4.1). In fact, it is not difficult to verify that the spectral norm is the only unitarily invariant norm on Cn×n that is axis-oriented. By the singular value decomposition theorem [23, Theorem 2.6.3], for all A ∈ Cn×n , there are unitary matrices U, V ∈ Cn×n such that A = U ΣV ∗ , where Σ = diag(σ1 , . . . , σn )

On the norm of normal matrices

201

is a diagonal matrix whose entries are the singular values of A listed in non-increasing order. Hence, if k · k is a unitarily invariant norm satisfying (4.1), then kAk = kU ΣV ∗ k = kΣk = kdiag(σ1 , . . . , σn )k = max{|σ1 |, . . . , |σn |} = σ1 = kAk2→2 . The following result is an immediate consequence of (4.2) and the assumption (4.1). Proposition 4.2. For all normal matrix A ∈ Cn×n and all polynomial in two variables p,  kp(A, A∗ )k2→2 = max |p(λ1 , λ1 )|, . . . , |p(λn , λn )| .

§ 4.4.

More general norms

If k · k is not unitarily invariant, then some upper and lower estimates come into play. A given matrix U acts as an operator on Cn×n by left or right multiplication. Hence, we naturally consider the left operator norm induced by k · k   kU Xk n×n kU kℓ := sup (4.3) :X∈C , X 6= 0 , kXk and the right operator norm induced by k · k   kXU k n×n (4.4) :X∈C , X 6= 0 . kU kr := sup kXk In other words, kU kℓ and kU kr are respectively the best constant such that (4.5)

kU Xk ≤ kU kℓ kXk,

X ∈ Cn×n ,

kXU k ≤ kU kr kXk,

X ∈ Cn×n .

and (4.6)

If U is a unitary matrix, we can replace U by U ∗ in the above inequalities and obtain the corresponding bounds for U ∗ . Then, depending on the case, replacing X by U X or XU , we obtain the lower bounds (4.7)

kU Xk ≥

1 kXk, kU ∗ kℓ

X ∈ Cn×n ,

kXU k ≥

1 kXk, kU ∗ kr

X ∈ Cn×n .

and (4.8)

In the important case where k · k is submultiplicative, we have kAkℓ ≤ kAk and kAkr ≤ kAk for every A ∈ Cn×n . The reverse inequalities trivially hold if k · k satisfies kIk = 1. From these observations, we can deduce the following result.

202

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

Proposition 4.3. If k · k is a matrix norm on Cn×n satisfying kIk = 1, then the left operator norm and the right operator norm coincide with the norm itself. Although there is a vast class of norms for which k · kℓ ≡ k · kr ≡ k · k, and chief among them will be the induced matrix norms, this equivalence does not always occur. One only needs to took at the left and right operator norms induced by the Frobenius norm. For the identity matrix I, for instance, we have √ kIkℓ = kIkr = 1 < n = kIkF . It should furthermore be noted that if kXk = kX ∗ k for all X ∈ Cn×n , then kU kℓ = kU ∗ kr and kU ∗ kℓ = kU kr . And if k · k is a unitarily invariant norm, then clearly kU kℓ = kU kr = 1. Equipped with these estimations, we may derive a result analogous to Proposition 4.2 that equally holds for unitarily invariant and non-unitarily invariant norms. The price to pay for having greater generality is that we no longer have an exact expression for kp(A, A∗ )k, but rather some lower and upper bounds. Proposition 4.4. Let k · k be a norm on Cn×n which satisfies (4.1) and let
A ∈ Cn×n be a normal matrix which admits the decomposition U diag λ1 , . . . , λn U ∗ . Then  kp(A, A∗ )k ≤ kU kℓ kU ∗ kr max |p(λ1 , λ1 )|, . . . , |p(λn , λn )| and kp(A, A∗ )k ≥

 1 max |p(λ , . . . , |p(λ , λ )|, λ )| . 1 1 n n kU kr kU ∗ kℓ

Proof. The proof is based on a judicious use of estimations developed above. First, by (4.2),
p(A, A∗ ) = U p(Λ, Λ∗ )U = U diag p(λ1 , λ1 ), . . . , p(λn , λn ) U ∗ . Then by (4.5) (applied for U ) and (4.6) (applied for U ∗ ), we obtain
kp(A, A∗ )k ≤ kU kℓ kU ∗ kr kdiag p(λ1 , λ1 ), . . . , p(λn , λn ) k. The result now follows upon using (4.1). The other part is proved similarly. □
∗ Note that the decomposition A = U diag λ1 , . . . , λn U is not unique, and some decomposition may lead to more precise estimations than others in Proposition 4.4. It is easily verified that these estimations are sharp in general since we can replace both inequalities by an equality when k · k is the spectral norm (see Proposition 4.2). Moreover, it is trivial to show that both estimations becomes an equality if A is a diagonal matrix. However, it is unknown in general when it is possible to replace the inequalities of Proposition 4.4 by equalities.

On the norm of normal matrices

Open question 4.5. Proposition 4.4.

203

Characterize the cases of equality in both estimations of

Remark. In (4.3) and (4.4), we defined kU kℓ and kU kr as     kXU k kU Xk n×n n×n sup :X∈C , X 6= 0 , & sup :X∈C , X 6= 0 , kXk kXk respectively. These definitions were deemed natural, since we considered the matrix U acting as an operator on Cn×n by left or right multiplication. We then restricted ourselves to the cases where U is a unitary matrix. With this assumption, it is in many ways more natural to view U as an operator acting on the unitary group U(n), i.e., the group of n × n unitary matrices. The associated norms would thus be   kU Xk kU kℓ∗ := sup : X ∈ Un kXk and  kU kr∗ := sup

 kXU k : X ∈ Un . kXk

Of course, these norms are not appropriate for our study and they do not help us obtain a better result than Proposition 4.4. Nonetheless, we propose the following question which is interesting in its own right. Open question 4.6. kU kℓ .

§ 4.5.

Find necessary and sufficient conditions such that kU kℓ∗ =

Application to circulant matrices

With the help of the tools presented in the previous subsections, we now turn our attention to the circulant matrices. Let A be an arbitrary circulant matrix and recall the fundamental representation (3.4) of A, i.e.,
A = Wn diag PA (1), PA (ωn ), PA (ωn2 ), . . . , PA (ωnn−1 ) Wn∗ , where Wn is the adjoint of the Discrete Fourier Transform matrix and PA is the associated polynomial of A. Then, by Proposition 4.4,  kAk = kPA (Cn )k ≤ kWn kℓ kWn∗ kr max |PA (1)|, . . . , |PA (ωnn−1 )| , and kAk = kPA (Cn )k ≥

 1 n−1 max |P (1)|, . . . , |P (ω )| . A A n kWn kr kWn∗ kℓ

204

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

In particular, if the norm is unitarily invariant (i.e., if it is the spectral norm) then  kAk2→2 = max |PA (1)|, . . . , |PA (ωnn−1 )| ,

(4.9)

which corresponds to a special case of Proposition 4.2. More generally, if the matrix norm is the induced p-norm, then Proposition 4.3 reveals that the above estimations reduce to kAkp→p ≤ kWn kp→p kWn kq→q max |PA (ωnk )|

(4.10)

1≤k≤n

and kAkp→p ≥

(4.11)

1 max |PA (ωnk )|. 1≤k≤n kWn kp→p kWn kq→q

It is not immediately clear how precise these estimations are and naturally wonders if they are sharper than those presented in Theorem 3.7, in the particular case where A = A(n, −a, b). To answer this question, we first need to compute the value of kWn kp→p . Lemma 4.7.

Let Wn be defined as in (3.3) and assume that 1 ≤ p ≤ ∞. Then kWn kp→p = kWn∗ kp→p = n|1/p−1/2| .

Proof. Let us suppose that 1 ≤ p ≤ 2. Since Wn is unitary, we have (4.12)

kWn xkp ≤ n p − 2 kWn xk2 = n p − 2 kxk2 ≤ n p − 2 kxkp . 1

1

1

1

1

1

It thus follows that kWn kp→p ≤ n1/p−1/2 . Considering the vector x = (1, 0, . . . , 0)⊺ , we have p Pn−1 Pn−1 Pn−1 p −ij p x ω kWn xkp i=0 j=0 j p n i=0 |1| = = = p/2 = n1− 2 . P p n−1 p p/2 kxkp n n np/2 i=0 |xi | Hence, we conclude kWn kp→p = n1/p−1/2 . The same method can then be used to show 1 1 that kWn∗ kp→p = n p − 2 . Therefore, the conclusion follows after a bit of simplification by noticing that kWn∗ kp→p = kWn kq→q

&

kWn kp→p = kWn∗ kq→q ,

where q denote the H 旦 lder conjugate of p. Remark. (4.13)

Let cp and Cp be the the best constants such that cp kxkp ≤ kU xkp ≤ Cp kxkp ,

□

205

On the norm of normal matrices

where U varies over all unitary matrices and x ∈ Cn . It is not difficult to show that we have cp Cp = 1. Also, observe that (4.12) and its analog for 2 ≤ p ≤ ∞ are also valid for any unitary matrix U . In particular, we have kU xkp ≤ n| p − 2 | kxkp 1

1

for any n × n unitary matrix U and x ∈ Cn . Moreover, we have shown in the previous proof that there exist some x ∈ Cn such that kWn xkp = n|1/p−1/2| kxkp . Therefore, we cannot replace n|1/p−1/2| by a smaller constant in the inequality kU xkp ≤ Cp kxkp and thus, we have shown that Cp = n|1/p−1/2| = c−1 p . Now, we know from (4.9) that max1≤k≤n |PA (ωnk )| = kAk2→2 and Lemma 4.7 ensures that kWn kp = n|1/p−1/2| . Therefore, we see that (4.10) and (4.11) become n−| p −1| kAk2→2 ≤ kAkp→p ≤ n| p −1| kAk2→2 . 2

2

Compare this to the upper bound kAkp→p ≤ n| p − 2 | kAk2→2 . 1

1

of Theorem 3.6. While the form is obviously very similar, it is interesting to note that their derivation was quite different. In fact, it is an easy exercise to show that for every p ∈ [1, ∞], the latter (i.e., the upper bound obtained by interpolation) is sharper. Hence, the new approach detailed in this section, despite being novel and general, does not improve the previous estimations of the induced p-norm of A = A(n, −a, b) for a, b ≥ 0. As for the lower bound, remark that since the induced p-norm is a matrix norm, we have ρ(A) ≤ kAkp→p for every A ∈ Cn×n . Moreover, notice that the |PA (ωnk )|’s are in fact the eigenvalues of A. Thus, it follows that n−| p −1| kAk2→2 = n−| p −1| max |PA (ωnk )| = n−| p −1| ρ(A). 2

2

2

1≤k≤n

Hence, we easily see that the lower bound is not sharp, except if n−| p − 2 | = 1. As n ≥ 2, this only happens if p = 2. 1

§ 5.

1

An optimization-oriented approach

In this section, we discuss some of our partial findings relative to the Open question 3.8: Is it possible to provide a simple closed formula for kA(n, −a, b)kp→p ? Here, we adopt a more direct approach using essentially tools from elementary calculus. Accordingly, the results obtained are more limited in scope, and their proofs are quite

206

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

technical in nature. Although, once more, we derive some results that cover a wide scope, the key difference is that we eventually focus our attention on the particular case of kA(n, −a, b)kp→p .

§ 5.1.

A more precise lower bound

We begin with the following proposition, which provides a more refined lower bound for kA(n, −a, b)kp→p . This estimate will be used later on. Proposition 5.1. Let a, b ≥ 0, and let A = A(n, −a, b) be given by (2.1). Assume 1 < p < ∞. Then  p  2−p p 1 1 a + b (n − 1) 1−p + (n − 1) 1−p a + b (n − 1) p−1 − n + 2 kAkpp→p ≥ . 1 1 + (n − 1) 1−p In particular, we also have kAkp→p ≥ a + b, with equality if and only if either n = 2, or (n − 2)b ≤ 2a and p = 2. 1 . Consider the vector x = (−η ρ , 1, 1, . . . , 1)⊺ . Proof. Fix η := n − 1 and ρ := p−1 A simple computation gives a + bη 1−ρ p + η −ρ |a + b (η ρ − η + 1)|p kAxkpp , = kxkpp 1 + η −ρ −1

proving the first part of the statement. Now, let f (z) := |z|p and let t := (1 + η −ρ ) Then we can rewrite the above equation as

.


kAxkpp = tf a + bη 1−ρ + (1 − t)f (a + b (η ρ − η + 1)) . p kxkp Hence, the convexity of f ensures us that

kAxkpp ≥ f t a + bη 1−ρ + (1 − t) (a + b (η ρ − η + 1)) p kxkp

= f a + bt η −ρ + 1 (η − η ρ ) + b (η ρ − η + 1) = f (a + b (η − η ρ ) + b (η ρ − η + 1)) = f (a + b) = (a + b)p . It follows that kAkp→p ≥ a + b. Moreover, since f is never linear, the inequality fact an equality if and only if a + bη 1−ρ = a + b (η ρ − η + 1). This occurs if and
if (η ρ + 1) η 1−ρ − 1 = 0, if and only if n = 2 or p = 2. It is trivial to verify kAkp→p = a + b for every p ∈ [1, ∞] if n = 2 and we know from Theorem 3.2 kAk2→2 = a + b if and only if (n − 2)b ≤ 2a. This concludes the proof.

is in only that that □

207

On the norm of normal matrices

Remark. If p tends to 1 (resp. to ∞), then the inequality given in Proposition 5.1 becomes a + (n − 1)b, which is precisely the value of kAk1→1 (resp. of kAk∞→∞ ). As usual, let a, b ≥ 0, and let A = A(n, −a, b) be given by (2.1). The maximizing ∥Ax∥ real vectors of ∥x∥pp are the vectors x for which kAxkp = kAkp→p . kxkp As a direct application to the previous proposition, we obtain the following result. This corollary will be used in the following section, which provides further details on the ∥Ax∥ maximizing real vectors of ∥x∥pp . Corollary 5.2. Let a, b ≥ 0, and let A = A(n, −a, b) be given by (2.1). Assume ∥Ax∥ that x ∈ Rn is a maximizing real vector for ∥x∥pp which satisfies x1 + · · · + xn = 0. Then either n = 2, or (n − 2)b ≤ 2a and p = 2. Proof. If x1 +· · ·+xn = 0, a direct computation reveals that ∥Ax∥p ∥x∥p ,

∥Ax∥p ∥x∥p

= a+b. Hence,

we find kAkp→p = a + b. Proposition 5.1 since x is a maximizing real vector for then ensures that this occurs if and only if either n = 2, or (n − 2)b ≤ 2a and p = 2. □

§ 5.2.

The maximizing vectors for

∥Ax∥p ∥x∥p

Let a, b ≥ 0, and let A = A(n, −a, b) be given by (2.1). We now focus on the ∥Ax∥ maximizing real vectors for ∥x∥pp . It is proved in [8] that, in the particular case where p = 2 and (n − 2)b ≤ 2a, the maximizing real unit vectors are those whose entries verify x1 + x2 + · · · + xn = 0. Whereas in the case where p = 2 but (n − 2)b ≥ 2a, the maximizing real unit vectors are x = ± √1n (1, 1, . . . , 1)⊺ . ∥Ax∥

Finding the maximizing real unit vectors for ∥x∥pp when 1 ≤ p ≤ ∞ and p 6= 2 is a daunting task. However, in what follows, we show that the entries of a given maximizing real vector always form a set of cardinality at most three. The proof of this result is somewhat convoluted. Prior to presenting a detailed demonstration, we need to establish the following two technical lemmas. We also need to define x[p] := sgn(x)|x|p = x|x|p−1 , which is the derivative of 1 p+1 when 1 < p < ∞. p+1 |x| Lemma 5.3. valued function

Assume 1 ≤ p ≤ ∞. Let a, b, c1 , c2 , d ∈ R, and consider the real[p−1]

f (x) := a (c1 x − c2 )

− bx[p−1] + d.

Then either f is identically zero or it has at most three distinct roots.

208

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

Proof. Suppose that f is not identically zero. A computation reveals that   p−2 p−2 f ′ (x) = (p − 1) ac1 |c1 x − c2 | − b |x| . If abc1 ≤ 0, then f ′ is either always non-negative or always non-positive. In the special case where f ′ ≡ 0, we find that f has no roots since f 6≡ 0. Otherwise, f is either strictly increasing or strictly decreasing, and thus f has exactly one root. If abc1 > 0, then f ′ changes sign precisely at the following two points:
1
1 ac1 p−2 ac1 p−2 c c2 2 b b . x1 = & x2 = 1 1

ac1 p−2 ac1 p−2 c + c − 1 1 1 1 b b Hence, f is decreasing on (−∞, min{x1 , x2 }), increasing on (min{x1 , x2 }, max{x1 , x2 }) and decreasing on (max{x1 , x2 }, ∞), or vice versa. It follows immediately that f has at most three distinct roots, as stated. □ Lemma 5.4. Let a, b ≥ 0, with at least one of them non-zero. Let A = A(n, −a, b) be given by (2.1). Suppose n > 2. Assume 1 < p < ∞. If x ∈ Rn is a maximizing real ∥Ax∥ vector for ∥x∥pp , then, for every 1 ≤ j, k, l ≤ n, (5.1)

[p−1]

wk

[p−1]


− wj

[p−1]

xk

[p−1]


− xl

=

[p−1]

wk

[p−1]


− wl

[p−1]

xk

[p−1]


− xj

where wi = (a + b)xi − b(x1 + x2 + · · · + xn ). Proof. We have kAkpp→p = = =

sup kAxkpp

∥x∥p =1

sup

n X

∥x∥p =1 i=1 n X

sup

∥x∥p =1 i=1

p

|(a + b)xi − b(x1 + · · · + xn )| |wi |p .

Since the unit circle is compact, we can define the Lagrange multiplier
L(x, λ) := kAxkpp − λ kxkpp − 1 and since for 1 < p < ∞, the derivative of |x|p exist everywhere, we find that the maximum of kAxkpp on the set of unit vectors is obtained if and only if we have ∂L ∂λ = 0 ∂L ∂L and ∂xk = 0, for every k ∈ {1, 2, . . . , n}. It is then easily verified that we have ∂xk = 0 if and only if (5.2)

(a +

[p−1] b) wk

−

n X i=1

[p−1]

bwi

[p−1]

= λxk

.

209

On the norm of normal matrices

Now, if any two of xj , xl and xk are equal (1 ≤ j, k, l ≤ n) equation (5.1) is directly satisfied. Otherwise, choose any j, k, l such that xj 6= xk 6= xl . Subtracting the equation (5.2) associated with the coefficient j (resp. l) to the one associated with the coefficient k and simplifying, we get [p−1]

wk

[p−1] xk

[p−1]

− wj −

[p−1] xj

λ = a+b

[p−1]

&

wk

[p−1] xk

[p−1]

− wl −

[p−1] xl

=

λ . a+b

Note that we can safely divide by a+b since a, b ≥ 0, with at least one of them non-zero. Hence, equalling the left hand side of both equations and simplifying yield [p−1]
[p−1] [p−1]
[p−1] [p−1]
[p−1] [p−1]
[p−1] wk − wj xk − xl = wk − wl xk − xj and we are done.

□

We now have everything in hand to demonstrate that the following holds true. Proposition 5.5. Let a, b ≥ 0, with at least one of them non-zero. Let A = A(n, −a, b) be given by (2.1). Suppose n > 2. Assume 1 < p < ∞, with p 6= 2. If ∥Ax∥ x ∈ Rn is a maximizing real vector for ∥x∥pp , then the entries of x form a set of cardinality at most three. Proof. Suppose without any loss of generality, even if it means rearranging the coefficients of x, that x1 6= x2 and for simplicity, define ρ := p − 1. Then Lemma 5.4 ensures us that [ρ] [ρ]
[ρ] [ρ]
[ρ] [ρ]
[ρ] [ρ]
wk − w1 xk − x2 = wk − w2 xk − x1 for any k, where wi = (a + b)xi − b(x1 + x2 + · · · + xn ). Since n > 2 and p 6= 2, Corollary 5.2 ensures us that x1 + · · · + xn 6= 0. Hence, we can define yi := xi /(x1 + · · · + xn ) and zi := wi /(x1 + · · · + xn ) = (a + b)yi − b. A simple division by x1 + · · · + xn 6= 0 then allows us to show that [ρ] [ρ]
[ρ] [ρ]
[ρ] [ρ]
[ρ] [ρ]
z k − z 1 yk − y2 = zk − z2 yk − y1 , (1 ≤ k ≤ n). This can be rewritten using the function
[ρ] [ρ] [ρ]
[ρ] [ρ]
[ρ] [ρ] [ρ] [ρ]
(5.3) f (x) := y2 − y1 (a + b)x − b + z1 − z2 x[ρ] + y1 z2 − y2 z1 as f (yk ) = 0 for every 1 ≤ k ≤ n. Now, let α := y1 and β := y2 . An application of Lemma 5.3 reveals that the function f has at most three distinct roots. Moreover, we easily verify that f (α) = f (β) = 0. Therefore, the roots of f are α, β and possibly another value, call it γ. Since f (yk ) = 0 for every 1 ≤ k ≤ n, it follows that for any 1 ≤ k ≤ n, we either have yk = α, β or γ. Hence, yk can take at most three distinct values and thus, xk can also take at most three distinct values. □

210

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

Remark. Note that, even though the proof given in the previous proposition does not apply for p = 1, 2, ∞, it is not actually a problem since the norm of A for all of these values are known and are almost trivial problems. Moreover, in all of these cases (with the exception of p = 2 and (n − 2)b ≤ 2a), the entries of the maximizing vectors actually only have at most two distinct values. ∥Ax∥

If x is a maximizing real vector for ∥x∥pp , then Proposition 5.5 tells us that x has at most three distinct coefficients. Suppose that x does indeed have exactly three distinct coefficient and define once again yi := xi /(x1 + · · · + xn ) and zi := wi /(x1 + · · · + xn ) = ((a + b)yi − b). Moreover, assume that two of the three distinct coefficients of y are α, occurring m times, and β, occurring k times. Then the third distinct coefficient of y, say γ, is precisely the unique root of f which is not equal to α or β, where f is given by (5.3). Moreover, by construction, the sum of the coefficients of y must be equal to 1, i.e., they must satisfy the equation mα + kβ + (n − m − k)γ = 1 and thus we also find that γ = 1−mα−kβ n−m−k . Remark that this is two completely different way of determining the value of γ. However, it is natural to assume that the two functions determining the value of γ are independent and thus that we have no reason to expect the two methods to give the same result. This observation motivated the following conjecture, which is strongly backed up with numerical evidence for several values of a, b, n and p. Conjecture 5.6. Let a, b ≥ 0, with at least one of them non-zero. Let A = A(n, −a, b) be given by (2.1). Assume 1 ≤ p ≤ ∞, with p 6= 2. If x ∈ Rn is a ∥Ax∥ maximizing real vector for ∥x∥pp , then the entries of x form a set of cardinality at most two.

§ 5.3.

The special case of I − n1 K

Our interest in the matrices A(n, a, b) stems from other studies on the geometry of the Birkhoff polytope and originated with the matrix I − n1 K which corresponds to the 1 1 matrix −A(n, 1−n n , n ). Here, we provide a description of kI − n Kkp→p , assuming the validity of Conjecture 5.6. We then show that the conjecture is valid in the special case n = 3. Proposition 5.7. Let 1 ≤ p ≤ ∞, with p 6= 2. Let xp be the unique root of the function   

2−p 1 x 7−→ (p − 1) 1 + (x − 1) p−1 1 − (x − 1)p−2 + 1 − (x − 1) p−1 1 + (x − 1)p−1     in the interval [1, 2], and let m1 := xnp and m2 := xnp . Suppose that Conjecture 5.6 is valid. Then  1− p1 p1  1
p−1
p−1 n n +1 +1

m −1 m −1

I − 1 K = max (5.4) . n n p→p m∈{m1 ,m2 }

m

211

On the norm of normal matrices

Proof. Let x be the maximizing real vector with coefficients α, occurring m times, and β, occurring (n − m) times. We know from Corollary 5.2 that the sum of the coefficients of x cannot be 0. Hence, without any loss of generality, suppose that mα + (n − m)β = n, i.e., that β = n−mα n−m . In this case, a direct computation reveals that k(I

− n1 K)xkpp kxkpp

p

m (α − 1)

=

mαp +



+ (n − m) m−mα n−m p  (n − m) n−mα n−m

p =: fm (α).

Taking the derivative relative to α, we get 2−p

′ fm (α)

(5.5)

⇐⇒

= 0

α =

n (n − m) p−1 2−p

1

m (n − m) p−1 − m p−1

,

and we easily verify that this critical point is, in fact, a local maximum. Putting this value in fm (α) and simplifying, we get ( )p−1 )( 1 p−1 n n  −1) +1 −1) p−1 +1 ( (  m m  if m 6= 0, n p ) (m fm (α) =    1 if m = 0. Therefore,

(5.6)



I − 1 K p = n p→p

 max

n m

−1


p−1

1≤m≤n

 +1

n m
n p m

−1

1
p−1

p−1 +1 .

Remark that, since α and β are interchangeable, we can consider only the maximum for n2 ≤ m ≤ n. Moreover, defining  g(x) :=

p−1

(x − 1)

 +1

(x − 1)

xp

1 p−1

p−1 +1 ,

we have that the right hand side of (5.6) correspond to max1≤m≤n g(n/m). Since n 2 ≤ m ≤ n, finding the maximum of g(n/m) is closely related to finding the maximum of g in [1, 2]. Let xp be the unique root of the function   

2−p 1 p−2 p−1 p−1 x 7−→ (p − 1) 1 + (x − 1) 1 − (x − 1) + 1 − (x − 1) 1 + (x − 1)p−1 in [1, 2]. Then, an analysis of g reveal that it is increasing on [1, xp ] and decreasing on [xp , 2]. Therefore, the value of m ≥ n2 for which the maximum of g(n/m) is attained must be one of the two closest values of n/m to xp . More precisely, it can be verified     that it must be either xnp or xnp . □

212

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

p 0

2

3

4

5

Figure 2: Values of xp between p = 1 and p = 5.25.

Despite its complicated description, we may use various methods to obtain approximations of xp and evaluate it numerically. It is also possible to show that xp varies 2x between 1 (when p → 1, ∞) and ≈ 1.090776, which is the unique root of ln (x − 1)− x−2 in [1, 2] (when p → 2). This observation alone can greatly reduce the number of cases needed to verify in (5.6) to compute the norm of I − n1 K (still assuming the validity of Conjecture 5.6), even without precisely knowing xp for every value of the parameter p.

§ 5.4.

A special case of the Conjecture 5.6

We now turn our attention to proving that Conjecture 5.6 is valid in the particular case n = 3. To do this, we first need to prove the following two technical lemmas.
p−1 p−1 2−p −t , where t ∈ ( 12 , 1] and 2 p−1 ≤ γ < 1. Lemma 5.8. Let J(t) := t + γ−1 3 If p > 2 is a rational number with an even numerator and an odd denominator, then J(t) < J(1 − t). Proof. We have !  p−2 γ − 1 J ′ (t) = (p − 1) t+ − tp−2 . 3
p−2 p−2 Therefore, J ′ (t) < 0 if and only t + γ−1 < t . Because of our hypothesis on the 3 < |t|, which is equivalent to form of p, this inequality is satisfied if and only if t + γ−1 3 0
2 is a rational number with an even numerator and an odd denominator, then
p−1
1 2p−1 + 1 2 p−1 + 1 , max max Fγ (t) = s≤γ≤1 0≤t≤1 3p where the equality is attained only if t = 21 . Proof. Case 1 (γ = 1): If γ = 1, then Fγ (t) = 1 for each t ∈ [0, 1]. Case 2 (γ 6= 1): Computing the derivative of Fγ , we find Fγ′ (t)

= p

t+



p−1 γ−1 p−1 − 1 − t + γ−1 3 3 p tp + (1 − t) + γ p

p−1

tp−1 − (1 − t) − Fγ (t) p p t + (1 − t) + γ p

! .

214

´ de ´ric Morneau-Gue ´rin Ludovick Bouthat, Javad Mashreghi, and Fre

By direct verification, Fγ′ (t) = 0 if t = 1/2. If t 6= 1/2, then tp−1 − (1 − t) 6= 0 and we have !

γ−1 p−1 γ−1 p−1 p−1 p−1 t + − 1 − t + t − (1 − t) 3 3 Fγ′ (t) = p p − Fγ (t) , p p−1 p p−1 t + (1 − t) + γ t − (1 − t) p−1

which vanishes if and only if t+



γ−1 p−1 γ−1 p−1 1 − t + − 3 3 p−1 p−1 t − (1 − t)

= Fγ (t).

We now show that we always have
p−1
γ−1 p−1 t + γ−1 − 1 − t + 3 3 (5.8) < 1. p−1 p−1 t − (1 − t) It will then follow that if Fγ′ (t) vanishes at a point (γ, t), with t 6= 1/2, then we must have Fγ (t) < 1. Therefore, by Case 1, the point (γ, t) cannot be a maximum, and we will be able to reject these points. First notice that by the symmetry along the x = 1/2 axis, we can suppose without any loss of generality that 1/2 < t ≤ 1 (since we supposed that t 6= 1/2). In that case, both p−1  p−1  γ−1 γ−1 p−1 − 1−t+ & tp−1 − (1 − t) t+ 3 3 are positive and thus (5.8) is equivalent to having  p−1  p−1 γ−1 γ−1 p−1 p−1 t+ (5.9) −t < 1−t+ − (1 − t) , 3 3 for 1/2 < t ≤ 1. Now, if we define

 p−1 γ−1 J(t) := t + − tp−1 , 3

the above inequality can be restated as J(t) < J(1 − t). Lemma 5.8 then ensures us that this inequality is satisfied and thus, (5.9) is valid and so is (5.8). Therefore, the only points (γ, t) which can give rise to a maximum of Fγ (t) occurs when t = 21 . Moreover, it is easily verified that for a fixed γ ∈ [s, 1], t = 12 is indeed a maximum of Fγ , and in particular that Fγ (t) < Fγ (1/2) for every t ∈ [0, 21 ) ∪ ( 12 , 1]. Hence, for t ∈ [0, 1] and γ 6= 1, we have max max Fγ (t) =

s≤γ max{||a||, ||b||}, the following inequality holds: hM (T a, T b) ≤ ||T ||hM (a, b). That is ||T a M T b|| ≤ ||T || ⊗ ||a M b|| or

s

||T a||2 − 2T a·T b + ||T b||2 ≤ ||T || ⊗ 1 − s22 T a·T b + s14 ||T a||2 ||T b||2

s

||a||2 − 2a·b + ||b||2 . 1 − s22 a·b + s14 ||a||2 ||b||2

The equality holds if and only if one of the following conditions is satisfied: (i) a = b (ii) T = 0 (iii) ||T a|| = ||a|| and ||T b|| = ||b||. Remark. ˆ The inequality ||T a M T b|| ≤ ||T (a M b)|| does not hold in general. ˆ By letting s → ∞, the following norm inequality can be recaptured

||T a − T b|| ≤ ||T ||||a − b||. Theorem 2.6. Let U, V be real inner product spaces and let T : U → V be a bounded real linear operator with ||T || ≤ 1. For any s > 0, the following identity holds: hM (T a, T b) = ||T ||. ||a||,||b|| 0, we denote the open s-ball of H centered at the origin by Hs = {v ∈ H; ||v|| < s}, 1

where ||v|| = hv, vi 2 . Let us recall the notion of the Schwarz-Pick system due to L.A. Harris.

¨ bius gyrovector spaces and functional analysis Mo

227

Definition 3.1 ([9]. See also [8]). Call any system which assigns a pseudometric to each domain in every normed linear space a Schwarz-Pick system if the following conditions hold: (i) The pseudometric assigned to the open unit disc D in the complex plain is the Poincar´e metric. (ii) If ρ1 and ρ2 are the pseudometrics assigned to domains D1 and D2 respectively, then ρ2 (h(x), h(y)) ≤ ρ1 (x, y) for all holomorphic h : D1 → D2 and all x, y ∈ D1 . Theorem 3.2 (cf.[8, Chapter 2, Section 15]). All Schwarz-Pick systems assign the same metric ρ to the open unit ball H1 of any complex Hilbert space H. Moreover, we know that 1

ρ(u, v) = tanh−1 (1 − σ(u, v)) 2 , where σ(u, v) =

(1 − ||u||2 )(1 − ||v||2 ) . |1 − hu, vi|2

for all u, v ∈ H1 . Next, let us recall some fundamental definitions and results related to the Einstein addition due to Ungar, where the carrier is a complex inner product space. Definition 3.3 ([15]). Let H be a complex inner product space. The abstract complex relativistic velocity addition is given by the equation   1 1 1 γu u ⊕E v = u+ (3.1) v+ 2 hv, uiu γu s 1 + γu 1 + ⟨v,u⟩ 2 s for any u, v ∈ Hs , where γv = r

1 ||v||2 1− 2 s

.

Remark. ˆ Although we use the same notation ⊕E as in the case where the carrier is a real inner product space, there will be no confusion. Because the notations for vectors and spaces are different as a ∈ V and u ∈ H etc. ˆ If H is the one dimensional complex plain C with the standard inner product, then (H1 , ⊕E ) is nothing but the classical Poincar´e disc (D, ⊕).

228

Keiichi Watanabe

Theorem 3.4 ([15]). Let H be a complex inner product space. (Hs , ⊕E ) is a gyrocommutative gyrogroup. Definition 3.5.

We use the notations u E v = u ⊕E (−v) hE (u, v) = tanh−1

||v E u|| . s

for any u, v ∈ Hs . Assume s = 1 for a while, for the sake of simplicity. Let H be a complex inner product space. The fol-

Theorem 3.6 ([15, (3.8)]). lowing identity holds ||u E v||2 =

||u||2 − 2Rehu, vi + ||v||2 + |hu, vi|2 − ||u||2 ||v||2 = 1 − σ(u, v) 1 − 2Rehu, vi + |hu, vi|2

for any u, v ∈ H1 . Therefore, hE = ρ holds. Equation [15, (3.8)] is expressed as γu⊕v

hv, ui = γu γv 1 + s2

using the gamma factors for general s > 0. We give a proof here for the convenience, which is just a straightforward calculation of inner product. Proof. 2

|1 + hv, ui| ||u ⊕E v||2   γu 1 γu 1 v+ hv, ui u, u + v+ hv, ui u = u+ γu 1 + γu γu 1 + γu = ||u||2 +

+

√ 1 − ||u||2 hu, vi +

1+

1 1−||u||2 √ 1 2 1−||u||

p 1 − ||u||2 hv, ui + (1 − ||u||2 )||v||2 + √

+

p

1+

1 1−||u||2 √ 1 2 1−||u||

hu, vi ||u||2 1 √ 1+

2

1 1−||u||2

|hu, vi| 

hv, ui ||u||2 +

1 1+ √

1 1−||u||2

|hu, vi| + 

√

2

1+

1 1−||u||2 √ 1 2 1−||u||

2  | hu, vi |2 ||u||2

¨ bius gyrovector spaces and functional analysis Mo

229

p 1 − ||u||2 hu, vi ||u||2 ||u||2 p p 2 (1 − p 1 − 1 − ||u||2 ) ||u|| 2 2 2 + 1 − ||u||2 hv, ui + (1 − ||u|| )||v|| + |hu, vi| 2 ||u|| p p p 1 − 1 − ||u||2 1 − ||u||2 (1 − 1 − ||u||2 ) 2 2 + ui ||u|| + hv, |hu, vi| 2 2 ||u|| ||u|| !2 p 1 − 1 − ||u||2 + | hu, vi |2 ||u||2 ||u||2

p 1− = ||u|| + 1 − ||u||2 hu, vi + 2

p p 1 − ||u||2 (1 − 1 − ||u||2 ) 2 = ||u|| + hu, vi + hv, ui + (1 − ||u|| )||v|| + 2 |hu, vi| 2 ||u|| p 1 − 2 1 − ||u||2 + (1 − ||u||2 ) + | hu, vi |2 ||u||2 ||u||4 2

2

2

2

= ||u||2 + 2Re hu, vi + ||v||2 + |hu, vi| − ||u||2 ||v||2 . □

As far as concerning the hyperbolic metric hE , the Lipschitz continuity of the restriction of contractive complex linear operators to the open unit ball of a complex Hilbert space is just a specific consequence of a generalization of Schwarz lemma related to holomorphic functions. Corollary 3.7. holds

Let H and K be complex Hilbert spaces. The following inequality

hE (f (u), f (v)) ≤ hE (u, v) for any holomorphic maps f : H1 → K1 and any u, v ∈ H1 . In particular, hE (T u, T v) ≤ hE (u, v) for any complex linear operator T : H → K with ||T || ≤ 1 and any u, v ∈ H1 . In the case that H and K are finite dimensional, the corollary above is nothing but [13, Theorem 8.1.4]. ¿From a viewpoint of theory of holomorphic mappings, the complex Einstein addition ⊕E due to Ungar can be regarded as the best binary operation on the open unit ball of a complex Hilbert space. One can consider possibly various gyro additions on open balls centered at the origin of a complex inner product space. We state some related results of [25] below.

230

Keiichi Watanabe

Definition 3.8. Let (H, h·, ·i) be a complex inner product space. It is wellknown as an elementary fact that one can regard the complex linear space H as a real linear space and define hu, viR = Re hu, vi for any u, v ∈ H, then (H, h·, ·iR ) is a real inner product space. Since hu, uiR = hu, ui, the norm of vectors and the open balls centered at the origin are identical in both spaces. For any s > 0, the Einstein and M¨obius additions are defined on Hs , because Hs is an open s-ball of a real inner product space in this way. They are given by the equations   1 1 1 γu (3.2) u ⊕RE v = u+ v+ 2 hu, viR u γu s 1 + γu 1 + s12 hu, viR 1 = 1 + s12 Rehu, vi (3.3)

u ⊕RM v =

=



1 γu 1 u+ v+ 2 Rehu, viu γu s 1 + γu





1+

2 1 2 u + 1 − s12 ||u||2 s2 hu, viR + s2 ||v|| 1 + s22 hu, viR + s14 ||u||2 ||v||2

1+

2 1 2 u + 1 − s12 ||u||2 s2 Rehu, vi + s2 ||v|| 1 + s22 Rehu, vi + s14 ||u||2 ||v||2






v


v

for any u, v ∈ Hs . Remark. ˆ In [24] and [25], we used the notations ⊕E , ⊕M or ⊕s , instead of the notations ⊕RE and ⊕RM as mentioned above. ˆ If H is the one dimensional complex plain C with the standard inner product, then (H1 , ⊕RM ) is nothing but the classical Poincar´e disc (D, ⊕).

Definition 3.9. We define the notations RE , hRE from ⊕RE (resp. RM , hRM from ⊕RM ) in the same manner that we defined the notations E , hE from ⊕E . The following theorem is directly derived from the results by Ungar and the fact that a complex inner product space is naturally a real inner product space. Although the author gave the proofs of [25, Theorem 2.1, Theorem 5.1] by calculation, they are not necessary. Theorem 3.10.

Let H be a complex inner product space and let s > 0. Then

(1) (Hs , ⊕RE ) and (Hs , ⊕RM ) are gyrocommutative gyrogroups.

¨ bius gyrovector spaces and functional analysis Mo

231

(2) The following identities hold ||u RE v||2 =

||u||2 − 2Rehu, vi + ||v||2 + (Rehu, vi)2 − ||u||2 ||v||2 1 − s22 Rehu, vi + s14 (Rehu, vi)2

||u RM v||2 =

||u||2 − 2Rehu, vi + ||v||2 1 − s22 Rehu, vi + s14 ||u||2 ||v||2

for any u, v ∈ Hs . (3) (Hs , hRE ) and (Hs , hRM ) are metric spaces. If H is a Hilbert space, then (Hs , hRE ) and (Hs , hRM ) are also complete as metric spaces. (4) The following identity holds 2 ⊗ (u ⊕RM v) = (2 ⊗ u) ⊕RE (2 ⊗ v) for any u, v ∈ Hs . The following gyro scalar multiplication by complex numbers can be naturally introduced as an extension of the scalar multiplication by real numbers due to Ungar. Definition 3.11. Let H be a complex inner product space and let s > 0. For any complex number α with its polar form α = |α|eiθ , we define a scalar multiplication ⊗ by the equation   v −1 ||v|| iθ α ⊗ v = e s tanh |α| tanh s ||v|| for any nonzero vector v ∈ Hs and α ⊗ 0 = 0. Theorem 3.12. Let H be a complex inner product space and let s > 0. Then the triplet (Hs , ⊕RM , ⊗) possesses the following properties: (i) Re hgyr[u, v]w1 , gyr[u, v]w2 i = Rehw1 , w2 i (ii) 1 ⊗ v = v (iii) (r1 + r2 ) ⊗ v = r1 ⊗ v ⊕RM r2 ⊗ v (iv) (αβ) ⊗ v = α ⊗ (β ⊗ v) (v) gyr[u, v](r ⊗ w) = r ⊗ gyr[u, v]w (vi) gyr[r1 ⊗ v, r2 ⊗ v] = id. (vii) ||α ⊗ v|| = |α| ⊗ ||v||

232

Keiichi Watanabe

(viii) ||u ⊕RM v|| ≤ ||u|| ⊕ ||v|| for any real numbers r1 , r2 , r ∈ R, complex numbers α, β ∈ C and elements u, v, w, w1 , w2 ∈ Hs . Here, id. is the identity map on Hs . The triplet (Hs , ⊕RE , ⊗) also possesses the similar properties. Remark. In this remark, let (D, ⊕) be the classical Poincar´e disc. Let ⊗ : R×D → D be the multiplication of real numbers due to Ungar. In [2], Dr. Abe showed that it is impossible to extend ⊗ to a map ⊗C : C × D → D so that the extension satisfies the following identities (α + β) ⊗C a = (α ⊗C a) ⊕ (β ⊗C a) (αβ) ⊗C a = α ⊗C (β ⊗C a) for any complex numbers α, β ∈ C and element a ∈ D. As a personal opinion of the author of the present article, this result is significant in the context of relationship of linearity and gyro structure, and it should be published in an appropriate journal. A binary operation on the open balls of a complex inner product space satisfying a set of fundamental properties is uniquely determined to be the M¨obius addition ⊕RM given by equation (3.3). Requirements 3.13. Let H be a complex inner product space and let s > 0. For any binary operation ⊕ : Hs × Hs → Hs , consider the following conditions (R1)–(R3): (R1) u ⊕ v = (R2) u ⊕ v =

u+v 1 + sλ2 ||u||2 1+

whenever v = λu for some λ ∈ C


u + 1 − s12 ||u||2 v 1 + s14 ||u||2 ||v||2

1 2 s2 ||v||




(R3) u ⊕ (v ⊕ w) = (u ⊕ v) ⊕ w Theorem 3.14.

whenever u ⊥ v whenever u ⊥ w and v ⊥ w.

Let H be a complex inner product space and let s > 0. Then

(1) The M¨ obius addition ⊕RM given by equation (3.3) satisfies the requirements (R1)– (R3). (2) Conversely, suppose that a binary operation ⊕ : Hs × Hs → Hs satisfies the requirements (R1)–(R3). Then, ⊕ coincides with the M¨ obius addition ⊕RM given by equation (3.3). Next, we state some results in [24].

¨ bius gyrovector spaces and functional analysis Mo

233

Theorem 3.15. Let H, K be complex inner product spaces, let u, v be elements in H and let T be a bounded linear operator from H into K. If s > max{||u||, ||v||} and ||T || ≤ 1, then the following inequality holds: hRM (T u, T v) ≤ ||T ||hRM (u, v). That is ||T u RM T v|| ≤ ||T || ⊗ ||u RM v|| or

s

||T u||2 − 2RehT u, T vi + ||T v||2 ≤ ||T || ⊗ 1 − s22 RehT u, T vi + s14 ||T u||2 ||T v||2

s

||u||2 − 2Rehu, vi + ||v||2 . 1 − s22 Rehu, vi + s14 ||u||2 ||v||2

The equality holds if and only if one of the following conditions is satisfied: (i) u = v (ii) T = 0 (iii) ||T u|| = ||u|| and ||T v|| = ||v||. Remark. ˆ The inequality ||T u RM T v|| ≤ ||T (u RM v)|| does not hold in general. ˆ By letting s → ∞, the following norm inequality can be recaptured

||T u − T v|| ≤ ||T ||||u − v||. Theorem 3.16. Let H, K be complex inner product spaces and let T : H → K be a bounded linear operator with ||T || ≤ 1. For any s > 0, the following identity holds: sup ||u||,||v|| 0. Definition 3.17. Let H be a complex inner product space. We define a binary operation ⊕M on the open unit ball H1 by the equation u ⊕M v = p

|c| − (1 − ||u||2 )(1 − ||v||2 ) |c|2 − (1 − ||u||2 )2 (1 − ||v||2 )2

·

c (1 + 2hv, ui + ||v||2 )u + (1 − ||u||2 )v · 1p |c| |c|2 − (1 − ||u||2 )2 (1 − ||v||2 )2 2

for any u, v ∈ H1 , where c = (1 + ||u||2 )(1 + ||v||2 ) + 4hu, vi.

234

Keiichi Watanabe

Let H be a complex inner product space. Then the following

Theorem 3.18. identity holds

2 ⊗ (u ⊕M v) = (2 ⊗ u) ⊕E (2 ⊗ v)

(3.4) for any u, v ∈ H1 . Definition 3.19.

For the simplicity of notations, we put

a = ||u||2 ,

b = ||v||2

and

z = hu, vi

for any u, v ∈ H1 . Lemma 3.20.

The following identities hold

2 2⊗u= u, 1+a

γ2⊗u

1+a = 1−a

and

p 1 − 1 − ||w||2 1 ⊗w = w 2 ||w||2

for any u, w ∈ H1 . Lemma 3.21.

The following identity holds 2

||(1 + 2z + b)u + (1 − a)v|| = (a + b)(1 + ab) + 2Rez(1 + a)(1 + b) + 4|z|2 =

o 1n 2 |(1 + a)(1 + b) + 4z| − (1 − a)2 (1 − b)2 4

for any u, v ∈ H1 . Lemma 3.22.

If we put

w=

2 {(1 + 2z + b)u + (1 − a)v} , (1 + a)(1 + b) + 4z

then the following identity holds 2

|(1 + a)(1 + b) + 4z| (1 − ||w||2 ) = (1 − a)2 (1 − b)2 . Proof of Theorem 3.18. calculation shows

By using formula (3.1) and Lemma 3.20, a straightforward

(2 ⊗ u) ⊕E (2 ⊗ v)  γ2⊗u 2⊗u+ (2 ⊗ v) + h2 ⊗ v, 2 ⊗ ui (2 ⊗ u) γ2⊗u 1 + γ2⊗u ) (   1+a 1 2 2 2 2 2 1−a E u + 1+a v+ v, u u 1+a 1+a 1+b 1+b 1+a 1+a 1 + 1−a u 1−a

1 = 1 + h2 ⊗ v, 2 ⊗ ui D

= 1+ =

1 2 2 1+b v, 1+a



1

2 {(1 + 2z + b)u + (1 − a)v} . (1 + a)(1 + b) + 4z

¨ bius gyrovector spaces and functional analysis Mo

235

Therefore, we can obtain p 1 − 1 − ||w||2 1 1 ⊗ {(2 ⊗ u) ⊕E (2 ⊗ v)} = ⊗ w = w 2 2 ||w||2

=

1− 4 |(1+a)(1+b)+4z|2

(1−a)(1−b) |(1+a)(1+b)+4z| 2

||(1 + 2z + b)u + (1 − a)v|| ·

=

2 {(1 + 2z + b)u + (1 − a)v} (1 + a)(1 + b) + 4z

(1 + 2z + b)u + (1 − a)v |(1 + a)(1 + b) + 4z| − (1 − a)(1 − b) |(1 + a)(1 + b) + 4z| · · 2 ||(1 + 2z + b)u + (1 − a)v|| (1 + a)(1 + b) + 4z ||(1 + 2z + b)u + (1 − a)v|| |c| − (1 − ||u||2 )(1 − ||v||2 )

=p

|c|2 − (1 − ||u||2 )2 (1 − ||v||2 )2

·

c (1 + 2 hv, ui + ||v||2 )u + (1 − ||u||2 )v · 1p |c| |c|2 − (1 − ||u||2 )2 (1 − ||v||2 )2 2

= u ⊕M v as desired.

□

Thus, via identity (3.4), the binary operation ⊕M corresponds to the complex Einstein addition ⊕E defined by equation (3.1) due to Ungar. So we might be able to call ⊕M the complex M¨ obius addition. Entire work with details including the following theorem will be published elsewhere. Theorem 3.23.

Let H be a complex inner product space and let s > 0. Then

(1) (Hs , ⊕M ) is a gyrocommutative gyrogroup. (2) (Hs , hM ) is a metric space. If H is a Hilbert space, then (Hs , ⊕M ) is also complete as metric space. (3) The triplet (Hs , ⊕M , ⊗) satisfies the properties of “inner product gyrovector spaces”. Acknowledgements The author would like to thank Dr. Abe for kindly sending me the reference [2]. The author also thanks to the referee for his/her careful reading the original manuscript and giving valuable suggestions. This work was supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University. It means that the author attended the online conference by RIMS with no financial support.

236

Keiichi Watanabe

References [1] Abe, T., Normed gyrolinear spaces: A generalization of normed spaces based on gyrocommutative gyrogroups, Math. Interdiscip. Res. 1 (2016), 143–172. [2] Abe, T., Complex scalar multiplications on the Poincar´e disk, preprint. [3] Abe, T. and Hatori, O., Generalized gyrovector spaces and a Mazur-Ulam theorem, Publ. Math. Debr. 87 (2015), 393–413. [4] Abe, T. and Watanabe, K., Finitely generated gyrovector subspaces and orthogonal gyrodecomposition in the M¨ obius gyrovector space, J. Math. Anal. Appl. 449 (2017), 77–90. [5] Frenkel, P.E., On endomorphisms of the Einstein gyrogroup in arbitrary dimension, J. Math. Phys. 57 (2016), 032301, 3 pp. [6] Ferreira, M. and Ren, G., M¨ obius gyrogroups: A Clifford algebra approach, J. Algebra 328 (2011), 230–253. [7] Ferreira, M. and Suksumran, T., Orthogonal gyrodecompositions of real inner product gyrogroups, Symmetry 2020, 12(6), 941. [8] Goebel, K. and Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984. [9] Harris, L.A., Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, Advances in Holomorphy, North Holland, Amsterdam, 1979, pp.345–406. [10] Hatori, O., Examples and applications of generalized gyrovector spaces, Results Math. 71 (2017), 295–317. [11] Honma, T. and Hatori, O., A gyrogeometric mean in the Einstein gyrogroup, Symmetry 2020, 12(8), 1333. [12] Moln´ ar, L. and Virosztek, D., On algebraic endomorphisms of the Einstein gyrogroup, J. Math. Phys. 56 (2015), 082302, 5 pp. [13] Rudin, W., Function Theory in the Unit Ball of Cn , Springer-Verlag, New York, 1980. [14] Suksumran, T. and Wiboonton, K., Einstein gyrogroup as a B-loop, Rep. Math. Phys. 76 (2015), 63–74. [15] Ungar, A.A., The abstract complex Lorentz transformation group with real metric. I. Special relativity formalism to deal with the holomorphic automorphism group of the unit ball in any complex Hilbert space, J. Math. Phys. 35 (1994), 1408–1426. [16] Ungar, A.A., Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity, World Scientific Publishing Co. Pte. Ltd., Singapore, 2008. [17] Watanabe, K., A confirmation by hand calculation that the M¨ obius ball is a gyrovector space, Nihonkai Math. J. 27 (2016), 99–115. [18] Watanabe, K., Orthogonal gyroexpansion in M¨ obius gyrovector spaces, J. Funct. Spaces 2017, Article ID 1518254, 13 pp. [19] Watanabe, K., A Cauchy type inequality for M¨ obius operations, J. Inequal. Appl. 2018, Paper No. 97, 9 pp. [20] Watanabe, K., A Cauchy-Bunyakovsky-Schwarz type inequality related to the M¨ obius addition, J. Math. Inequal. 12 (2018), 989–996. [21] Watanabe, K., Cauchy-Bunyakovsky-Schwarz type inequalities related to M¨ obius operations, J. Inequal. Appl. 2019, Paper No. 179, 19 pp. [22] Watanabe, K., Continuous quasi gyrolinear functionals on M¨ obius gyrovector spaces, J. Funct. Spaces 2020, Article ID 1950727, 14 pp. [23] Watanabe, K., On quasi gyrolinear maps between M¨ obius gyrovector spaces induced from finite matrices, Symmetry 2021, 13(1), 76.

¨ bius gyrovector spaces and functional analysis Mo

237

[24] Watanabe, K., On Lipschitz continuity with respect to the Poincar´e metric of linear contractions between M¨ obius gyrovector spaces, J. Inequal. Appl. 2021, Paper No. 166, 15 pp. [25] Watanabe, K., On a notion of complex M¨ obius gyrovector spaces, Nihonkai Math. J. 32 (2021), 111–131. [26] Watanabe, K., On inner product gyrovector spaces, preprint. [27] Watanabe, K., Quasi gyrolinear mappings between inner product gyrovector spaces, preprint. [28] Watanabe, K., On the Lipschitz continuity of contractive linear operators with respect to the M¨ obius metric (Japanese), manuscript for Proceedings of Conference on Function Algebras 2021.

RIMS Kˆ okyˆ uroku Bessatsu B93 (2023), 239–250

On isometries of Wasserstein spaces By

´r∗, Tam´as Titkos∗∗ and D´aniel Virosztek∗∗∗ Gy¨orgy P´al Gehe

Abstract It is known that if p ≥ 1, then the isometry group of the metric space (X, ϱ) embeds into the isometry group of the Wasserstein space Wp (X, ϱ). Those isometries that belong to the image of this embedding are called trivial. In many concrete cases, all isometries are trivial, however, this is not always the case. The aim of this survey paper is to provide a structured overview of recent results concerning trivial and different types of nontrivial isometries.

Contents § 1. Introduction § 2. Trivial isometries § 3. Nontrivial isometries § 3.1. Shape-preserving isometries § 3.2. Exotic isometries § 3.3. Mass-splitting isometries Received March 28, 2022. Revised December 14, 2022. 2020 Mathematics Subject Classification(s): Primary: 54E40; 46E27 Secondary: 60B05 Key Words: Wasserstein space, shape-preserving isometries, exotic isometries. Geh´ er was supported by the Leverhulme Trust Early Career Fellowship (ECF-2018-125), and also by the Hungarian National Research, Development and Innovation Office (Grant no. K115383); Titkos was supported by the Hungarian National Research, Development and Innovation Office NKFIH (grant no. K134944); Virosztek was supported by the Momentum Program of the Hungarian Academy of Sciences (grant no. LP2021-15/2021) and partially supported by the Hungarian National Research, Development and Innovation Office – NKFIH (grant no. K124152). ∗ Department of Mathematics and Statistics,University of Reading,United Kingdom e-mail: [email protected] ∗∗ Alfr´ ed R´ enyi Institute of Mathematics, Hungary. BBS University of Applied Sciences, Alkotm´ any u. 9., Budapest H-1054, Hungary. e-mail: [email protected] ∗∗∗ Alfr´ ed R´ enyi Institute of Mathematics, Hungary. e-mail: [email protected]

© 2023 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

240

´r and T. Titkos and D. Virosztek Gy.P. Gehe

References

§ 1.

Introduction

Let D be a subset of all Borel probability measures P(X) over a complete separable metric space (X, ϱ). In our considerations, (X, ϱ) will be a very concrete object, rather than a general complete separable metric space: the unit interval, the real line or in general a Euclidean space; a high dimensional sphere, a high-dimensional torus, or a nice manifold; a countable discrete metric space, or in general a graph metric space. We will refer to these spaces as underlying spaces. Similarly, D will be a reasonable subset of P(X), rather than an arbitrary one. Having the set D at hand, one can endow it with a metric in quite numerous ways depending on the nature of the problem under consideration. For example, to metrize the weak convergence of measures, one can use the L´evy-Prokhorov metric (see e.g. Theorem 6.8 in [3]), or the p-Wasserstein metric (see Theorem 6.9 in [25]). In this paper, we will consider the latter one, assuming that D is the collection of all probability measures with finite p-th moment. (See the precise definitions later.) When working in a metric setting, a natural question arises: can we describe the structure of distance preserving maps? 1 In recent years, there has been a lot of activity concerning this question, see e.g. [2, 5, 6, 9–15, 18, 19, 21, 24, 26]. It turned out that isometries of these spaces of measures are strongly related to self-maps of the underlying space X. Concerning the Kolmogorov-Smirnov distance, Dolinar and Moln´ar showed in [6] that there is a one-to-one correspondence between all isometries of P(R) and all homeomorphisms of the real line. Concerning the L´evy-Prokhorov metric, the first and the second author showed in [10] that P(X) endowed with the L´evy-Prokhorov distance is more rigid: assuming that X is a real separable Banach space, a self-map of X induces an isometry on P(X) if and only if it is itself an isometry. For a more detailed overview of similar results we refer the reader to the survey [26], where the case of the Kolmogorov-Smirnov [6] and the Kuiper distances [9] are also discussed. Last but not least, we mention a paper of Dolinar, Kuzma and Mitrovi´c. In [5] they described the structure of all w∗ -continuous isometries of P(R) with respect to the total variation distance: all these isometries are induced by continuous bijections. However, if one considers a smaller domain D ⊆ P(R), then the structure of isometries can be 1 As

Hermann Weyl said in [27]: “ Whenever you have to do with a structure–endowed entity Σ try to determine its group of automorphisms, the group of those element–wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way. ”

241

On isometries of Wasserstein spaces

much more involved. The aim of this paper is to show that the case of the p-Wasserstein metric is similarly colourful: the structure of isometries depends in an interesting way both on some characteristics of the underlying space X and on the value of p.

§ 2.

Trivial isometries

This section aims to introduce all the necessary notions and to take a closer look at a very important class of isometries, the so-called trivial isometries. Besides that these are the most natural isometries in this setting, it seems that it is a very rare phenomenon that a p-Wasserstein space possesses nontrivial isometries (only a few such examples are known). After the precise definitions, we will give a short overview of all known cases where the isometry group of a p-Wasserstein space consists of only trivial isometries. The cases where nontrivial isometries occur will be discussed in Section 3. Let us recall first what a p-Wasserstein space is. Let p ≥ 1 be a fixed real number, and let (X, ϱ) be a complete and separable metric space. We denote by P(X) the set of all Borel probability measures on X and by ∆1 (X) the set of all Dirac measures: ∆1 (X) = {δx | x ∈ X}. A probability measure π on X × X is called a coupling for µ, ν ∈ P(X) if the marginals of π are µ and ν, that is, π (A × X) = µ(A) and π (X × B) = ν(B) for all Borel sets A, B ⊆ X. The set of all couplings is denoted by C(µ, ν). By means of couplings, we can define the p-Wasserstein distance and the corresponding p-Wasserstein space as follows: the p-Wasserstein space Wp (X, ϱ) is the set of all µ ∈ P(X) that satisfy Z (2.1) ϱ(x, x ˆ)p dµ(x) < ∞ X

for some (and hence all) x ˆ ∈ X, endowed with the p-Wasserstein distance  (2.2)

dp (µ, ν) := 

ZZ inf

1/p ϱ(x, y)p dπ(x, y)

.

π∈C(µ,ν) X×X

It is known (see e.g. Theorem 1.5 in [1] with c = ϱp ) that the infimum in (2.2) is in fact a minimum in this setting. Those couplings that minimize (2.2) are called optimal transport plans. As the terminology suggests, all notions introduced above are strongly related to the theory of optimal transportation. Indeed, for given sets A and B the quantity π(A, B) is the weight of mass that is transported from A to B as µ is RR transported to ν along the transport plan π, while the quantity ϱ(x, y)p dπ(x, y) is X×X

the cost of the transport assuming that the cost of moving one unit of mass from x to y is ϱ(x, y)p . In other words, the p-Wasserstein distance measures the minimal effort required to morph one probability mass into another, when the cost of transporting

242

´r and T. Titkos and D. Virosztek Gy.P. Gehe

mass is the p-th power of the distance. Due to this connection and many nice geometric features of the p-Wasserstein distance, p-Wasserstein spaces have strong connections to many flourishing areas in pure and applied sciences. We mention here only a few papers [4,8,16,17,20,22], for a comprehensive overview and for more references we refer the reader to Ambrosio’s, Figalli’s, Santambrogio’s, and Villani’s textbooks [1,7,23,25]. After a small detour, we continue by introducing the key notions that are needed in the sequel. A self-map ψ : X → X is called an isometry if it is surjective and preserves the distance, that is, ϱ(ψ(x), ψ(y)) = ϱ(x, y) for all x, y ∈ X. The symbol Isom(X) will stand for the group of all isometries. Similarly, the group of all distance preserving bijective self-maps of Wp (X, ϱ) will be denoted by Isom(Wp (X, ϱ)). For an isometry ψ ∈ Isom(X) the induced push-forward map ψ# : P(X) → P(X)
is defined by ψ# (µ) (A) = µ(ψ −1 [A]) for all Borel set A ⊆ X and µ ∈ P(X)), where ψ −1 [A] = {x ∈ X | ψ(x) ∈ A}. We call ψ# (µ) the push-forward of µ with ψ. A very important feature of p-Wasserstein spaces is that Wp (X, ϱ) contains an isometric copy of (X, ϱ). Indeed, since C(δx , δy ) has only one element (the Dirac measure δ(x,y) ), we have that  ZZ dp (δx , δy ) =

1/p ϱ(u, v)p dδ(x,y) (u, v) = ϱ(x, y),

X×X

and thus the embedding

(2.3)

ι : X → Wp (X, ϱ),

ι(x) := δx

is distance preserving. Furthermore, the set of finitely supported measures (in other words, the collection of all finite convex combinations of Dirac measures) is dense in Wp (X, ϱ) (see e.g. Example 6.3 and Theorem 6.18 in [25]). Another very important observation is that isometries of the underlying space appear in Isom(Wp (X)) by means of a natural group homomorphism

(2.4)

# : Isom(X) → Isom (Wp (X, ϱ)) ,

ψ 7→ ψ# .

To see that ψ# ∈ Isom(Wp (X, ϱ)) whenever ψ ∈ Isom(X), let us introduce the map b b y) := (ψ(x), ψ(y)). Then observe that if π ∈ C(µ, ν) is a ψ : X × X → X × X as ψ(x, coupling, then ψb# π belongs to C(ψ# (µ), ψ# (ν)). Now using that ψ −1 is an isometry as

On isometries of Wasserstein spaces

243

well, change of variables (ψ(x) := u and ψ(y) := v) gives ZZ dpp (µ, ν)

=

ϱp (x, y) dπ(x, y)

inf π∈C(µ,ν) X×X

ZZ

=

inf π∈C(µ,ν) X×X

ZZ

=

inf π∈C(µ,ν) X×X

≥

ϱp (ψ −1 (u), ψ −1 (v)) dψb# π(u, v) ϱp (u, v) dψb# π(u, v) ZZ

inf

ϱp (u, v) dη(u, v)

η∈C(ψ# (µ),ψ# (ν)) X×X

= dpp (ψ# (µ), ψ# (ν)). The reverse inequality can be proved along the same lines. Indeed, since ψ −1 is an isometry, the above inequality gives
dpp (µ′ , ν ′ ) ≥ dpp (ψ −1 )# (µ′ ), (ψ −1 )# (ν ′ ) for all µ′ , ν ′ ∈ Wp (X, ϱ). Using that ψ# is surjective and that (ψ −1 )# = (ψ# )−1 , substituting µ′ := ψ# (µ) and ν ′ := ψ# (ν) gives dpp (ψ# (µ), ψ# (ν)) ≥ dpp (µ, ν). As we have just seen, # embeds Isom(X) into Isom(Wp (X, ϱ)). We call an isometry of Wp (X, ϱ) a trivial isometry if it belongs to the image of this embedding. If the embedding is surjective (i.e., if all isometries are trivial), we call Wp (X, ϱ) isometrically rigid. We close this section by collecting all the known isometrically rigid p-Wasserstein spaces.2 Bertrand and Kloeckner showed in [2] that if (X, ϱ) is a negatively curved geodesically complete Hadamard space then W2 (X, ϱ) is isometrically rigid. In [24], Santos-Rodr´ıguez proved that rigidity holds for 2-Wasserstein spaces over closed Riemannian manifolds with strictly positive sectional curvature. Furthermore, for compact rank one symmetric spaces (CROSSes), he was able to prove isometric rigidity not only for the p = 2 case, but for general p-Wasserstein spaces with 1 < p < ∞. In [15] we showed that p-Wasserstein spheres Wp (Sn , ∢) and p-Wasserstein tori Wp (Tn , r) are rigid for all n ≥ 1 and p ≥ 1. Here Sn denotes the unit sphere of Rn+1 and ∢ denotes the geodesic distance ∢(x, y) = arccoshx, yi (x, y ∈ Sn ), while Tn is the n-dimensional
n torus, that is, the set Rn /Zn ' [−1/2, 1/2)/∼ (the equivalence relation ∼ denotes 2 We

remark that one can define the p-Wasserstein space for∫∫0 < p < 1 as well. In that case the pWasserstein distance is defined as dp (µ, ν) := inf π∈C(µ,ν) ϱ(x, y)p dπ(x, y). In [14] we proved X×X

that if 0 < p < 1 then the p-Wasserstein space Wp (X, ϱ) is isometrically rigid no matter what the underlying space (X, ϱ) is.

244

´r and T. Titkos and D. Virosztek Gy.P. Gehe

the identification of −1/2 and 1/2) equipped with the usual metric n X

r (x, y) =

! 12 |(xk − yk )mod 1 |

2

,

k=1

where x = (x1 , . . . , xn ) ∈ Rn and y = (y1 , . . . , yn ) ∈ Rn . In [18] we proved isometric rigidity in the discrete setting, namely when (X, ϱ) is a graph metric space. This class contains several important metric spaces: any countable set with the discrete metric (see also [11]), the set of natural numbers and the set of integers with the usual | · |-distance; d-dimensional lattices endowed with the l1 -metric, finite strings with the Hamming distance (the classical 1-Wasserstein distance with respect to the Hamming metric is called Ornstein ’s distance), just to mention a few. The last special case what we mention is when (X, ϱ) ∈ {([0, 1], |·|), (R, |·|), (E, k·k)}, where E is a real Hilbert space space with 2 ≤ dim E ≤ ∞. Bullets in the table below indicate that the corresponding p-Wasserstein space is isometrically rigid. For more details we refer the reader to [12] and [14]. Wp ([0, 1], | · |) p=1 p=2 p 6= 1, 2

• •

Wp (R, | · |) •

Wp (E, k · k) (dimE ≥ 2) •

•

•

We will see in the next section that the missing cases Wp ([0, 1], | · |), W2 (R, | · |) and W2 (E, k · k) with dim E ≥ 2 are basically all the known examples where various types of nontrivial isometries appear.

§ 3.

Nontrivial isometries

Now we turn to those cases where nontrivial isometries exist. We will discuss three types of nontrivial isometries (see the precise definitions later): 1) shape-preserving nontrivial isometries, 2) exotic isometries, 3) mass-splitting isometries. shape-preserving isometries

exotic isometries

trivial isometries

mass-splitting isometries

As we will see, the set of trivial isometries is a proper subset of shape-preserving isometries if p = 2 and the underlying space is a Euclidean space. Isometries that do not

On isometries of Wasserstein spaces

245

preserve shape are termed as exotic isometries. We will see that W2 (R, | · |) possesses such isometries, however, even these isometries of W2 (R, | · |) leave the set of Dirac masses invariant. Those exotic isometries which do not leave the set of Dirac measures invariant are called mass-splitting.

§ 3.1.

Shape-preserving isometries

Let Wp (X, ϱ) be a p-Wasserstein space. An isometry Φ : Wp (X, ϱ) → Wp (X, ϱ) is called shape-preserving if for all measures µ there exists an isometry ψµ of X (depending on µ) such that Φ(µ) is the push-forward of µ with respect to ψµ . Of course, every trivial isometry is shape-preserving. In that case, there exists a ψ ∈ Isom(X) such that ψµ = ψ for all µ ∈ Wp (X, ϱ). Kloeckner in [19] proved that W2 (Rn , k · k) has shape-preserving isometries which are nontrivial. In [14] we extended his result to the infinite dimensional case. Combining these two results, we have the following characterization of isometries. Let E be a separable real Hilbert space of dimension at least two, and let Φ be an isometry of W2 (E, k · k). Then Φ can be written as the following composition:   µ (µ ∈ W2 (E)), (3.1) Φ(µ) = ψ ◦ tm(µ) ◦ R ◦ t−1 m(µ) #

where ψ : E → E is an affine isometry, R : E → E is a linear isometry, and tm(µ) : E → E is the translation on E by the barycenter m(µ) of µ. Recall that the barycenter of µ is the point m(µ) ∈ E such that Z (3.2) hm(µ), zi = hx, zidµ(x) E

holds for all z ∈ E. We will see in the next subsection that the assumption dimE ≥ 2 is essential.

246

´r and T. Titkos and D. Virosztek Gy.P. Gehe

§ 3.2.

Exotic isometries

All isometries that we have seen until this point were related to isometries of the underlying space. Now we turn to the case when such a connection does not exist, and an isometry can distort the shape of measures. Following Kloeckner’s terminology we call these isometries exotic, rather than nonshape-preserving. All known exotic isometries are related to two concrete p-Wasserstein space: W2 (R, | · |) and W1 ([0, 1], | · |). In case of (X, ϱ) = (R, | · |), the cumulative distribution function of a measure µ ∈ W2 (R, | · |) is defined as (3.3)

Fµ : R → [0, 1],

x 7→ Fµ (x) := µ ((−∞, x]) .

The quantile function of µ (or the right-continuous generalized inverse of Fµ ) is denoted by Fµ−1 and is defined as Fµ−1 : (0, 1) → R, y 7→ Fµ−1 (y) := sup {x ∈ R | Fµ (x) ≤ y} .

In case of X, ϱ = [0, 1], | · | we shall handle the cumulative distribution and the quantile functions (with the convention sup{∅} := 0) of a µ ∈ P([0, 1]) as [0, 1] → [0, 1] functions. The quantile function in this case is defined by right-continuity at 0 and it takes the value 1 at 1. A very important feature of p-Wasserstein spaces over the interval and the real line is that they embed isometrically into the corresponding Lp (0, 1) space by means of the map µ 7→ F −1 (µ). In particular, d2 (µ, ν) = kFµ−1 − Fν−1 k2 for all µ, ν ∈ W2 (R, | · |) and d1 (µ, ν) = kFµ−1 − Fν−1 k1 for all µ, ν ∈ W1 ([0, 1], | · |).

(3.4)

First we sketch the very surprising result of Kloeckner on the isometry group of the quadratic Wasserstein space over the real line. For the details we refer the reader to [19]. Let us introduce the notations (3.5)

∆1 (R) := {δx | x ∈ R},

∆2 (R) := {λδx + (1 − λ)δy | x, y ∈ R, λ ∈ [0, 1]}.

One can prove that if Φ is an isometry of W2 (R, | · |), then there exists an isometry ψ ∈ Isom(R) such that for all x ∈ R: Φ(δx ) = δψ(x) . Furthermore, Φ maps the set ∆2 (R) onto itself. The next task is to understand what happens with measures supported on two points. Consider the following parametrization of ∆′2 (R) := ∆2 (R) \ ∆1 (R): any such measure µ can be written as (3.6)

µ(m, σ, r) :=

er e−r r −r . δ + δ m−σe e−r + er e−r + er m+σe

In probabilistic terms, if µ is the law of a random variable then m is its expected value and σ 2 is its variance. Since   (3.7) d22 µ(m1 , σ1 , r1 ), µ(m2 , σ2 , r2 ) = (m1 − m2 )2 + σ12 + σ22 + 2σ1 σ2 e|r1 −r2 | ,

247

On isometries of Wasserstein spaces

it is obvious that for any ψ ∈ Isom(R), the map Φψ : µ(m, σ, r) 7→ µ(m, σ, ψ(r)) is distance preserving on ∆′2 (R), and is not a restriction of a shape-preserving isometry of W2 (R, | · |), unless ψ(r) = −r. Indeed, a push-forward isometry can modify the support, but it does not modify the weights of a µ ∈ ∆′2 (R). But for the map Φψ defined above, µ and Φψ (µ) typically have different weights at their supporting points.

-

µ

- -- - -