Banach-Space Operators On C*-Probability Spaces Generated by Multi Semicircular Elements 1032199016, 9781032199016

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Banach-Space Operators On C ∗-Probability Spaces Generated by Multi Semicircular Elements

Chapman & Hall/CRC Monographs and Research Notes in Mathematics Banach-Space Operators On C *-Probability Spaces Generated by Multi Semicircular Elements Series Editors: John A. Burns, Thomas J. Tucker, Miklos Bona, Michael Ruzhansky About the Series This series is designed to capture new developments and summarize what is known over the entire field of mathematics, both pure and applied. It will include a broad range of monographs and research notes on current and developing topics that will appeal to academics, graduate students, and practitioners. Interdisciplinary books that appeal to the mathematical community, engineers, physicists, and computer scientists are encouraged. This series will maintain the highest editorial standards, publishing well-developed monographs as well as research notes on new topics that are final, but not yet refined into a formal monograph. The notes are meant to be a rapid means of publication for current material where the style of exposition reflects a developing topic. The Center and Focus Problem Algebraic Solutions and Hypotheses M.N. Popa & V.V. Pricop Abstract Calculus A Categorical Approach Francisco Javier Garcia-Pacheco Noncommutative Polynomial Algebras of Solvable Type and Their Modules Basic Constructive-Computational Theory and Methods Huishi Li Fixed Point Results in W-Distance Spaces Vladimir Rakoˇcevi´c Analysis of a Model for Epilepsy Application of a Max-Type Difference Equation to Mesial Temporal Lobe Epilepsy Candace M. Kent, David M. Chan Double Sequence Spaces and Four-Dimensional Matrices Feyzi Ba¸sar, Medine Ye¸silkayagil Sava¸scı Banach-Space Operators On C *-Probability Spaces Generated by Multi Semicircular Elements Ilwoo Cho For more information about this series please visit: https://www.crcpress.com/Chapman–HallCRC-Monographs-and-Research-Notesin-Mathematics/book-series/CRCMONRESNOT

Banach-Space Operators On C ∗-Probability Spaces Generated by Multi Semicircular Elements

Ilwoo Cho St. Ambrose University, USA

First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2023 Ilwoo Cho CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Cho, Ilwoo, author. Title: Banach-space operators on C ∗ -probability spaces generated by multi semicircular elements / authored by: Ilwoo Cho. Description: First edition. | Boca Raton : C&H/CRC Press, 2022. | Series: Chapman & Hall/CRC monographs and research notes in mathematics | Includes bibliographical references and index. Identifiers: LCCN 2021061428 (print) | LCCN 2021061429 (ebook) | ISBN 9781032199016 (hardback) | ISBN 9781032204185 (paperback) | ISBN 9781003263487 (ebook) Subjects: LCSH: Banach spaces. | Free probability theory. | Operator theory. Classification: LCC QA322.2 .C46 2022 (print) | LCC QA322.2 (ebook) | DDC 515/.732–dc23/eng20220412 LC record available at https://lccn.loc.gov/2021061428 LC ebook record available at https://lccn.loc.gov/2021061429 ISBN: 9781032199016 (hbk) ISBN: 9781032204185 (pbk) ISBN: 9781003263487 (ebk) DOI: 10.1201/9781003263487 Typeset in Minion by codeMantra

Contents AUTHOR

ix

Chapter 1 ■ Introduction

1

1.1

MOTIVATION AND BACKGROUND

2

1.2

OVERVIEW

3

Chapter 2 ■ Preliminaries

5

2.1

FREE PROBABILITY

5

2.2

SEMICIRCULAR ELEMENTS

6

2.3

GROUP DYNAMICAL SYSTEMS AND CROSSED PRODUCT ALGEBRAS

7

Chapter 3 ■ Joint Free Distributions of Multi Semicircular Elements

9

Chapter 4 ■ A C ∗ -Probability Space of |Z|-Many Semicircular Elements 25

Chapter 5 ■ C ∗ -Probability Spaces (XN , φN )

37

5.1

FREE-DISTRIBUTIONAL DATA ON XN

40

5.2

RESTRICTED ACTIONS αN OF λ ON XN

41

5.3

ACTION OF ΛN ON XN

47

5.4

FREE-HOMOMORPHIC RELATIONS ON {XN }N ∈N∞

>1

54

v

vi ■ Contents

Chapter 6 ■ Adjointable Banach-Space Operators Acting on XN 57 6.1

PROJECTIONS OF ΛN

58

6.2

PARTIAL ISOMETRIES ON XN

70

6.3

FREE-DISTRIBUTIONAL DATA AFFECTED BY PARTIAL ISOMETRIES

76

PARTIAL ISOMETRIES OF ΛN FOR N < ∞

83

6.4

Chapter 7 ■ Free-Probabilistic Information on XN Affected by Partial Isometries 91

Chapter 8 ■ Application I. Shift Operators Acting on X∞

Chapter 9 ■ Application II. The Circular Law

97

105

9.1

CIRCULAR ELEMENTS

105

9.2

DEFORMED CIRCULAR LAWS ON XN BY PARTIAL ISOMETRIES OF λN

111

Chapter 10 ■ Application III. Free Poisson Distributions

119

10.1

FREE POISSON ELEMENTS

119

10.2

DEFORMED FREE POISSON DISTRIBUTIONS ON XN

123

Chapter 11 ■ Examples

129

11.1

PARTIAL ISOMETRIES OF λ5 ⊂ Λ5 ON X5

129

11.2

PARTIAL ISOMETRIES OF λ∞ ⊂ Λ∞ ON X∞

132

Chapter 12 ■ The Group-Dynamical System Γ

139

12.1

DYNAMICS ON Xφ

140

12.2

THE Γ-SEMICIRCULAR ∗-PROBABILITY SPACE X

141

12.3

DISCUSSION

144

Contents ■ vii

Chapter 13 ■ On The Γ-Semicircular ∗-Probability Space X

145

13.1

GENERATING FREE RANDOM VARIABLES OF X

145

13.2

FREE-DISTRIBUTIONAL DATA ON X

149

13.3

MORE ABOUT FREE-DISTRIBUTIONAL DATA ON X

155

Chapter 14 ■ Operator-Theoretic Properties on X

157

14.1

OPERATOR-THEORETIC PROPERTIES ON G

157

14.2

OPERATOR-THEORETIC PROPERTIES OF FREE REDUCED WORDS OF X IN G

162

14.3

CERTAIN OPERATOR SUMS OF X

171

14.4

FREE DISTRIBUTIONS OF SOME SELF-ADJOINT OPERATORS

180

Chapter 15 ■ Free Probability on XN = (XN ⊗C ΛN , τN )

185

15.1

BANACH ∗-PROBABILITY SPACES XN

186

15.2

FREE-DISTRIBUTIONAL DATA ON XN

189

15.3

FREE-HOMOMORPHIC RELATIONS ON {XN }N ∈N∞

>1

Chapter 16 ■ Operator-Theoretic Properties on XN 16.1

OPERATOR-THEORETIC PROPERTIES ON ΛN

16.2

OPERATOR-THEORETIC PROPERTIES OF XJS

16.3

192

195 195

⊗T

IN XN

201

OPERATOR-THEORETIC PROPERTIES OF W ⊗ T IN XN

205

Chapter 17 ■ Summary

211

Bibliography

213

Index

217

Author Ilwoo Cho is a Professor at St. Ambrose University, Iowa. Dr Cho earned his PhD in Mathematics from University of Iowa in 2005 and his master’s degree from Sungkyunkwan University in 1999. Dr. Cho’s research interests include free probability, operator algebra and theory, combinatorics, and groupoid dynamical systems.

ix

CHAPTER

1

Introduction

HE MAIN PURPOSES OF THIS MONOGRAPH are (i) to consider the C ∗ -probability space X = (X, φ) generated by the free semicircular family X = {xj }j∈Z , and its free-probabilistic substructures, XN = (XN , φN ) generated by free semicircular sub-families N XN = {xj }j=1 , for

T

def

N ∈ N∞ >1 = (N \ {1}) ∪ {∞},  (ii) to construct suitable ∗-isomorphisms λ = β k k∈Z acting on X,  and their restrictions λN = β k |XN k∈Z , which are ∗-homomorphisms acting on XN , and the adjointable Banach-space operators generated by λ and λN , respectively, for N ∈ N∞ >1 , (iii) to study certain types of adjointable Banach-space operators of (ii), and to consider how they deform the free probability on X, and those on XN , (iv) to characterize the deformations on the C ∗ -probability spaces from the Banach-space operators of (iii), and (v) to study dynamics induced by our Banachspace operators. In particular, we are interested in certain types of “adjointable” Banach-space operators, induced by λN acting on XN , especially, where they are projections or partial isometries. Different from the Banachspace operators induced by λ acting on X, such operators from λN distort the free probability on XN , in general. We characterize such distortions on XN occurred by projections, and partial isometries.

DOI: 10.1201/9781003263487-1

1

2 ■ Banach-Space Operators On C ∗ -Probability Spaces

1.1

MOTIVATION AND BACKGROUND

The study of semicircular elements is one of the major topics (e.g., [20,21,29,30]), not only in both commutative function theory and noncommutative free probability theory but also in various applied fields, including quantum statistical physics (e.g., [5–8,10,11,12]). The semicircular law, which is the free distributions of semicircular elements, is well characterized analytically, and combinatorially, in classical function theory and in free probability theory (e.g., [1,17,18,21,28–30]). In particular, it is playing a key role in freeprobabilistic operator algebra theory (and hence, in quantum physics) by the (free) central limit theorem(s) (e.g., see [2,17,19,28–30]), i.e., it becomes a noncommutative analytic analogue of the classical Gaussian (or the normal) distribution (in commutative function theory). From combinatorial approaches (e.g., [17,22,23]), the semicircular ∞ law is universally characterized by the Catalan numbers {ck }k=1 , where 1 cn = n+1



2n n



 =

1 n+1



(2n)! n!(2n − n)!

 =

(2n)! , n!(n + 1)!

for all n ∈ N0 = N ∪ {0}. i.e., the semicircular law is characterized by the free-moment sequence, ∞

(ωn cn )n=1 = (0, c1 , 0, c2 , 0, c3 , . . .) , where

(1.1.1) ωn

( 1 = 0

def

if n is even if n is odd,

and ck are the Catalan numbers, for all k ∈ N. From the analysis on p-adic number fields Qp (e.g., [26,27]), one can construct semicircular elements (e.g., [5,12]). By generalizing the constructions of [5,12], semicircular elements are constructed whenever there are |Z|-many orthogonal projections in a C ∗ -algebra (e.g., [6–8,10] and [11]), different from earlier works (e.g., [20,25,29,30]). In this new approach, the semicircular elements are understood as Banach-space operators acting on a given C ∗ -algebra, by regarding the C ∗ -algebra as a Banach space equipped with its C ∗ -norm (e.g., [13,14]).

Introduction ■ 3

Independently, the joint free distributions of mutually free, multi semicircular elements were re-characterized in [9] (See Section 3 below) combinatorially and analytically. Especially, there, the analytic characterization provided natural estimations, and asymptotic estimations of such joint free distributions. i.e., we have a tool to verify how our adjointable Banach-space operators deform the original free-distributional data on {X} ∪ {XN }N ∈N∞ . >1

1.2

OVERVIEW

This monograph consists of two parts. The first part is covered from Section 2 to Section 11. The second part is considered from Sections 12–18. In the first part, we study how certain Banach-space operators act on mutually free, multi semicircular elements. Deformed freedistributional data are characterized under actions of the operators. In the second part, motivated by the main results of the first part, we consider dynamics determined by our Banach-space operators. In Section 2, we introduce concepts and terminology used in text briefly. A combinatorial formula of joint free distributions of mutually free, multi semicircular elements is re-characterized in Section 3. In Section 4, the free probability on the C ∗ -probability space X generated by the free semicircular family {xj }j∈Z is considered; and  certain free-isomorphisms λ = β j j∈Z (which are the ∗-isomorphisms preserving the free probability) on X are studied. In Section 5, we show that C ∗ -probability spaces generated by mutually free, N -many semicircular elements, are free-isomorphic to a free-probabilistic substructure XN of X (under additional conditions) for N ∈ N∞ >1 , and  l N consider how the restrictions λN = βN l=−N of the free-isomorphisms λ act on XN . In Section 6, by fixing N ∈ N∞ >1 , we concentrate on studying how the restrictions of λN deform the free probability on XN , by acting certain adjointable Banach-space operators induced by λN . To do that, we focus on the operator-theoretic properties of the ∗-homomorphisms l βN , by regarding them as Banach-space operators acting on XN . It is shown that they are partial isometries on XN . By using the partial-isometry-property of them, we study how they deform the free-distributional data on XN in Section 7, in detail.

4 ■ Banach-Space Operators On C ∗ -Probability Spaces

Section 8 is devoted to study a special case where N = ∞ in N∞ >1 , and the main results of Section 7 are applied there. Some additional interesting results are obtained. In Sections 9 and 10, we study how our partial isometries deform the circular law, and free Poisson distributions on XN . Certain examples based on the main results of Sections 5–10 are observed in Section 11. Based on the main results of Section 4–11, we consider suitable dynamics on X, and those on XN , determined by our Banachspace operators in Section 12–18. We concentrate on studying freeprobabilistic data affected by the dynamics and the analytic properties of certain elements induced by dynamical systems. Readers can realize the connections and differences between the main results of Sections 4–8, and those of Sections 12–18. Finally, in Section 19, we summarize the our results, re-explain, and emphasize the meanings of them.

CHAPTER

2

Preliminaries

I

N THIS SECTION, we briefly introduce concepts used in text.

2.1

FREE PROBABILITY

In general, free probability is the noncommutative operator-algebraic analogue of classical measure theory (including probability theory), e.g., see [3,15–17,28–30]. It is not only a major field in operator algebra theory (e.g., [9,17,20–22,28,29]), but also an interesting application in other applied fields (e.g., [4–8,10–12,24,25]). In this monograph, the combinatorial approach of [17,22,23] is used. Without introducing detailed definitions, or combinatorial backgrounds, the (joint) free moments are computed to verify free distributions of operators; freeprobabilistic free product of algebras are used to study freeness condition. As usual, if A is a topological ∗-algebra (a C ∗ -algebra, or a von Neumann algebra, or a Banach ∗-algebra, etc.), and if φ : A → C is a bounded linear functional on A, then the pair (A, φ) is said to be a (noncommutative) topological ∗-probability space (a C ∗ -probability space, respectively, a W ∗ -probability space, respectively, a Banach ∗-probability space, etc.). It is a free-probabilistic counterpart of a measure space. Operators a ∈ A are called free random variables, if we understand a as elements of (A, φ). Two free random variables are free in (A, φ), if all mixed free cumulants of them vanish (e.g., [17,22,23]).

DOI: 10.1201/9781003263487-2

5

6 ■ Banach-Space Operators On C ∗ -Probability Spaces

Let a1 , . . . , as ∈ (A, φ) be free random variables, for s ∈ N. Then the free distributions of them are characterized by the joint free moments, ( ! )! n Y ∞ ∪ ∪ φ arikk : r1 , . . . , rn ∈ {1, ∗} , n n=1

(i1 ,...,in )∈{1,...,s}

k=1

(2.1.1) or by the joint free cumulants,   
∞ r1 rn ∪ ∪ kn ai1 , . . . , ain : r1 , . . . , rn ∈ {1, ∗} , n n=1

(i1 ,...,in )∈{1,..,s}

by the M¨ obius inversion of [17], where k• (. . .) is the free cumulant on A in terms of the linear functional φ (e.g., [17,22,23]).

2.2

SEMICIRCULAR ELEMENTS

Let (A, φ) be a topological ∗-probability space of a topological ∗algebra A, and a bounded linear functional φ on A. A free random variable a ∈ (A, φ) is self-adjoint, if a is self-adjoint in A in the sense that: a∗ = a in A, where a∗ is the adjoint of a. Note that the free distribution of a self-adjoint free random variable a is fully characterized by ∞

the free moment sequence (φ (an ))n=1 ,

(2.2.1)

and by 

∞



the free cumulant sequence kn a, a, . . . .., a {z } | ntimes

,

n=1

under the M¨ obius inversion of [17], by (2.1.1). Definition 2.1. A self-adjoint free random variable x ∈ (A, φ) is semicircular, if φ (xn ) = ωn cn , for all n ∈ N, (2.2.2) where ωn are in the sense of (1.1.1), and ck are the k-th Catalan numbers.

Preliminaries ■ 7

By the M¨ obius inversion, a self-adjoint free random variable x is semicircular in (A, φ), if and only if kn (x, . . . , x) = δn,2 ,

(2.2.3)

for all n ∈ N, where δ is the Kronecker delta. i.e., the free distributions of “all” semicircular elements are characterized by the free-moment sequence, (0, c1 , 0, c2 , 0, c3 , . . .) , (2.2.4) equivalently, by the free-cumulant sequence, (0, 1, 0, 0, 0, . . .) ,

(2.2.5)

by (2.2.1), (2.2.2) and (2.2.3). By the universality (2.2.4), or (2.2.5), the free distributions of all semicircular elements are called “the” semicircular law.

2.3

GROUP DYNAMICAL SYSTEMS AND CROSSED PRODUCT ALGEBRAS

Let A be a C ∗ -algebra, and let λ be a group with its identity β0 . Assume that this group λ acts on A through a group-action α. i.e., for any β ∈ λ, the image α(β) becomes a ∗-isomorphism in Aut (A), where Aut (A) = {g : A → A | g is a ∗ -isomorphism} is the automorphism group of A. Definition 2.2. Suppose a C ∗ -algebra A, and a group λ are given as above. Then the mathematical triple, Γ

denote

=

(A, λ, α)

(2.3.1)

is called the group dynamical system of λ acting on A through an action α. Let Γ be a group dynamical system (2.3.1). Then one can define a crossed product Banach ∗-algebra, AΓ

denote

=

A ⋊α λ,

(2.3.2)

8 ■ Banach-Space Operators On C ∗ -Probability Spaces

by the Banach ∗-algebra generated by the Cartesian product set A × λ satisfying the α-relation: (a1 , β1 ) (a2 , β2 ) = (a1 α(β1 ) (a2 ) , β1 β2 ) and

(2.3.3)
∗ (a1 , β1 ) = α(β1 ) (a∗1 ) , β1−1 ,

for all a1 , a2 ∈ A, and β1 , β2 ∈ λ, where β1−1 is the group-inverse of β1 in λ. Suppose now a given C ∗ -algebra A induces a C ∗ -probability space (A, ψ) for a bounded linear functional ψ on A. Then, on the crossed product Banach ∗-algebra AΓ of (2.3.2), one can define a well-defined linear functional ψΓ by the morphism satisfying that def

ψΓ ((a, β)) = ψ (α(β) (a))

(2.3.4)

for all a ∈ (A, ψ), and β ∈ λ. Definition 2.3. Let Γ be a group dynamical system (2.3.1), and AΓ , the corresponding crossed product Banach ∗-algebra (2.3.2) under the α-relation (2.3.3). Assume that there exists a well-defined linear functional ψΓ of (2.3.4) on AΓ . Then the Banach ∗-probability space, AΓ

denote

=

(AΓ , ψΓ ),

(2.3.5)

is called the Γ-dynamical (crossed-product-Banach-)∗-probability space. One of the main purposes of this monograph is to consider free probability, and operator theory on such group-dynamical ∗-probability spaces of (2.3.5).

CHAPTER

3

Joint Free Distributions of Multi Semicircular Elements

N THIS SECTION, we consider joint free distributions of mutually free, multi semicircular elements in a C ∗ -probability space (A, φ). If a given C ∗ -probabilistic structure (A, φ) is replaced to be a W ∗ probability space, or a Banach ∗-probability space, the following results would be same up to topology. Let X be an arbitrary finite set, and let N C (X) be the lattice of all non-crossing partitions over X, equipped with its partial ordering (≤) defined by

I

def

π ≤ θ ⇐⇒ ∀V ∈ π, ∃U ∈ θ s.t., V ⊆ U, for “V 0X 1X

noncrossing partitions π, θ ∈ N C (X), where “V ∈ π” means that is a block of π.” Then this lattice NC (X ) has its minimal element whose blocks have a single element of X, and its maximal element which is the single-block partition. For instance, if Y = {a, b, c} ,

then N C (Y ) = {π1 , . . . , π5 } ,

DOI: 10.1201/9781003263487-3

9

10 ■ Banach-Space Operators On C ∗ -Probability Spaces

set-theoretically, where π1 = 0Y = {(a) , (b) , (c)} , π2 = {(a, b) , (c)} , π3 = {(a) , (b, c)} , π4 = {(a, c) , (b)} , and π5 = 1Y = {(a, b, c)} , where the expressions (. . .) means the blocks of the partitions consisting of elements inside (. . .) (e.g., [22,23]). It is shown that 0Y = π1 ≤ πk ≤ π5 = 1Y , ∀k = 2, 3, 4. It is well know that |N C (X)| = c|X| , ∀finite sets X (e.g., [22,23]), where |Z| is the cardinality of a set Z, and ck are the kth Catalan numbers for all k ∈ N0 = N∪{0}. So, in the above example, indeed, one can check that |N C (Y )| = c|Y | = c3 = 5. Now, let (B, ψ) be an arbitrary topological ∗-probability space, and y1 , . . . , yn ∈ (B, ψ), free random variables, which are not necessarily mutually distinct from each other, for some n ∈ N. Suppose N C (n)

denote

=

N C ({1, . . . , n})

is the noncrossing-partition lattice over {1, . . . , n}, and π ∈ N C (n). Then, for a fixed partition π, one can define so-called the (noncrossing-) partition (free-)moment, ψπ (y1 , y2 , . . . , yn ) of the free random variables y1 , . . . , yn by Y ψπ (y1 , . . . , yn ) = ψV (y1 , . . . , yn ) , V ∈π

Joint Free Distributions ■ 11

where

(3.1)   |V | Y ψV (y1 , . . . , yn ) = ψ  yil  , l=1

whenever
V = i1 , . . . , i|V | in π. For instance, if one has mutually distinct free random variables y1 , y2 , y3 ∈ (B, ψ) (which are not necessarily distinct from each other), and assume π1 = {(1, 2) , (3)} , π2 = {(1, 3) , (2)} ∈ N C (3) . Then ψπ1 (y1 , y2 , y3 ) = ψ (y1 y2 ) ψ (y3 ) , and ψπ2 (y1 , y2 , y3 ) = ψ (y1 y3 ) ψ (y2 ) , etc. Now, let x ∈ (B, ψ) be a semicircular element, i.e.,   ψ (xn ) = ωn c n2 , and knψ y, y, . . . ., y  = δn,2 , | {z } n-times

for all n ∈ N, by (2.2.2) and (2.2.3), where k•ψ (..) is the free cumulant on B with respect to the linear functional ψ, and δ is the Kronecker delta. Observe that    X Y ψ   ωn c n2 = ψ (xn ) = k|V | x, x, x, . . . .., x  {z } | π∈N C(n)

V ∈π

|V |-times

by the M¨ obius inversion of [23] ! =

X

Y

π∈N C(n)

V ∈π

δ|V |,2

! =

X

Y

θ∈N C2 (m)

U ∈θ

1

12 ■ Banach-Space Operators On C ∗ -Probability Spaces

by the semicircularity (2.2.3), where N C2 (n) = {θ ∈ N C (n) : |U | = 2, ∀U ∈ θ} is the subset of N C (n) consisting of all “pair” noncrossing partitions whose blocks have only two elements, and hence, it goes to  X  X = 1#(θ) = 1 θ∈N C2 (m)

θ∈N C2 (m)

where #(θ) is the number of blocks of a partition θ = |N C2 (n)| ,

(3.2)

for all n ∈ N. Lemma 3.1. Let x be a semicircular element of a topological ∗-probability space (B, ψ). Then  n  (3.3) ωn c n2 = ψ (xn ) = |N C2 (n)| = N C , 2 for all n ∈ N, where  ωk =

1 0

if k is even if k is odd,

for all k ∈ N. Proof. The formula (3.3) is shown by (2.2.2) and (3.2).

■

The above formula (3.3) implies that, indeed, all “odd” free moments of a semicircular element vanish (See (2.2.2)) determined up to {ωk }k∈N of (3.3), since n , whenever n is odd, N C2 (n) = ∅ = N C 2 where ∅ is the empty set. Corollary 3.2. Under the same hypothesis of Lemma 3.1,   ψ1n x, x, x, . . . .., x = ωn c n2 = |N C2 (n)| , ∀n ∈ N, | {z }

(3.4)

n-times

where 1n = 1{1,...,n} = {(1, . . . , n)} is the maximal element of N C (n).

Joint Free Distributions ■ 13

Proof. Note that, by (3.1), one immediately obtain that   ψ1n x, x, . . . . . . ., x = ψ (xn ) , ∀n ∈ N, | {z } n-times

because 1n = {((1, . . . , n))} in N C (n), and hence, the formula (3.4) holds by (3.3). ■ Now, let (A, φ) be a fixed C ∗ -probability space, and suppose there are N -many mutually free semicircular elements x1 , . . . , xN ∈ (A, φ), for N ∈ N∞ >1 . By [23], all “mixed” free cumulants of x1 , . . . , xN vanish by the freeness on them. And, by the self-adjointness of x1 , . . . , xN in A, the free distribution ρ = ρx1 ,...,xN (3.5) of these semicircular elements is characterized by the joint free moments, ( !)! n Y ∞ ∪ ∪ φ xil , (3.6) n n=1

(i1 ,...,in )∈{1,...,N }

l=1

by (2.1.1) (e.g., see [17,22,23]). To verify the joint free distribution ρ of (3.5), we focus on computing free-distributional data of (3.6). For any s ∈ N, fix an s-tuple s

Is = (i1 , . . . , is ) ∈ {1 , . . . , N } .

(3.7)

For example, I8 = (1, 1, 3, 1, 3, 2, 2, 1) 8

is an 8-tuple of {1, 2, 3, 4, 5} . From the sequence Is of (3.7), define a set [Is ] by [Is ] = {i1 , . . . , is } ,

(3.8)

without considering repetitions of quantities in {1, . . . , N }. i.e., the set [Is ] induced by Is is a set with its cardinality s, because we ignore the repetitions of quantities of {1, . . . , N } in Is . For instance, if I8 is as above, then [I8 ] = {i1 , i2 , . . . , i8 } ,

14 ■ Banach-Space Operators On C ∗ -Probability Spaces

with its cardinality 8, satisfying i1 = i2 = i4 = i8 = 1 , and i3 = i5 = 3, and i6 = i7 = 2 , without considering repetition; i.e., for instance, we regard all 1’s in I8 as mutually distinct elements i1 , i2 , i4 and i8 in the set [I8 ]. Then, from the set [Is ] of (3.8), one can define a “noncrossing” partition π(i1 ) of the noncrossing-partition lattice, NC ([Is ]) = NC ({i1 , i2 , . . . , is }) (e.g., [17,22,23]), such that: (i) starting from i1 , construct the maximal block U1 of π(i1 ), satisfying   U1 = ij1 = i1 , ij2 , . . . , ijk ⇐⇒ ij1 = ij2 = . . . = ijk = i1 , (3.9) in {1, . . . , N }, where k =| U1 | is the cardinality of the block U1 (as a set), (ii) construct the next noncrossing block U2 from the very next element, il ∈ [Is ] \ {ij1 , . . . , ijk } = [Is ] \ U1 ; as in (3.9); do such processes until end, and (iii) the resulted partition π(i1 ) of (ii) is “maximal” for the partial ordering (≤) on the lattice NC ([Is ]) satisfying both (i) and (ii). For example, if I8 and [I8 ] are as above, then there exists a noncrossing partition, π(i1 ) = {(i1 , i2 , i4 , i8 ) , (i3 ) , (i5 ) , (i6 , i7 )} , in NC ([I8 ]), satisfying the above conditions (i), (ii) and (iii). Remark here that, even though i3 = i5 = 3 , one cannot have the block (i3 , i5 ) in π(i1 ), because this block (i3 , i5 ) has a crossing with the block U1 = (i1 , i2 , i4 , i8 ) of (3.9), so, one has to take two separated blocks (i3 ) and (i5 ) to satisfy the conditions (i), (ii) and (iii). Now, let π(i1 ) ∈ N C ([Is ]) be the noncrossing partition induced by the first entry i1 of Is , and let π(i1 ) = {U1 , . . . , Ut } ,

Joint Free Distributions ■ 15

where t ≤ s, and Uk ∈ π(i1 ) are the blocks satisfying (i), (ii) and (iii), for k = 1 , . . . , t. For example, if I8 is as above, then π(i1 ) = {U1 , U2 , U3 , U4 } with t = 4 < 8 = s, where U1 = {i1 , i2 , i4 , i8 } , U2 = {i3 } , and U3 = {i5 } , U4 = {i6 , i7 } . This partition π (i1 ) is regarded as the joint partition, π(i1 ) = 1{U1 } ∨ 1{U2 } ∨ . . . ∨ 1{Ut } ,

(3.10)

where 1{Uk } in (3.10) are the maximal elements, the one-block partitions, of NC (Uk ), for all k = 1 , . . . , t, by regarding Uk as independent discrete sets. For example, if π(i1 ) for the first entry i1 of the above 8-tuple I8 is given as above, then π(i1 ) = 1{i1 ,i2 ,i4 ,i8 } ∨ 1{i3 } ∨ 1{i5 } ∨ 1{i6 ,i7 } . Now, construct a new noncrossing partition π(i2 ) ∈ NC ([Is ]) similarly for the second entry i2 of Is , as a noncrossing partition satisfying the conditions: (I) for the fixed entry i2 , construct the maximal block,   B1 = ik1 , . . . , ikm = i2 , . . . , ikl , where l = |B1 | as in (3.9), satisfying ik1 = . . . = ikm = . . . = ikl = i2 , in {1, . . . , N }, (II) do such processes for other entries step-by-step until end, and (III) the resulted partition π(i2 ) is maximal in N C ([Is ]) satisfying (I) and (II) altogether. For example, for the above 8-tuple I8 = (1, 1, 3, 1, 3, 2, 2, 1), we have π(i2 ) = {(i1 , i2 , i4 , i8 ) , (i3 ) , (i5 ) , (i6 , i7 )} . In this case, one can realize that this partition π(i2 ) is identically same with π(i1 ) from the above example.

16 ■ Banach-Space Operators On C ∗ -Probability Spaces

Construct such noncrossing partitions for all rest entries i3 , . . . , is of the s-tuple Is of (3.7). For example, from the above 8-tuple I8 , one has π(i1 ) = π(i2 ) = π(i4 ) = π(i8 ), as above, π(i3 ) = π(i5 ) = {(i3 , i5 ) , (i1 , i2 , i8 ) , (i4 ) , (i6 , i7 )} , and π(i6 ) = π(i7 ) = {(i6 , i7 ) , (i1 , i2 , i4 , i8 ) , (i3 ) , (i5 )} . Without loss of generality, one may write π(i1 ) = π(i2 ) = π(i4 ) = π(i8 ) π(i3 ) = π(i5 ) and π(i6 ) = π(i7 )

denote

=

denote

=

denote

=

π(1), (3.11)

π(3), π(2)!

Observation. It is not difficult to check that ij1 = ij2 in {1, . . . , N } =⇒ π(ij1 ) = π(ij2 ) in N C ([Is ]) s

from the very constructions of our noncrossing partitions {π(il )}l=1 . However, the converse does not hold in general. For instance, if let

I6 = (1, 1, 1, 1, 2, 2) = (i1 , . . . , i6 ) , then π (1) = {(i1 , i2 , i3 , i4 ) , (i5 , i6 )} = π (2) , in N C ({i1 , . . . , i6 }).

□ s

By the above observation, if Is = (i1 , . . . , is ) ∈ {1, . . . , N } is an s-tuple, and if it contains mutually distinct entries, ij1 < ij2 < . . . < ijn in {1, . . . , N } , where n ≤ N , then one may/can construct mutually distinct noncrossing partitions, π (ij1 ) , . . . , π (ijn ) in N C ([Is ]) ,

(3.12) 8

satisfying the above conditions. For example, if I8 ∈ {1, . . . , 5} is given as above, then one has mutually distinct partitions π(1), π(2) and π(3) in the sense of (3.11) (or, more generally, in the sense of (3.12)).

Joint Free Distributions ■ 17 s

Notation 3.1. (1) For a fixed s-tuple Is ∈ {1, . . . , N } of (3.7), let π (ij1 ) , . . . , π (ijn ) be the noncrossing partitions (3.12) in N C ([Is ]), for n ≤ N . Denote the subset of all such partitions by Π ([Is ]) = {π (ij1 ) , . . . , π (ijn )} , in N C ([Is ]). (2) We denote the subset of Π ([Is ]) consisting of all partitions whose blocks have only even-many elements by Πe ([Is ]), i.e., Πe ([Is ]) = {θ ∈ Π ([Is ]) : |V | ∈ 2N, ∀V ∈ θ} , where 2N = {2n : n ∈ N}.
(3) Let π ∈ N C ([Is ]) and suppose V = ik1 , . . . , ik|V | ∈ π is a block. We say that “V has identical entries of Is ,” if ik1 = ik2 = . . . = ik|V | in {1, . . . , N } . i.e., a block V ∈ π has identical entries of Is , if and only if all elements of V are identically same in {1, . . . , N }. □ For example, if I8 is given as above, then Π ([I8 ]) {π(1), π(2), π(3)} is obtained as in (3.11). But,

=

Πe ([Is ]) = ∅, because all distinct partitions π (1), π (2) and π (3) of Π ([I8 ]) contain blocks with odd-cardinalities; also, one can check that all blocks of the partitions in Π ([I8 ]) have identical entries of I8 . Let Is = (i1 , . . . , is ) be an s-tuple (3.7), and let xi1 , . . . , xis ∈ {x1 , . . . , xN } be the corresponding semicircular elements of (A, φ) induced by Is . Define a free random variable X [Is ] by def

X [Is ] =

s Y

xil ∈ (A, φ) .

(3.13)

l=1

For example, if I8 is as above, then X[I8 ] = x1 x1 x3 x1 x3 x2 x2 x1 = x21 x3 x1 x3 x22 x1 , which is a free reduced word with its length-6 in (A, φ) (e.g., [23,30]).

18 ■ Banach-Space Operators On C ∗ -Probability Spaces

If X [Is ] is a free random variable (3.13), then ! X

Y

π∈N C([Is ])

V ∈π

φ (X [Is ]) =

kV

by the M¨ obius inversion of [17,22], where   kV = k|V | xik1 , . . . , xik|V | ,
whenever V = ik1 , . . . , ik|V | is a block of a partition π, where k• (. . .) is the free cumulant on A with respect to φ, and hence, ! X Y = kV (3.14) θ∈P([Is ])

V ∈θ

where P ([Is ]) = {π ∈ N C ([Is ]) : V ∈ π has identical entries of Is , ∀V ∈ π} , in the sense of Notation 3.1, by the mutual-freeness of x1 , . . . , xN in (A, φ) (i.e., all mixed free cumulants of x1 , . . . , xN vanish by their mutual-freeness!) ! X Y = δn,2 θ∈P([Is ])

V ∈θ

by the semicircularity (2.2.3) ! X

Y

θ∈P([Is ])∩N C2 ([Is ])

V ∈θ

=

1 ,

(3.15)

by (3.14), where def

N C2 (Y ) = {π ∈ N C (Y ) : V ∈ π ⇐⇒| V |= 2} is the subset of the noncrossing-partition lattice N C(Y ) over finite sets Y . i.e., π ∈ N C2 (Y ), if and only if all blocks of π have two elements. By (2.2.3), (2.2.5), (3.13) and (3.15), if there exists at least one k0 ∈ {1 , . . . , t}, such that | Uk0 | is odd in N, then φ (X [Is ]) = 0.

(3.16)

Joint Free Distributions ■ 19

So, the formula (3.15) is nonzero, only if (i) s ∈ N is even, and (ii) the s-tuple Is = (i1 , . . . , is ) contains mutually distinct entries ik1 , . . . , ikn (n ≤ N ), and each distinct entry appeared in Is even number of times. s

Lemma 3.3. Let Is ∈ {1, . . . , N } be a fixed s-tuple (3.7) and let X[Is ] ∈ (A, φ) be the corresponding free random variable (3.13) N induced by the mutually free semicircular elements {xj }j=1 . Then φ (X[Is ]) = ωs |P2 ([Is ])| ,

(3.17)

where P2 ([Is ])

denote

=

P ([Is ]) ∩ N C2 ([Is ]) ,

and ωs and P ([Is ]) are in the sense of (1.1) and (3.14), respectively, for all s ∈ N. Proof. Suppose s is odd in N. Then the formula (3.17) is obtained by (3.16). Meanwhile, if s is even in N, then ! X Y φ (X [Is ]) = 1 θ∈P([Is ])∩N C2 ([Is ])

V ∈θ

by (3.15) =

X θ∈P([Is ])∩N C2 ([Is ])



 1#(θ) =

X

1,

θ∈P2 ([Is ])

where #(θ) is the number of blocks in θ ∈ N C ([Is ]). Therefore, the formula (3.17) holds true. ■ Note that, even though s ∈ N is even, it is possible that P2 ([Is ]) is empty. In such a case, φ (X [Is ]) = 0, anyway, by (3.17). Recall now the subset Π ([Is ]) of Notation 3.1, consisting of all mutually distinct partitions of (3.12). s

Lemma 3.4. Suppose Is = (i1 , . . . , is ) ∈ {1, . . . , N } be an s-tuple of (3.7) for s ∈ 2N, having its mutually distinct entries, ik1 < ik2 < . . . < ikn with 1 ≤ n ≤ N, Let P2 ([Is ]) be in the sense of (3.17), and let Πe ([Is ]) be the subset of Notation 3.1 in N C ([Is ]).

20 ■ Banach-Space Operators On C ∗ -Probability Spaces

(3.18) If P2 ([Is ]) ̸= ∅, and π ∈ P2 ([Is ]), then there exists θ ∈ Πe ([Is ]) such that π ≤ θ in the lattice N C ([Is ]). (3.19) If Πe ([Is ]) ̸= ∅, then θ ∈ Πe ([Is ]) has at least one π ∈ P2 ([Is ]), such that π ≤ θ in N C ([Is ]), if and only if all blocks of θ contains even-many elements. Proof. Under hypothesis, suppose P2 ([Is ]) = P ([Is ]) ∩ N C2 ([Is ]) is not empty. Then, the subset Πe ([Is ]) of Π ([Is ]) is not empty by the very construction of Π ([Is ]) and its subset Πe ([Is ]). Moreover, for any π ∈ P2 ([Is ]), there exists θ ∈ Πe ([Is ]), such that π ≤ θ in N C ([Is ]), by the constructions of P2 ([Is ]). Therefore, the statement (3.18) holds. Remark that if Πe ([Is ]) ̸= ∅, then P2 ([Is ]) ̸= ∅, because, for any θ ∈ Πe ([Is ]), one can take at least one π ∈ P2 ([Is ]) such that π ≤ θ in N C ([Is ]), by the definition of P ([Is ]) of (3.14). Thus, the statement (3.19) holds, too. ■ The above lemma shows that if either P2 ([Is ]), or Πe ([Is ]) is not empty in N C ([Is ]), then all elements of P2 ([Is ]) are covered by elements of Πe ([Is ]) (in the sense of (3.18)), and conversely, all elements of Πe ([Is ]) cover elements of P2 ([Is ]). It means that if nonempty, Πe ([Is ]) classify P2 ([Is ]), i.e., if Πe ([Is ]) = {θ1 , . . . , θn } , with 1 ≤ n ≤ N, then

n

P2 ([Is ]) = ⊔ P2 ([Is ] , θl ) , l=1

with

(3.20) P2 ([Is ] , θl ) = {π ∈ P2 ([Is ]) : π ≤ θl } ,

by (3.18) and (3.19), for all l = 1, . . . , n, where ⊔ is the disjoint union. Proposition 3.5. Let P2 ([Is ]) be in the sense of (3.14), and let s Πe ([Is ]) be in the sense of Notation 3.1, for Is ∈ {1, . . . , N } , for s ∈ N. Then P2 ([Is ]) = ⊔ {π ∈ P2 ([Is ]) : π ≤ θ} . (3.21) θ∈Πe ([Is ])

Joint Free Distributions ■ 21

Proof. The decomposition (3.21) of P2 ([Is ]) classified by the elements of Πe ([Is ]) is shown by (3.20). ■ Using the above decomposition (3.21), the formula (3.17) is refined by the following result. Theorem 3.6. Let Is be an s-tuple (3.7), and X [Is ] ∈ (A, φ), the corresponding free random variable (3.13), and let Πe ([Is ]) be the subset of N C ([Is ]) in the sense of Notation 3.1. Then ! X Y φ (X[Is ]) = c |V | , (3.22) θ∈Πe ([Is ])

V ∈θ

2

where Πe ([Is ]) is in the sense of Notation 3.1. Clearly, one has that φ (X [Is ]) = 0 if and only if Πe ([Is ]) = ∅. Proof. Under hypothesis, one has φ (X [Is ]) = ωs |P2 ([Is ])| by (3.17) = ωs ⊔ P2 ([Is ] , θ) θ∈Πe ([Is ]) by (3.21), where P2 ([Is ] , θ) are in the sense of (3.20)   X = ωs  |P2 ([Is ] , θ)| θ∈Πe ([Is ])



 = ωs 

X 

θ∈Πe ([Is ])

 = ωs 

X

 X



θ∈Πe ([Is ])

1

π∈P2 ([Is ]),π≤θ



 = ωs 

 X

1#(π) 

π∈P2 ([Is ]),π≤θ

 X

θ∈Πe ([Is ])

! X

Y

π∈P2 ([Is ]),π≤θ

V ∈π



1 

22 ■ Banach-Space Operators On C ∗ -Probability Spaces

 = ωs 

!

 X

X 

θ∈Πe ([Is ])

δ|V |,2 

V ∈π

π∈P2 ([Is ]),π≤θ

 = ωs 

Y 

X

φθ (xi1 , . . . , xis ) ,

(3.23)

θ∈Πe ([Is ])

by the semicircularity (2.2.3), where φθ (•) are the partition moments of (3.1). And, by (3.1), (3.3) and (3.4), one can get that Y Y φθ (xi1 , . . . , xis ) = φV (xi1 , . . . , xis ) = c |V | , (3.24) V ∈θ

V ∈θ

2

for all θ ∈ Πe ([Is ]). Therefore, the free-distributional data (3.22) holds by (3.23) and (3.24). So, by (3.22), if Πe ([Is ]) = ∅, then φ (X [Is ]) = 0. Conversely, if φ (X [Is ]) = 0, then Πe ([Is ]) is empty in Π ([Is ]) by (3.23) and (3.24). ■ Example 3.1. (1) Let W =x21 x42 x1 x22 x1 ∈ (A, φ) be in the sense of (3.13). Then one can take the 10-tuple, let

IW = (1 , 1 , 2 , 2 , 2 , 2 , 1 , 3 , 3 , 1 ) = (i1 , . . . , i10 ) , as in (3.7). Then the noncrossing partitions, π(1) = {(i1 , i2 , i7 , i10 ) , (i3 , i4 , i5 , i6 ) , (i8 , i9 )} , and π (2) = π (3) = π (1) in N C ([IW ]) , of (3.12) are obtained. So, we have Π ([IW ]) = {π (1)} = Πe ([IW ]) in the sense of Notation 3.1. Therefore, φ (W ) = c 24 c 42 c 22 = c22 c1 = 4, by (3.22).

Joint Free Distributions ■ 23

(2) Let Y = x12 x3 x1 x3 x22 x1 ∈ (A, φ). Then one can take an 8-tuple, let

IY = (1, 1, 3, 1, 3, 2, 2, 1) = (i1 , . . . , i8 ) of (3.7). Thus, the corresponding partition, π(1) = {(i1 , i2 , i4 , i8 ) , (i3 ) , (i5 ) , (i6 , i7 )} , π(2) = π (1) , and π (3) = {(i1 , i2 ) , (i3 , i5 ) , (i4 ) , (i6 , i7 ) , (i8 )} , of (3.12) are obtained, and they construct Π ([IY ]) = {π (1) , π (3)} , however, Πe ([IY ]) = ∅ in Π ([IY ]) , as in Notation 3.1, because all distinct partitions π(1) and π (3) contain odd blocks. Thus φ (Y ) = 0, by (3.22). (3) Let U = x1 x2 x21 x2 x21 x2 x21 x2 x1 ∈ (A, φ). Then one can take a 12-tuple, let

IU = (1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1) = (i1 , . . . , i12 ) of (3.7). Thus, one obtains the noncrossing partitions, π (1) = {(i1 , i3 , i4 , i6 , i7 , i9 , i10 , i12 ) , (i2 ) , (i5 ) , (i8 ) , (i11 )} , and π (2) = {(i2 , i5 , i8 , i11 ) , (i1 , i12 ) , (i3 , i4 ) , (i6 , i7 ) , (i9 , i10 )} , constructing Π ([IU ]) = {π (1) , π (2)} , with Πe ([IU ]) = {π (2)} , since π (1) contains four odd blocks. Thus, we have φ (U ) = c 42 c 22 c 22 c 22 c 22 = c2 c41 = 2, by (3.22).

□

24 ■ Banach-Space Operators On C ∗ -Probability Spaces

Note that the joint-free-moment formula (3.22) can be re-written as follows: !  X Y φ (X [Is ]) = ω|V | c |V | , (3.25) π∈Π([Is ])

by Notation 3.1.

V ∈π

2

CHAPTER

4

A C ∗ -Probability Space of |Z|-Many Semicircular Elements

ET (A, φ) BE A C ∗ -PROBABILITY SPACE, containing a family X = {xj }j∈Z of mutually free, |Z|-many semicircular elements xj ’s. Such a topological ∗-probability space (A, φ) does exist naturally (e.g., [5,12]), and artificially-but-canonically (e.g., [6–11]).

L

Definition 4.1. A topological ∗-probability space (A1 , φ1 ) is said to be free-homomorphic to a topological ∗-probability space (A2 , φ2 ), if there is a ∗-homomorphism Ω : A1 → A2 , such that φ2 (Ω (a)) = φ1 (a) , ∀a ∈ (A1 , φ1 ) . In such a case, the ∗-homomorphism Ω is called a free-homomorphism from (A1 , φ1 ) to (A2 , φ2 ). We denote this free-homomorphic relation by free−homo (A1 , φ1 ) −→ (A2 , φ2 ) . (4.1) If the free-homomorphism Ω is a ∗-isomorphism from A1 onto A2 , then it is called a free-isomorphism. In this case, (A1 , φ1 ) is said to be freeisomorphic to (A2 , φ2 ). This free-isomorphic relation is denoted by (A1 , φ1 )

free−iso

=

(A2 , φ2 ) .

(4.2)

By the free-homomorphic relation (4.1), if (A1 , φ1 )

free−homo

DOI: 10.1201/9781003263487-4

−→

(A2 , φ2 ) , 25

26 ■ Banach-Space Operators On C ∗ -Probability Spaces

then (Ω (A1 ) , φ2 ) is understood as a free-probabilistic sub-structure of (A2 , φ2 ). Also, by the free-isomorphic relation (4.2), if (A1 , φ1 )

free−iso

=

(A2 , φ2 ) ,

then they are regarded as equivalent topological ∗-probability spaces in the sense of Voiculescu (e.g., [28–30]), i.e., they are same freeprobabilistic structures. Let (A, φ) be a fixed C ∗ -probability space, containing a free semicircular family X = {xj }j∈Z . Assume further that the C ∗ -algebra A contains the unity (or the multiplication-identity) 1A , satisfying φ (1nA ) = φ (1A ) < ∞. Then one can construct a unital C ∗ -subalgebra X of A, generated by the family X, having its unity 1X = 1A . i.e., X = C ∗ (X) = C [X] in A,

(4.3)

where C [Z] are the polynomial algebras in Z over C = C · 1X , and Z are the C ∗ -norm- topology closures of all subsets Z of A. By (4.3), one obtains a free-probabilistic sub-structure, X

denote

=

(X, φ = φ |X ) of (A, φ) .

(4.4)

Proposition 4.1. Let X be the C ∗ -probability space (4.4). Then the C ∗ -algebra X of (4.3) satisfies that ∗−iso

X =

⋆



j∈Z

C[{xj }]



∗−iso





= C ⋆ {xj } in (A, φ) , j∈Z

(4.5)

∗−iso

where “ = ” means “being ∗-isomorphic,” and the free product (⋆) in the first ∗-isomorphic relation of (4.5) is the free-probabilistic free product, and the free product (⋆) in the second ∗-isomorphic relation of (4.5) is the pure-algebraic free product generating noncommutative free words in X = ∪ {xj }. j∈Z

Proof. the C ∗ -subalgebra X = C ∗ (X) is ∗-isomorphic to  By (4.3),  ⋆ C[{xj }] in (A, φ), since the generating set X is a free family

j∈Z

A C ∗ -Probability Space of |Z| ■ 27

consisting of mutually free, self-adjoint generators X = {xj }j∈Z in (A,φ) (e.g., see [22,28,30]), satisfying that C ∗ ({xj }) = C

xj , x∗j





= C [{xj }],

in A over C = C · 1X , for all j ∈ Z. So, the first ∗-isomorphic relation of (4.5) holds. Note that all noncommutative free words in X = ∪ {xj } have their j∈Z

unique free “reduced” words of X, under the operator-multiplication on X inherited from that on A. Thus, the second ∗-isomorphic relation of (4.5) holds, too (e.g., [28–30]). ■ By (4.5), the free probability on the C ∗ -probability space X of (4.4) is characterized by the joint free moments of the generating elements in the free semicircular family X = {xj }j∈Z . Corollary 4.2. Let Is = (i1 , . . . , is ) be an arbitrary s-tuple in Z, for s Q any s ∈ N, and let X[Is ] = xil ∈ X be the corresponding free random l=1

variable, where xi1 , . . . , xis ∈ X. Then the free-distributional data of X[Is ] is characterized by (3.22), or (3.25). i.e., ! X Y φ (X [Is ]) = c |V | , θ∈Πe ([Is ])

V ∈θ

2

or, equivalently, ! φ (X [Is ]) =

X θ∈Π([Is ])

Y V ∈θ

ω|V | c |V |

,

2

where Πe ([Is ]) and Π ([Is ]) are in the sense of Notation 3.1. Proof. The free-distributional data of X[Is ] ∈ Xφ is obtained by (3.22), (3.25) and (4.5), for all s-tuples Is ∈ Zs in Z, for all s ∈ N, by the semicircularity on the generator set X, under mutual-freeness. ■ The above corollary characterizes the free-distributional data on X because all elements of X are the limits of linear combinations of free reduced words in X = {xj }j∈Z by (4.5) (e.g., [9,22,23,30]).

28 ■ Banach-Space Operators On C ∗ -Probability Spaces

On the index set Z of the generating family X, define a bijection h : Z → Z, by def h(j) = j + 1, for all j ∈ Z. (4.6) Then it has its inverse h−1 : Z → Z, h−1 (j) = j − 1, for all j ∈ Z.

(4.7)

Define now a “multiplicative” linear transformation β on the C ∗ algebra X of (4.3) by the linear morphism satisfying, β (xj ) = xh(j) = xj+1 , for all xj ∈ X,

(4.8)

where h is the bijection (4.6). Suppose W = (i1 , . . . , iN ) ∈ ZN is an alternating N -tuple for N ∈ N, in the sense that: i1 ̸= i2 , i2 ̸= i3 , . . . , iN −1 ̸= iN in Z, and let

(4.9)

N Y X[W ] = xnill ∈ X l=1

be a free reduced word with its length-N , induced by the alternating N -tuple W , for n1 , . . . , nN ∈ N. If X[W ] is a free reduced word (4.9) of X in X, then N N N Y Y
Y n β xnill = β (X[W ]) = β (xil ) l = xnill+1 , l=1

l=1

(4.10)

l=1

by the multiplicativity of β on X, and (4.8). Also, by the bijectivity (4.7) of h, the restriction β |X is bijective on the generating set X of X, implying that this multiplicative linear transformation β is bounded and bijective on the C ∗ -algebra X, by (4.5). Indeed, one can take the inverse β −1 on X by the multiplicative linear transformation satisfying β −1 (xj ) = xh−1 (j) = xj−1 , ∀xj ∈ X,

(4.11)

and hence, β

−1

N Y (X[W ]) = xnill−1 , on X, l=1

(4.12)

A C ∗ -Probability Space of |Z| ■ 29

where X[W ] ∈ X is a free random variable (4.9). Observe now that, for any t ∈ C, and xj ∈ X ⊂ X,
∗
∗ ∗ β (txj ) = β tx∗j = tβ (xj ) = txj+1 = (txj+1 ) = (β (txj )) , implying that ∗

β (T ∗ ) = (β(T )) , ∀T ∈ X,

(4.13)

by (4.5), (4.8), (4.10) and (4.12). i.e., the multiplicative linear transformation β acting on the C ∗ -probability space X is a ∗homomorphism by (4.13). Lemma 4.3. The morphism β of (4.8) is a ∗-isomorphism on X. Proof. By (4.13), this multiplicative linear morphism β is a ∗homomorphism on X. By (4.5), (4.11) and (4.12), it is bijective and bounded on X. Therefore, it is a ∗-isomorphism on X. ■ The above lemma shows that the inverse β −1 of β, in the sense of (4.11), is a well-defined ∗-isomorphism too, i.e., β −1 β = 1X = 1A = ββ −1 , on X, where the product is the multiplication on ∗-homomorphisms. Lemma 4.4. The ∗-isomorphism β of (4.8) is a free-isomorphism on X. Proof. By the above lemma, the morphism β is a ∗-isomorphism on X. Consider that, if X[W ] ∈ X is an any arbitrary free reduced word (4.9), then one can take the s-tuple   Is = i1 , . . . i1 , . . . , iN , . . . iN  | {z } | {z } n1 −times

in Zs , as in (3.3), where s =

N P

(4.14)

nN −times

nl ∈ N.

l=1

The image β (X[W ]) ∈ X induces the corresponding s-tuple,   Is′ = i1 + 1, . . . i1 + 1, . . . , iN + 1, . . . iN + 1 {z } {z } | | n1 −times

nN −times

(4.15)

30 ■ Banach-Space Operators On C ∗ -Probability Spaces

in Zs , as in (3.3). The s-tuples Is and Is′ provide the corresponding sets Πe ([Is ]) and Πe ([Is′ ]) in the sense of Notation 3.1, and they not only satisfy |Πe ([Is ])| = |Πe ([Is′ ])| in {0, 1, . . . , N } , but also have the same partition-structures by (4.14) and (4.15) in the sense that: for any θ ∈ Πe ([Is ]), there exists a unique ϑ ∈ Πe ([Is′ ]) such that θ and ϑ have exactly same block structures, and vice versa. It implies that         φθ xi1 , . . . , xiN  = φϑ xi1 +1 , . . . , xiN +1  , {z } {z } | | s-many

s-many

for all θ ∈ Πe ([Is ]) (respectively, for all ϑ ∈ Πe ([Is′ ])), and hence, φ (β (X[W ])) = φ (X[W ]) , by (3.22), (3.25), (4.9) and (4.10), under the semicircularity (2.2.4). Since X[W ] is arbitrary in X, this relation guarantees φ (β (T )) = φ (T ) , ∀T ∈ X, by (4.5). Therefore, β is a free-isomorphism on X, by (4.16).

(4.16) ■

By the above lemma, the inverse β −1 of β is a free-isomorphism on X, too. i.e., free−iso free−iso β (X) = X = β −1 (X) . (4.17) Define new multiplicative linear transformations β n on X by   if n = 0 1X      β ...β  | {z } n def (4.18) β = if n > 0 n−times   −1 −1  β ...β   {z }   | |n|−times if n < 0, for all n ∈ Z. By the definition (4.18), these morphisms {β n }n∈Z are ∗-isomorphisms on X.

A C ∗ -Probability Space of |Z| ■ 31

Theorem 4.5. The ∗-isomorphisms β n of (4.18) are free-isomorphisms on X, i.e., free−iso β n (X) = X, for all n ∈ Z. (4.19) Proof. Since β (resp., β −1 ) is a free-isomorphism by (4.17), the iterated products β n (resp., β −n ) of (4.18) are free-isomorphisms on X, for all n ∈ N. It is not difficult to check that the identity map 1X = β 0 is a trivial free-isomorphism on X, since it is a ∗-isomorphism on X, satisfying
φ (T ) = φ β 0 (T ) , ∀T ∈ X. Therefore, the morphisms {β n }n∈Z of (4.18) are free-isomorphisms on X. i.e., the free-isomorphic relation (4.19) holds. ■ Now, take the family, def

λ = {β n : n ∈ Z} ,

(4.20)

where β n are the free-isomorphisms (4.18) on X. Let Aut (X) be the automorphism group of all ∗-isomorphisms on X, equipped with the multiplication (or the composition) on ∗isomorphisms. Then one obtains the following structure theorem of the set λ of (4.20). Theorem 4.6. The family λ of (4.20) is a subgroup of Aut (Xφ ). Moreover, Group λ = (Z, +) , (4.21) Group

where “ = ” means “being group-isomorphic.” Proof. It is not difficult to check that β n1 β n2 = β n1 +n2 on X, for all n1 , n2 ∈ Z, by (4.18). So, the multiplication on λ (inherited from that on Aut (Xφ )) is well-defined. Moreover, one can check the associativity; (β n1 β n2 ) β n3 = β n1 +n2 +n3 = β n1 (β n2 β n3 ) , for all n1 , n2 , n3 ∈ Z. i.e., under multiplication, λ forms a semigroup.

32 ■ Banach-Space Operators On C ∗ -Probability Spaces

One can take β 0 = 1X ∈ λ, satisfying that β n β 0 = β n+0 = β n = β 0+n = β 0 β n , ∀n ∈ Z, and hence, 1X = β 0 ∈ λ is the multiplication-identity of the semigroup λ, and hence, λ is a monoid. Finally, by (4.12) and (4.18), for any β n ∈ λ, there exists a unique −n β ∈ λ, such that β n β −n = β n+(−n) = β 0 = 1X = β −n β n , on λ, for all n ∈ Z. Therefore, the set λ of (4.20) is a subgroup of Aut (X). It is trivial to check that β n1 β n2 = β n1 +n2 = β n2 +n1 = β n2 β n1 , on λ, for all n1 , n2 ∈ Z. i.e., this subgroup λ is an abelian subgroup of Aut (X). Define now a function Φ : λ → Z, by def

Φ (β n ) = n in Z, ∀β n ∈ λ. Then, by (4.18), it is a well-defined bijection. Moreover, Φ (β n1 β n2 ) = n1 + n2 = Φ (β n1 ) + Φ (β n2 ) , in Z, for all β n1 , β n2 ∈ λ. Therefore, this bijection Φ is a grouphomomorphism, and hence, it is a group-isomorphism. Thus, the subgroup λ of Aut (X) is isomorphic to the infinite cyclic abelian group (Z, +), proving the relation (4.21). ■ By (4.21), the group λ = {β n }n∈Z is understood as an infinite cyclic abelian subgroup of the automorphism group Aut (X) of X, i.e., λ = ⟨β⟩, the cyclic group generated by β. Definition 4.2. The infinite cyclic abelian subgroup λ of the automorphism group Aut (X), in the sense of (4.20), is called the integer-shift group (acting) on X. All free-isomorphisms β n ∈ λ are called the n(-integer)-shifts on X.

A C ∗ -Probability Space of |Z| ■ 33

Let λ = {β n }n∈Z be the integer-shift group on the C ∗ -probability space X. Then one can define the subspace Λ of the operator space B (X), consisting of all Banach-space operators acting on X (e.g., [13]), by def Λ = spanC (λ) in B (X) , (4.22) where spanC (Y ) are the pure-algebraic subspaces spanned by Y over C, and Y mean the operator-norm-topology closures of Y , for all subsets Y of B (X). Note that, since the generating set λ of the subspace Λ of (4.22) is a group, in fact, Λ = C[λ] in B (X) , where C[Y ] are the polynomial algebras generated by subsets Y of B (X) over C. i.e., Λ forms a Banach algebra embedded in the topological vector space B (X). By the definition (4.22), every element T ∈ Λ has its expression, X T = tn β n , with tn = tβ n ∈ C, (4.23) β n ∈λ

where

P

is the infinite (or limits of finite) sum(s) over λ.

Definition 4.3. The Banach algebra Λ of (4.22) is said to be the (integer-)shift-operator algebra (acting) on X. All elements T ∈ Λ of (4.23) are called the (integer-)shift operators on X. The shift-operator algebra Λ acts naturally on X; for any T ∈ Λ of (4.23), one can have X T (S) = tn β n (S) in X, for all S ∈ X. (4.24) β n ∈λ

It is easily verified that, even though the integer-shift group λ preserves the free probability on X by (4.19), the shift-operator algebra Λ deform the free probability on X, in general, by (4.24). Lemma 4.8. Let t ∈ C, and β k ∈ λ, for some k ∈ Z, and let T = tβ k ∈ Λ be a shift operator on X. Then, for any semicircular elements xj ∈ X ⊂ X, one has n

φ ((T (xj )) ) = ωn tn c n2 , for all n ∈ N.

(4.25)

34 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. Let T = tβ k ∈ Λ be a given shift operator. Then, for xj ∈ X,

n φ ((T (xj )) ) = φ tn xnj+k = tn ωn c n2 , for all n ∈ N, by the semicircularity of xj+k ∈ X in X. Therefore, the deformed free- distributional data (4.25) is obtained. ■ Let T ∈ Λ be a shift operator (4.23) acting on X by (4.24). Define the support Supp(T ) of T by def



Supp(T ) =

j ∈ Z: tj = tβ j ̸= 0 in C in Z.

Then the expression (4.23) can be re-written by X

T =

tj β j in Λ.

(4.26)

j∈Supp(T )

Observe that, for k ∈ N, k

 X

Tk = 

tj β j 

j∈Supp(T )

X

k Y

(j1 ,...,jk )∈Supp(T )k

l=1

=

! tjl β jl

by (4.26) =

X

k Y

)∈Supp(T )k

l=1

(j1 ,...,jk

k P

! tjl

jl

β l=1 .

(4.27)

So, if k1 , k2 ∈ N, and xj ∈ X ⊂ Xφ , then  



X

k1 Y

(j1 ,...,jk1 )∈Supp(T )k1

l=1

T k1 xkj 2 = 

=

X

k1 Y

(j1 ,...,jk1 )∈Supp(T )k1

l=1

! tjl

k1 P

 jl

  β l=1  xkj 2

 ! P k1  jl  tjl β l=1 xkj 2 

A C ∗ -Probability Space of |Z| ■ 35

by (4.27) k1 Y

X

=

(j1 ,...,jk1 )∈Supp(T )k1

!



xk2

tjl

j+

l=1

.

k1 P

(4.28)

jl

l=1

By (4.28), one can get the following free-probabilistic information on X. Theorem 4.9. Let T ∈ Λ be a shift operator (4.26), and xj ∈ X, a semicircular element of X. Then there exists zT,k1 ∈ C, such that    φ T k1 xkj 2 (4.29) = ωk2 zT,k1 c k2 , ∀k1 , k2 ∈ N. 2

In particular, X

k1 Y

(j1 ,...,jk1 )∈Supp(T )k1

l=1

zT,k1 =

! tjl

in C.

(4.30)

Proof. Observe that  



φ T k1 xkj 2



X

k1 Y

(j1 ,...,jk1 )∈Supp(T )k1

l=1

= φ

!



xk2

tjl

j+

k1 P

 jl

l=1

by (4.28) k1 Y

X

=

(j1 ,...,jk1 )∈Supp(T )k1

l=1

X

k1 Y

(j1 ,...,jk1 )∈Supp(T )k1

l=1

=

! 



φ xk2

tjl

j+

k1 P

 jl

l=1

! tjl



φ xkj 2



by (4.19)  



X

k1 Y

(j1 ,...,jk1 )∈Supp(T )k1

l=1

= ωk2 c k2  2

! tjl  ,

by the semicircularity of xj ∈ X in Xφ , for all k1 , k2 ∈ N. Therefore, the C-quantity zT,k1 of (4.30), satisfying the freedistributional data (4.29), exists, for all k1 , k2 ∈ N. ■

36 ■ Banach-Space Operators On C ∗ -Probability Spaces

The above theorem illustrates how our shift-operator algebra Λ deform the semicircular law for the free semicircular family X on X, by (4.29). Theorem 4.10. Let Is = (i1 , . . . , is ) ∈ Zs be an arbitrary s-tuple for s Q s ∈ N, and let X[Is ] = xil ∈ Xφ be the corresponding free random l=1

variable. Suppose φ (X[Is ]) = ϖs in C,

(4.31)

determined by Corollary 4.2 (or, (3.22)). If T ∈ Λ be a shift operator (4.28), then there exists zT,k ∈ C, such that
φ T k (X[Is ]) = zT,k ϖs , (4.32) with zT,k =

X

k Y

)∈Supp(T )k

l=1

(j1 ,...,jk

! tjl

in C,

for all k ∈ N. Proof. For an arbitrary s-tuple Is ∈ Zs , the corresponding free random variable X[Is ] ∈ Xφ satisfies its free-distributional data (4.31), characterized by Corollary 4.2. Observe that  !  P k k X Y jl
φ T k (X[Is ]) = tjl φ β l=1 (X[Is ]) (j1 ,...,jk )∈Supp(T )k

l=1

X

k Y

(j1 ,...,jk )∈Supp(T )k

l=1

=

! tjl

φ ((X[Is ]))

by (4.19)  = ϖs 

X

k Y

)∈Supp(T )k

l=1

(j1 ,...,jk

! tjl  .

by (4.31). So, the free-distributional data (4.32) holds.

■

The above formula (4.32) fully characterizes the deformations of original free-distributional data on X, under the action of the shiftoperator algebra Λ.

CHAPTER

5

C ∗ -Probability Spaces (XN , φN )

N SECTION 4, WE STUDIED A C ∗ -probability space X = (X, φ) of (4.4), generated by the free semicircular family X = {xj }j∈Z , and how the integer-shift group λ and the shift-operator algebra Λ act on the Banach space X. In this section, we consider new C ∗ - probability spaces,

I

XN

denote

=

(XN , φN )

for all N ∈ N∞ >1 , generated by mutually free, N -many semicircular elements, N XN = {xj }j=1 . Let (B, ψ) be a C ∗ -probability space containing its unity (or the multiplication-identity) 1B , and suppose there are mutually free, N many semicircular elements, Y = {y1 , . . . , yN }

in

(B, ψ) ,

∗ for N ∈ N∞ >1 . Then one can define the C -subalgebra Y of B by

Y = C ∗ (Y ) = C [Y ]

in B.

(5.1)

For this C ∗ -subalgebra Y, one can have the corresponding freeprobabilistic sub-structure, Y

denote

=

(Y, ψ = ψ |Y ) of (B, ψ) ,

(5.2)

N

generated by the free semicircular family Y = {yj }j=1 . DOI: 10.1201/9781003263487-5

37

38 ■ Banach-Space Operators On C ∗ -Probability Spaces

Lemma 5.1. Let Y be a C ∗ -subalgebra (5.1) of B. Then     N ∗−iso ∗−iso N in B, Y = ⋆ C [{yj }] = C ⋆ {yj } j=1

j=1

(5.3)

where (⋆) in the first ∗-isomorphic relation of (5.3) is the freeprobabilistic free product, and (⋆) in the second ∗-isomorphic relation of (5.3) is the pure-algebraic free product, inducing the N

noncommutative free words in Y = ∪ {yj }. j=1

Proof. By the assumption that Y is a free family in (B, ψ), the first ∗isomorphic relation of (5.3) holds. Since all noncommutative free words in Y have their unique free reduced word forms, as operators of B, the second ∗-isomorphic relation of (5.3) holds. ■ Also, one can get the following result. Lemma 5.2. Let Y be the C ∗ -probability space (5.2), and s let Is = (i1 , . . . , is ) be an s-tuple of {1, . . . , N } , for s ∈ N. If s Q yil ∈ Y is the corresponding free random variable of Y , then Y [Is ] = l=1

the free-distributional data of Y [Is ] is determined by (3.22), or (3.25). Proof. The structure theorem (5.3) proves this lemma by (3.22).

■

By the above lemmas, one can get the following relation. Theorem 5.3. Let Y be a C ∗ -probability space (5.2), and X, the C ∗ probability space (4.4). If φ (1X ) = ψ (1B ) in C, then Y

free−homo

−→

(5.4)

X.

Proof. Let Y be the C ∗ -algebra (5.1), and let X be the C ∗ -algebra (4.3). Then, by (4.5) and (5.3), one can define a multiplicative linear morphism, Ψ : Y → X, satisfying Ψ (yi ) = xi , (5.5) for all i = 1, . . . , N , where yi ∈ Y are the generating semicircular elements of Y, and X = {xi }i∈Z is the free semicircular family generating our C ∗ -probability space X of (4.4).

C ∗ -Probability Spaces (XN , φN ) ■ 39

Then this morphism Ψ of (5.5) is an embedding (and hence, injective) ∗- homomorphism from Y into X. Moreover, it satisfies that φ (Ψ (1B )) = φ (1X ) = ψ (1B ) , by assumption, and φ (Ψ (yin )) = φ (xni ) = ωn c n2 = ψ (yin ) , for all n ∈ N, for all i = 1, . . . , N . Therefore, φ (Ψ (T )) = ψ (T ) , ∀T ∈ Y, by Corollary 4.2, and Lemma 5.2. More precisely, this freehomomorphism Ψ forms a free-isomorphism from N

B onto ⋆ C [{xj }] j=1

free−homo

−→

X.

Therefore, the embedding ∗-homomorphism Ψ of (5.5) is a freehomomorphism from Y into X, equivalently, the relation (5.4) holds. ■ The proof of the structure theorem (5.4) shows that there exists a free-probabilistic sub-structure, ! N

⋆ C[{xi }], φ | N

⋆ C[{xi }]

i=1

of X,

i=1

free-isomorphic to Y, whenever ψ (1B ) = φ (1X ). Define now a C ∗ -subalgebra XN of X by def N

XN = ⋆ C[{xi }], i=1

and the corresponding C ∗ -probability space, XN

denote

=

(XN , φN = φ |XN ) .

(5.6)

i.e., the C ∗ -probability space XN of (5.6) is a free-probabilistic sub-structure of X, generated by the free semicircular (sub-)family N XN = {xj }j=1 (of X). Remark that such a C ∗ -probability space XN contains its unity 1XN , identical to the unity 1X = 1A of X, for all N ∈ N∞ >1 .

40 ■ Banach-Space Operators On C ∗ -Probability Spaces

Corollary 5.1. Let (SN , ψN ) be a C ∗ -probability space generated by mutually free, N -many semicircular elements for N ∈ N∞ >1 , and let XN be in the sense of (5.6). If 1N is the unity of SN , and if ψN (1N ) = φN (1XN ) ,

(5.7)

then (SN , ψN )

free−iso

=

XN .

Proof. The proof is done similarly by that of (5.4), and by (5.6).

■

By (5.7), all C ∗ -probability spaces generated by mutually free, N many semicircular elements are free-isomorphic to the C ∗ -probability space XN of (5.6), whenever the free distributions of unities are ∗ identical, for all N ∈ N∞ >1 . Therefore, we now regard the C -probability spaces XN not only as free-probabilistic sub-structures of X, but also, as a representative of all such C ∗ -probability spaces.

5.1

FREE-DISTRIBUTIONAL DATA ON XN

In this section, we consider free-distributional data on the C ∗ probability spaces XN = (XN , φN ) of (5.6) generated by the free N semicircular families XN = {xj }j=1 , for all N ∈ N∞ >1 . s

Corollary 5.4. Let N ∈ N∞ >1 , and let Is = (i1 , . . . , is ) ∈ {1, . . . , N } s Q be an s-tuple in {1, . . . , N }, for s ∈ N. Let X[Is ] = xis be the l=1

corresponding free random variable (3.13) of XN in the generating free N semicircular family XN = {xj }j=1 . Then ! φN (X[Is ]) =

X

Y

θ∈Πe ([Is ])

V ∈θ

c |V |

,

2

determined by (3.22), where Πe ([Is ]) is in the sense of Notation 3.1. N

Proof. Since the generating elements {xj }j=1 are mutually free and semicircular, it is proven by (3.22), (3.25), (5.4), (5.7) and Lemma 5.2. ■

C ∗ -Probability Spaces (XN , φN ) ■ 41

The above corollary characterizes the free-distributional data on XN , and hence, it characterizes the free-probabilistic information on all C ∗ -probability spaces generated by mutually free, N -many semicircular elements, whose free distributions of unities are identical to φN (1XN ) = φ (1X ) = φ (1A ).

5.2

RESTRICTED ACTIONS αN OF λ ON XN

 Recall that, the integer-shift group λ = β k k∈Z of (4.20) acts naturally on the C ∗ -probability space X of (4.4) by (4.10), and all n-shifts β n ∈ λ are free-isomorphisms on X, for all n ∈ Z, by (4.19). Since free−homo XN −→ X, for all N ∈ N∞ (5.2.1) >1 , by (5.4), one can restrict the action α of λ on X to the actions αN of λ on XN , for all N ∈ N∞ >1 . ∞ Let’s fix N ∈ N>1 , and let XN be a C ∗ -probability space (5.6) N generated by the free semicircular family XN = {xj }j=1 , and let λ be the integer-shift group. Act λ on XN by the action, αN : β n ∈ λ 7−→ αN (β n ) ∈ Hom (XN ) , by the morphism satisfying ( β n (xj ) = xj+n def αN (β n ) (xj ) = 0N

(5.2.2)

if j + n ∈ {1, . . . , N } otherwise,

for xj ∈ XN , for all n ∈ Z, where 0N = 0XN is the zero element of XN . By (5.6) and (5.2.2), indeed, the images αN (β n ) are well-defined ∗-homomorphisms on XN , for all n ∈ Z. i.e., αN (β n ) ∈ Hom (XN ) , ∀β n ∈ λ, where Hom (XN ) is the homomorphism semigroup of all ∗-homomorphisms on XN , equipped with the multiplication on ∗-homomorphisms. n Notation. From below, we denote αN (β n ) of (5.2.2) by βN , for all n β ∈ λ. □ n Lemma 5.5. Suppose N < ∞ in N∞ >1 . A ∗-homomorphism βN is c identical to the ∗-homomorphism ON ∈ Hom (XN ), if and only if

either n ≥ N, or n ≤ −N,

(5.2.3)

42 ■ Banach-Space Operators On C ∗ -Probability Spaces

where c ON

 t · 1N (y) = 0N

if y = t · 1N , for t ∈ C otherwise,

where 1N = 1XN is the unity of XN . k Proof. Let N < ∞ in N∞ >1 . Since βN ∈ Hom (XN ) \ {ON }, where ON ∈ Hom (XN ) is the zero ∗-homomoprhism on XN , we have k βN (t · 1N ) = t · 1N , ∀t ∈ C,

for all k ∈ Z. Indeed, all nonzero ∗-homomorphisms on a unital topological ∗-algebra preserve the unity. (⇐) Suppose the condition (5.2.3) holds in Z. Then, for any generating semicircular elements xj ∈ XN of XN , for j = 1, . . . , N , one has β n (xj ) = xj+n “in X, ” satisfying j + n ∈ / {1, . . . , N }, and hence, n βN (xj ) = 0N in XN ,

by (5.2.2). So, if the condition (5.2.3) holds, then the restricted n-shift n βN is identified with the zero ∗-homomorphism ON of Hom (XN ). (⇒) Assume now that −N < n < N in Z. Then there always exists at least one j0 ∈ {1, . . . , N } such that j0 + n ∈ {1, . . . , N } , and hence, n βN (xj0 ) = xj0 +n ∈ XN in XN . n i.e., if the condition (5.2.3) does not hold, then βN ̸= ON in Hom (XN ). ■ n Meanwhile, if N = ∞ in N∞ >1 , then all restricted n-shifts {βN }n∈Z are nontrivial in Hom (XN ). n Lemma 5.6. Every restricted n-shift β∞ ∈ Hom (X∞ ) satisfies not n only β∞ ̸= ON , but also n c β∞ ̸= O∞ in Hom (X∞ ) ,

(5.2.4)

c where O∞ is in the sense of (5.2.3), where N = ∞. i.e., every restricted n n-shift β∞ has nonzero images on the generating free semicircular family XN = {xj }j∈N .

C ∗ -Probability Spaces (XN , φN ) ■ 43

Proof. It is not difficult to check that, for any arbitrarily fixed n ∈ Z, there always exists j ∈ N, such that j + n ∈ N, n by the Zorn’s lemma. It implies that, for any restricted shifts β∞ , there always exist generating semicircular elements xj ∈ X∞ of X∞ , such that n β∞ (xj ) = xj+n ∈ X∞ in X∞ ,

and hence, n c β∞ ̸= O∞ in Hom (X∞ ) .

Therefore, the relation (5.2.4) holds.

■

n The above two lemmas show that the restricted shifts βN of β n ∈ λ are contained in the subset, either n c {βN : n = −(N − 1), . . . , −1, 0, 1, . . . , N − 1} ∪ {ON },

(5.2.5)

or n {β∞ : n ∈ Z} ,

of Hom (XN ), for any N ∈ N∞ >1 , by (5.2.3) and (5.2.4). Definition 5.1. For N ∈ N∞ >1 , define a subset λN of Hom (XN ) by def

n λN = {βN = αN (β n ) ∈ Hom (XN ) : β n ∈ λ} .

(5.2.6)

We call the set λN of (5.2.6), the (N -)restricted-shift family (of λ on XN ). Notation. From below, we write each restricted-shift family λN of (5.2.6) set-theoretically, as follows;  k N λN = βN . k=−N This notation contains all set-theoretical information (5.2.5) of λN very −N N c well, because if N < ∞, then βN = ON = βN in λN (by (5.2.3)),  k N and hence, the above set-expression λN = βN k=−N satisfies the first case (5.2.5); meanwhile, if N = ∞, then the above notation  kof ∞ λ∞ = βN k=−∞ is nothing but the set λ∞ of the second case of (5.2.5). □

44 ■ Banach-Space Operators On C ∗ -Probability Spaces j −j By (5.2.2), if βN , βN ∈ λN in Hom (XN ), then new ∗-homomorphisms, j −j −j j βN βN , βN βN ∈ Hom (XN ) ,

are obtained under the product of ∗-homomorphisms, for j ∈ {1, . . . , N }, denote 0 where βN = 1N = 1XN , the identity map of Hom (XN ), satisfying −0 0 0 −0 0 βN βN = βN = βN βN , in Hom (XN ) .

If xj ∈ XN in XN , for j ∈ {1, . . . , N }, satisfying 1 ≤ 2j ≤ N , then −j j −j −j βN βN (xj ) = βN (xj+j ) = βN (x2j ) = x2j−j = xj ,

(5.2.7)

and j −j j j βN βN (xj ) = βN (xj−j ) = βN (0N ) = 0N ,

in XN . These computations (5.2.7) illustrate that j −j −j j βN βN ̸= βN βN in Hom (XN ) ,

(5.2.8)

and j −j −j j βN βN , βN βN ∈ / λN ,

in general (especially, where k ̸= 0), by (5.2.5) and (5.2.6). For example, −1 1 1 −1 β∞ β∞ (x1 ) = x1 , but β∞ β∞ (x1 ) = 0∞ ,

in X∞ . The relations of (5.2.8) implies that n1 n2 n2 n1 βN βN ̸= βN βN in Hom (XN ) ,

in general (especially, whenever neither n1 = 0 nor n2 = 0 in {0, ±1, . . . , ±N }) by (5.2.7), and hence, n1 n2 n2 n1 βN βN , βN βN ∈ / λN ,

in general (especially, whenever neither n1 = 0 nor n2 = 0 in {0, ±1, . . . , ±N }), for n1 , n2 ∈ {0, ±1, . . . , ±N }. Remark that, only if either n1 = 0, or n2 = 0 in {0, ±1, . . . , ±N }, then the above inclusion holds, because
0 βN = αN β 0 = IN ∈ Hom (XN ) ,

C ∗ -Probability Spaces (XN , φN ) ■ 45

where IN is the identity map on XN , and hence, n 0 n 0 n βN βN = βN = βN βN ∈ λN , ∀n ∈ {0, ±1, . . . , ±N } . n1 n2 n2 n1 However, in general, βN βN ̸= βN βN in Hom {XN }, and they are not contained in the restricted-shift family λN . i.e., our restricted-shift  k N family λN = βN does not form an algebraic sub-structure of k=−N the homomorphism semigroup Hom (XN ); it is simply a subset (5.2.5) of Hom (XN ) by (5.2.8).  k N Proposition 5.7. Let λN = βN be the restricted-shift family k=−N in Hom (XN ). Then n Y

kl βN

l=1

and

n Y

n Y k ̸= βNσ(l) in Hom (XN ) ,

(5.2.9)

l=1

kl βN ∈ / λN , for all n > 1 in N,

l=1

in Hom (XN ), where k1 , . . . , kn are mutually distinct from each other “in {±1, . . . , ±(N − 1)},” and σ ∈ Sn \ {en } are the “nontrivial” permutations over {1, . . . , n}, where en is the group identity of Sn . Proof. The relations in (5.2.9) are obtained inductively by the discussion of the very above paragraph, e.g., see (5.2.8). ■ The above proposition shows that, different from the integer-shift group λ ⊂ Aut (Xφ ), our restricted-shift family λN ⊂ Hom (XN ) is not an algebraic structure. However, since λN ⊂ Hom (XN ) ⊂ B (XN ) , where B (XN ) is the operator space of [13], consisting of all bounded linear transformations on XN (by regarding the C ∗ -algebra XN as a Banach space equipped with its C ∗ -norm), one can define the polynomial algebra C[λN ], embedded in the operator space B (XN ), generated by λN . Define ΛN by def

ΛN = C [λN ] in B (XN ) ,

(5.2.10)

46 ■ Banach-Space Operators On C ∗ -Probability Spaces

where Z are the operator-norm-topology closures of subsets Z of B (XN ). Since the products of restricted shifts in λN are ∗-homomorphisms of Hom (XN ), they are well-defined Banach-space operators of B (XN ) acting on the Banach space XN , and hence, the limits of linear combinations of such products in ΛN are well-defined operators on XN . i.e., the Banach algebra ΛN of (5.2.10) is well-determined in the operator space B (XN ). By (5.2.10), every element T ∈ ΛN has its expression,   ∞ X X 0  T = t0 βN + tK WK  , with t0 , tK ∈ C, n=1

K=(k1 ,...,kn )∈{±1,...,±N }n

(5.2.11) where WK =

n Y

kl βN ∈ Hom (XN ) ⊂ ΛN .

l=1

Define now a unary operation (∗) on the Banach algebra ΛN by   ∞ X X def 0 ∗  , (5.2.12) T ∗ = t0 βN + tK WK n=1

K=(k1 ,...,kn )∈{±1,...,±N }n

with ∗ WK

n Y

=

l=1

!∗ kl βN

def

=

n Y

−kn−l+1

βN

,

l=1

in ΛN , where T ∈ ΛN is in the sense of (5.2.11). By the very definition (5.2.12), one can check that T ∗∗ = T, for all T ∈ ΛN , and ∗

(tT ) = tT ∗ , ∀t ∈ C, T ∈ ΛN , and ∗

(T1 + T2 ) = T1∗ + T2∗ , ∀T1 , T2 ∈ ΛN , and ∗

(T1 T2 ) = T2∗ T1∗ , ∀T1 , T2 ∈ ΛN , showing that this operation (∗) is a well-defined adjoint on ΛN .

C ∗ -Probability Spaces (XN , φN ) ■ 47

Proposition 5.8. The set ΛN of (5.2.10) is a Banach ∗-algebra embedded in the operator space B (XN ). Equivalently, all elements T ∈ ΛN are adjointable (in the sense of [13]) on XN (or, in B (XN )). Proof. The subset ΛN is an embedded Banach algebra in B (XN ) by the very definition (5.2.10), and the adjoint (∗) of (5.2.12) is well-defined on ΛN . Therefore, the set ΛN forms a Banach ∗-algebra in B (XN ). Equivalently, every Banach-space operator T ∈ ΛN of (5.2.11) has its unique adjoint T ∗ of (5.2.12) “in ΛN ,” and hence, it is adjointable in ΛN (in the sense of [13]). ■ Definition 5.2. Let λN be the restricted-shift family (5.2.6), and let ΛN be the Banach ∗-algebra (5.2.10) generated by λN in the operator space B (XN ). Then we call ΛN , the (restricted-)shift-operator (Banach-∗-)algebra (acting on XN ). All Banach-space operators T ∈ ΛN of (5.2.11) are called the (restricted-)shift operators on XN (or, in B (XN )). c Let ON ∈ Hom (XN ) ⊂ B (XN ) be the Banach-space operator of (5.2.3). Then, by definition, it satisfies ∗

2

c c c (ON ) = ON = (ON ) on XN .

5.3

(5.2.13)

ACTION OF ΛN ON XN

In this section, we consider how the restricted-shift family λN of (5.2.6), and the corresponding shift-operator algebra ΛN of (5.2.10) act on the C ∗ -probability space XN = (XN , φN ), for an arbitrarily fixed N ∈ N∞ >1 . Lemma 5.9. If xj ∈ XN is a generating semicircular element of XN , then (


ωk c k = φN xkj if 1 ≤ j + n ≤ N n k 2 φN βN xj = (5.3.1) 0 otherwise, n for all βN ∈ λN , for all k ∈ N. n Proof. Let βN ∈ λN , for n ∈ {0, ±1, . . . , ±N }, and let xj ∈ XN in XN . Then (


φN xkj+n if 1 ≤ j + n ≤ N n k φN βN xj = φN (0N ) otherwise,

48 ■ Banach-Space Operators On C ∗ -Probability Spaces

by (5.2.2), for all k ∈ N. Therefore, the free-distributional data (5.3.1) holds. ■ The above lemma characterizes how our restricted-shift family λN deforms the semicircular law for the generating free semicircular family XN of XN . Also, one obtains the following result by Corollary 5.4 and Lemma 5.9. Theorem 5.10. Let Is = (i1 , . . . , is ) be an s-tuple in {1, . . . , N }, for s Q s ∈ N, and let X[Is ] = xil ∈ XN be the corresponding free random l=1

n variable in XN . If βN ∈ λN is a restricted shift, then

  φ (X[I ]) N s n φN (βN (X[Is ])) =   0

if 1 ≤ il + n ≤ N for all l = 1, . . . , s otherwise,

(5.3.2)

where φN (X[Is ]) is determined by Corollary 5.4. Proof. If i1 = i2 = . . . = is , say j, in {1, . . . , N }, then n n φN (βN (X[Is ])) = φN βN xsj





,

n is characterized by (5.3.1), for all s ∈ N. So, for any βN ∈ λN , the free-distributional data (5.3.2) holds. If i1 , . . . , is are mixed in {1, . . . , N }, for s ∈ N \ {1}, in the sense that: there exists at least one ip ∈ {i1 , . . . , is }, such that ip ̸= ik , for some k ̸= p in {1, . . . , s}. Then there exists an s-tuple Js = (i1 + n, . . . , is + n), such that n βN (X[Is ]) = X[Js ] in XN ,

where X[Js ] is in the sense of (3.13). Then X[Js ] is nonzero in XN , if and only if 1 ≤ il + n ≤ N, ∀l = 1, . . . , s, by (5.2.2) and (5.3.1). So, in such a mixed case, the free-distributional data (5.3.2) also holds, by Corollary 5.4. Therefore, the free-distributional data (5.3.2) holds. ■

C ∗ -Probability Spaces (XN , φN ) ■ 49

We now are interested in a certain type of shift operators T of ΛN , X n n ∈ C. T = tn βN ∈ ΛN , with tn = tβN (5.3.3) n ∈λ βN N

with their adjoints T ∗ , X

T∗ =

−n tn βN , in ΛN .

n ∈λ βN N

(Remark that, in general, shift operators of ΛN have their expressions (5.2.11).) Lemma 5.11. If T ∈ ΛN be a shift operator (5.3.3) on XN , and if xj ∈ XN is a generating semicircular element of XN , then xkj

T




=

N −j X

tl xkj+l ∈ XN , for all k ∈ N.

(5.3.4)

l=1−j

Proof. Observe that, for k ∈ N, N X


k

T xj =

! n tn βN

xkj




n=−N

=

N −j X

=

N X

n tn βN xkj




n=−N

tn xkj+n ,

n=1−j

in XN , since n βN

xkj




( xkj+n = 0N

if 1 ≤ j + n ≤ N otherwise,

in XN , for all k ∈ N, and n ∈ {0, ±1, . . . , ±N }. Therefore, the formula (5.3.4) holds.

■

By (5.3.4), we obtain the following free-distributional data on XN . Theorem 5.12. Let T ∈ ΛN be a shift operator (5.3.3), and let xj ∈ XN be a generating semicircular element of XN . Then   N −j X


φN T xnj = ωn c n2  tl  , ∀n ∈ N. (5.3.5) l=1−j

50 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. Observe that

φN

  N −j X


l T xnj = φN  tl βN xnj  l=1−j

by (5.3.4)

=

N −j X

tl φN xnj+l





=

N −j X

tl φN xnj





l=1−j

l=1−j

by (5.3.1) 
n

= φN xj 



N −j X

tl  ,

l=1−j

for all n ∈ N. So, the free-distributional data (5.3.5) holds by the semicircularity of xj ∈ XN in XN . ■ The above theorem shows how a shift operator T ∈ ΛN of (5.3.3) deforms the semicircular law for the free semicircular family XN in XN by (5.3.5). More generally, one can get the following result. s

Theorem 5.13. Let W = (i1 , . . . , is ) ∈ {1, . . . , N } be an s-tuple for s Q s ∈ N, and let XW = xil ∈ XN be the corresponding free random l=1

variable, satisfying φN (XW ) = ϖW in C determined by Corollary 5.4. If T ∈ ΛN is a shift operator (5.3.3), then there exists wT ∈ C, such that φN (T (XW )) = wT ϖW in C. (5.3.6) Proof. Consider that T (XW ) =

N X

n tn βN

n=−N

=

N X

n=−N

s Y

! xil

l=1

tn

t

s Y l=1

xil +n

|!

,

(5.3.7)

C ∗ -Probability Spaces (XN , φN ) ■ 51

where t

s Y

xil +n

 s  Q x i +n = l=1 l   0N

|

l=1

if −N ≤ il + n ≤ N for all l = 1, . . . , s otherwise,

in XN . Thus, one has that N X

φN (T (XW )) = φN

t s Y

tn

n=−N

xil +n

|!!

l=1

by (5.3.7) =

N X

tn φN

t s Y

n=−N

xil +n

|!

l=1

X

=

n∈{0,±1,...,±N },

tn φN (XW )

s

s Q

{ xil +n ̸=0N

l=1

by (5.3.2)  = ϖW



   

X

n∈{0,±1,...,±N },

s

s Q

{

xil +n ̸=0N

  tn  . 

(5.3.8)

l=1

Therefore, there exists wT =

X n∈{0,±1,...,±N },

tn ∈ C,

s

s Q

{

xil +n ̸=0N

l=1

such that φN (T (XW )) = wT ϖW in C, by (5.3.8). Therefore, the free-distributional data (5.3.6) holds.

■

The above theorem illustrates how our shift operators T ∈ ΛN of (5.3.3) deform the free probability on XN by (5.3.6).

52 ■ Banach-Space Operators On C ∗ -Probability Spaces

Now, consider a shift operator, n Y k kl k1 k2 kn W = βN = βN βN . . . βNn−1 βN ∈ ΛN ,

(5.3.9)

l=1

with its adjoint, ∗

W =

n Y

−kn−l+1

βN

−kn−1

−kn = βN βN

−k2 −k1 . . . βN βN ,

l=1

in ΛN , by (5.2.12). Lemma 5.14. If W ∈ ΛN is a shift operator (5.3.9), and xj ∈ XN is a generating semicircular element of XN , then  n P   if 1 ≤ j+ kl ≤ N,    l=i  
xs P for all i = 1, . . . , n, and n s W xj = (5.3.10) j+ kl n P  l=1  −N ≤ k ≤ N  l   l=1    0N otherwise, and

W∗

        xs
n P s xj = j− kl  l=1       0 N

i P

if 1 ≤ j−

kl ≤ N,

l=1

for all i = 1, . . . , n, and n P −N ≤ − kl ≤ N l=1

otherwise,

in XN , for all s ∈ N. Proof. The proof of (5.3.10) is done by the induction on (5.2.2). Indeed, 

 q y
k k k1 kn k1 W xsj = βN . . . βNn−1 βN xsj = βN . . . βNn−1 xsj+kn where q

xsj+kn

y

( xsj+kn = 0N

if 1 ≤ j + kn ≤ N otherwise,

C ∗ -Probability Spaces (XN , φN ) ■ 53

and hence, it goes to k

k1 = βN . . . βNn−2

r z Jxj+kn K+kn−1

where

r

z

Jxj+kn K+kn−1 =

     xs

j+kn +kn−1

    0

N

= ...         xs n P = j+ kl  l=1       0 N

for all s ∈ N. Similarly, one can get that         xs
n P ∗ s W xj = j− kl  l=1       0 N

if 1 ≤ j + kn ≤ N, and 1 ≤ j + kn + kn−1 ≤ N, and −N ≤ kn + kn−1 ≤ N otherwise, n P

if 1 ≤ j+

kl ≤ N,

l=i

for all i = 1, . . . , n, and n P −N ≤ kl ≤ N l=1

otherwise,

i P

if 1 ≤ j−

kl ≤ N,

l=1

for all i = 1, . . . , n, and n P −N ≤ − kl ≤ N l=1

otherwise,

for all s ∈ N. So, the formula (5.3.10) holds.

■

By the above lemma, we obtain the following result. Theorem 5.15. Let W ∈ ΛN be a shift operator (5.3.9). If xj ∈ XN is a generating semicircular element of XN , then  n P   if 1 ≤ j+ kl ≤ N,    l=i  

ωs c 2s for all i = 1, . . . , n, and (5.3.11) φN W xsj = n P   k ≤ N −N ≤  l   l=1   0 otherwise,

54 ■ Banach-Space Operators On C ∗ -Probability Spaces

and

φN W ∗

        ω c s

s 2 s xj =        0

i P

if 1 ≤ j−

kl ≤ N,

l=1

for all i = 1, . . . , n, and n P −N ≤ − kl ≤ N l=1

otherwise,

for all s ∈ N. Proof. The free-distributional data (5.3.11) are obtained by (5.3.10). ■ The above theorem characterizes how the shift operators of (5.3.9) deform the original free probability on XN , by (5.3.11). So, the freedistributional data (5.3.2), (5.3.6) and (5.3.11) illustrate how our shiftoperator algebra ΛN affect the free probability on XN , in terms of (5.2.11) and (5.2.12).

5.4

FREE-HOMOMORPHIC RELATIONS ON {XN }N ∈N∞

>1

N∞ >1 ,

∗

For any N ∈ one can have the C -probability space XN = (XN , φN ) of (5.6), generated by the free semicircular family N XN = {xj }j=1 , as a free-probabilistic sub-structures of the C ∗ probability space X = (X, φ) of (4.4), generated by the free semicircular family X = {xj }j∈Z . In this section, we study natural freehomomorphic relations on {XN }N ∈N∞ . >1

Theorem 5.16. If N1 ≤ N2 ≤ · · · ≤ Nn in N∞ >1 , for n ∈ N \ {1}, then XN1

free−homo

−→

XN 2

free−homo

−→

...

free−homo

−→

XNn .

Proof. By (4.19), (5.6) and (5.7), if N1 ≤ N2 ≤ · · · ≤ Nn in N∞ >1 , for n ∈ N \ {1}, then XN1

free−homo

−→

XN2

free−homo

−→

...

free−homo

−→

XNn .

(5.4.1)

C ∗ -Probability Spaces (XN , φN ) ■ 55

Indeed, for any Ni ≤ Ni+1 , one can define a ∗-homomorphism, Θi : XNi → XNi+1 by Θi (T ) = T in XNi+1 , ∀T ∈ XNi , as the canonical embedding, satisfying that φNi+1 (Θi (T )) = φNi+1 (T ) = φNi (T ) = φ (T ) , for all T ∈ XNi , by (5.6) and Corollary 5.4, for i = 1, . . . , n − 1.

■

The above free-homomorphic relation (5.4.1) can be naturally extendable to XN 1

free−homo

−→

...

free−homo

−→

XNn

free−homo

whenever N1 ≤ · · · ≤ Nn in N∞ >1 , for all n ∈ N.

−→

X,

CHAPTER

6

Adjointable Banach-Space Operators Acting on XN

I

N SECTION 4, WE STUDIED HOW THE INTEGERSHIFT-OPERATOR ALGEBRA Λ generated by the integershift group λ acts on the C ∗ -probability space X = (X, φ) generated by the free semicircular family X = {xj }j∈Z . In particular, we showed that all integer shifts β k ∈ λ are free-isomorphisms on X, as Banachspace operators on X. Meanwhile, in Section 5, it is shown that the k restrictions, the restricted shifts βN ∈ λN of β k ∈ λ “partially” preserve the original free-distributional data on the C ∗ -probability space XN = (XN , φN ), generated by the free semicircular family N XN = {xj }j=1 , which is a free-probabilistic sub-structure of X, for all N ∈ N∞ >1 . Especially, different from the integer-shift group λ, the restricted-shift family λN does not form an algebraic structure in the homomorphism semigroup Hom (XN ), and hence, we considered the restricted-shift-operator algebra ΛN generated by λN , in the operator space B (XN ), for N ∈ N∞ >1 . In this section, we fix N ∈ N∞ >1 and concentrate on studying how certain adjointable Banach-space operators of ΛN acts on XN , and how they deform the free probability on XN . In particular, we are interested in certain types of operators of ΛN .

DOI: 10.1201/9781003263487-6

57

58 ■ Banach-Space Operators On C ∗ -Probability Spaces

6.1

PROJECTIONS OF ΛN

Let ΛN be the (restricted-)shift-operator algebra, which is a Banach  k N ∗-algebra generated by the restricted-shift family λN = βN , k=−N embedded in the operator space B (XN ). It is considered that, for each j ∈ {1, . . . , N }, satisfying 1 ≤ 2j ≤ N , −j j βN βN (xj ) = xj = IN (xj ) ,

(6.1.1)

but j −j c βN βN (xj ) = 0N = ON (xj ) ,

by (5.2.8), where xj ∈ XN is the j -th generating semicircular element of XN , and IN ∈ Aut (XN ) ⊆ Hom (XN ) in B (XN ) c are the identity operator of B (XN ), and ON is in the sense of (5.2.3). By the adjoint (5.2.12) on ΛN , the above relation (6.1.1) can be restated by that: if j, 2j ∈ {1, . . . , N }, then  ∗  ∗ j j j j c βN βN (xj ) = IN (xj ) , βN βN (xj ) = ON (xj ) ,

in XN . More general to (6.1.1), we have the generalized result, i.e., k1 k2 k2 k1 βN βN ̸= βN βN in ΛN ⊂ B (XN ),

(6.1.2)

and k1 k2 k2 k1 βN βN , βN βN ∈ / λN in ΛN ⊂ B (XN ),

in general, by (5.2.9). In particular, if neither k1 = 0, nor k2 = 0 in {0, ±1, . . . , ±(N − 1)}, then the non-equality, and the non-inclusion of (6.1.2) always hold; while, if either k1 = 0, or k2 = 0, then they do not 0 hold as in (6.1.2), since βN = IN in B (XN ). Motivated by (6.1.2), we now consider a certain type of adjointable Banach-space operators of ΛN . Definition 6.1. Let B be an arbitrary topological ∗-algebra. An operator y ∈ B is a projection, if y ∗ = y = y 2 in B.

Adjointable Banach-Space Operators ■ 59

By definition, an operator y ∈ B is a projection, if and only if (i) it is self-adjoint, and (ii) it is idempotent in the sense that: y 2 = y in B. Remark that our projections of Definition 6.1 is a generalized concept of projections acting on Hilbert spaces (e.g., [14]). k Theorem 6.1. If βN ∈ λN is a generating operator of ΛN , for

k ≥ 0, in {0, ±1, . . . , ±N } , −k k k −k then βN βN and βN βN are projections in ΛN . i.e., −k k k k −k βN ∈ λN =⇒ βN βN , β N βN are projections in ΛN ,

(6.1.3)

for all k ∈ {0, ±1, . . . , ±N }. −0 0 Proof. Assume first that k = 0. Then βN = IN = βN , the identity operator of ΛN , and hence, −0 0 0 −0 βN βN = IN = βN βN ∗ 2 are projections in ΛN , since IN = IN = IN in ΛN . If N < ∞, and k c k = ±N , then βN = ON , and hence, (6.1.3) holds. The case where N = ∞ = N − 1 is covered by the following proof processes. k Suppose now that k > 0 in {0, ±1, . . . , ±(N − 1)}, and let β = βN ∈ λN in ΛN . Then, for any generating semicircular element xj ∈ X of XN , one has ( β ∗ (xj+k ) = xj if 1 ≤ j + k ≤ N ∗ β β (xj ) = (6.1.4) β ∗ (0N ) = 0N otherwise,

and

( ∗

ββ (xj ) =

if 1 ≤ j − k ≤ N otherwise,

β (xj−k ) = xj β (0N ) = 0N

in XN . Now, let T = β ∗ β ∈ ΛN . Then, by (5.6) and (6.1.4), one can get that T = IN , on

N −k

⋆ C [{xl }]

denote

l=1

=

[1,N −k]

XN

,

(6.1.5)

meanwhile, c T = ON , on

N

⋆

l=N −k+1

C [{xl }]

denote

=

[N −k+1,N ]

XN

,

60 ■ Banach-Space Operators On C ∗ -Probability Spaces

in ΛN . Similarly, if we let S = ββ ∗ ∈ ΛN , then S = IN , on

N

⋆

l=k+1

C [{xl }]

denote

=

[k+1,N ]

XN

,

(6.1.6)

meanwhile, k

c S = ON , on ⋆ C [{xl }] l=1

denote

=

[1,k]

XN ,

in ΛN . i.e., −k k βN βN

( IN = c ON

and k −k βN βN

[1,N −k]

on XN [N −k+1,N ] on XN ,

( IN = c ON

(6.1.7)

[k+1,N ]

on XN [1,k] on XN ,

for all k ∈ {±1, . . . , ±(N − 1)}, by (6.1.5) and (6.1.6). By (5.2.13) and (6.1.7), one can get that ( [1,N −k] ∗ 2 IN = IN = IN on XN ∗ 2 T =T =T = [N −k+1,N ] c ∗ c c 2 (ON ) = ON = (ON ) on XN ,

(6.1.8)

and ( ∗ 2 IN = IN = IN S∗ = S = S2 = ∗ c c c 2 (ON ) = ON = (ON )

[k+1,N ]

on XN [1,k] on XN .

Note that, by (5.6), N

XN = ⋆ C [{xj }], j=1

and hence,     [1,N −k] [N −k+1,N ] XN = XN ⋆ XN , and

    [1,k] [k+1,N ] XN = XN ⋆ XN .

Therefore, by (6.1.8) and (6.1.9), T ∗ = T = T 2 on XN ,

(6.1.9)

Adjointable Banach-Space Operators ■ 61

and S ∗ = S = S 2 on XN , −k k k −k where T = βN βN , and S = βN βN in ΛN . So, the shift operators T and S are projections in ΛN . i.e., the statement (6.1.3) holds. ■

The following corollary is a direct consequence of (6.1.3). k Corollary 6.2. Let βN ∈ λN be a restricted shift in ΛN , for k ≥ 0 in {0, ±1, . . . , ±N }. Then     −k k c βN βN = IN |X[1,N −k] ⋆ ON |X[N −k+1,N ] , (6.1.10) N

and

N

    k −k c βN βN = IN |X[k+1,N ] ⋆ ON |X[1,k] , N

N

are projections in ΛN , where T |Y are the restrictions of shift operators T ∈ ΛN to the ∗-subalgebras Y (which are subspaces) of (a Banach space) XN , and n2 [n ,n ] XN 1 2 = ⋆ C [{xl }] in XN , l=n1

for all n1 ≤ n2 ∈ {1, . . . , N }, and where T (a) = ((T1 |Y1 ) ⋆ (T2 |Y2 )) (a) = (T1 (a1 )) (T2 (a2 )) , whenever XN = Y1 ⋆ Y2 , and a = a1 a2 ∈ XN a free word in a1 ∈ Y1 , and a2 ∈ Y2 . Proof. The operator-equality (6.1.10) in ΛN is obtained by (6.1.3), (6.1.8) and (6.1.9). Remark that, if k = 0, then −0 0 0 −0 0 βN βN = βN βN = βN = IN |X[1,N ] = IN , N

in ΛN , satisfying (6.1.10), too.

■

−k k The operator-equality (6.1.10) illustrates not only that βN βN , ∈ ΛN are projections in ΛN but also that they have their domains (which are identical to their ranges in the opposite order), k −k βN βN

[1,N −k]

XN

[k+1,N ]

, and XN

, in XN ,

respectively, re-characterizing (6.1.3). Also, the equality (6.1.10) gives the following result.

62 ■ Banach-Space Operators On C ∗ -Probability Spaces k Corollary 6.3. Let βN ∈ λN be a restricted shift, as a shift operator of ΛN , for k ≥ 0 in {0, ±1, . . . , ±N }. Then −k −k k −k k k −k k βN = βN βN βN , and βN = βN βN βN ,

(6.1.11)

in ΛN . Proof. Observe that, for any k ≥ 0 in {0, ±1, . . . , ±N },
k
k k −k k k −k k βN βN βN = βN βN βN = Q[k+1,N ] βN = βN , or equivalently,

−k k k −k k k k k βN βN βN = βN βN βN = βN Q[1,N −k] = βN , on XN , by (5.2.5) and (6.1.10), where     c Q[k+1,N ] = IN |X[k+1,N ] ⋆ ON |X[1,k] , N

and

N

    c Q[1,N −k] = IN |X[1,N −k] ⋆ ON |X[N −k+1,N ] , N

N

by (6.1.10). Similarly, one can get that −k k −k −k βN βN βN = βN , in ΛN .

Therefore, the operator-equality (6.1.11) holds in ΛN .

■

kl Now, let k1 ̸= k2 ≥ 0 in {0, ±1, . . . , ±N }, and let βN ∈ λN be the generating shift operators of ΛN . Then they generate the projections, −kl kl kl −kl Tkl = βN βN , and Skl = βN βN in ΛN ,

for all l = 1, 2, by (6.1.3), and they are identified with     c Tkl = IN |X[1,N −kl ] ⋆ ON |X[N −kl +1,N ] , N

and

N

    c Skl = IN |X[kl +1,N ] ⋆ ON |X[1,kl ] , N

in ΛN , for all l = 1, 2, by (6.1.10) and (6.1.12).

N

(6.1.12)

Adjointable Banach-Space Operators ■ 63

Theorem 6.4. For 0 ≤ k1 ≤ k2 in {0, ±1, . . . , ±N }, let Tkl , Skl ∈ ΛN be the projections (6.1.12), for all l = 1, 2. Then Tk1 Tk2 = Tk2 Tk1 = Tk2 , and Sk1 Sk2 = Sk2 Sk1 = Sk2 , and     c Tk1 Sk2 = Sk2 Tk1 = IN |X]k2 +1,N −k1 [ ⋆ ON |XYN ,

(6.1.13)

N

and

    c Tk2 Sk1 = Sk1 Tk2 = IN |X]k1 +1,N −k2 [ ⋆ ON |XZN , N

where ( [kt + 1, N − ks ] ]kt + 1, N − ks [ = [N − ks , kt + 1]

if kt + 1 ≤ N − ks if kt + 1 ≥ N − ks ,

(6.1.14)

and XYN =

⋆

C [{xj }],

⋆

C [{xj }],

j∈{1,...,N }\]k2 +1,N −k1 [

(6.1.15)

and XZ N =

j∈{1,...,N }\]k1 +1,N −k2 [

in XN , with axiomatization: ı¿œ X¨N = {0N },

where ¨ı¿œ is the empty set in {1, . . . , N }, for all t ̸= s ∈ {1, 2}. Proof. In this proof, we will use the following terminology; if n1 < n2 , and n3 < n4 in {1, . . . , N }, then [n1 , n2 ] ∩ [n3 , n4 ] = [max{n1 , n3 }, min{n2 , n4 }] , and [n1 , n2 ] ∪ [n3 , n4 ] = [min{n1 , n3 }, max{n2 , n4 }], where [n, k] = {n, n + 1, . . . , k}, whenever n < k in {1, . . . , N }. Of course, if max{n1 , n3 } > min{n2 , n4 },

64 ■ Banach-Space Operators On C ∗ -Probability Spaces

then the first resulted interval for ∩ is identified with the empty set ¨ı¿œ. −kl kl Suppose 0 ≤ k1 < k2 in {0, ±1, . . . , ±N }, and let Tkl = βN βN , kl −kl and Skl = βN βN are the projections (6.1.12) of ΛN . Then    Tk1 Tk2 = IN |X[1,N −k1 ] IN |X[1,N −k2 ] N

N

= IN |X[1,N −k1 ]∩[1,N −k2 ] = IN |X[1,N −k2 ] = Tk2 = Tk2 Tk1 , N

N

[1,N −k2 ]

on XN

Tk1 Tk2

in ΛN because    c c = ON |X[N −k1 +1,N ] ON |X[N −k2 +1,N ] N

N

c c = ON |X[N −k1 +1,N ]∪[N −k2 +1,N ] = ON |X[N −k2 +1,N ] = Tk2 = Tk2 Tk1 , N

[N −k2 +1,N ]

on XN

N

in ΛN , by (6.1.10). Similarly, we have that

Sk1 Sk2 = IN |X[k1 +1,N ]∩[k2 +1,N ] = IN |X[k2 +1,N ] = Sk2 = Sk2 Sk1 , N

[1,N −k2 ]

on XN

N

, because

c c Sk1 Sk2 = ON |X[1,k1 1]∪[1,k2 ] = ON |X[1,k2 ] = Sk2 = Sk2 Sk1 , N

N

[N −k +1,N ]

on XN 2 in ΛN , by (6.1.10), whenever k1 < k2 in {1, . . . , N }. Also, one can get that    Tk1 Sk2 = IN |X[1,N −k1 ] IN |X[k2 +1,N ] N

N

= IN |X[1,N −k1 ]∩[k2 +1,N ] = IN |X]k2 +1,N −k1 [ = Sk2 Tk1 , N

N

]k +1,N −k1 [ XN 2 ,

on where ]n1 , n2 [ is in the sense of (6.1.14), for n1 , n2 ∈ {1, . . . , N }, because    c c Tk1 Sk2 = ON |X[N −k1 +1,N ] ON |X[1,N −k2 ] N

=

c ON |X[N −k1 +1,N ]∪[1,N −k2 ] = N

N

c ON |XYN = Sk2 Tk1 ,

on XYN , in ΛN , where Y is in the sense of (6.1.15). Similarly, we have that Tk2 Sk1 = IN |X[1,N −k2 ]∩[k1 +1,N ] = IN |X]k1 +1,N −k2 [ = Sk1 Tk2 , N

N

Adjointable Banach-Space Operators ■ 65 ]k +1,N −k2 [

on XN1

, because

c c Tk2 Sk1 = ON |X[N −k2 +1]∪[1,k1 ] = ON |XZN = Sk1 Tk2 , N

on

XZ N,

in ΛN , where Z is in the sense of (6.1.15). Therefore, the operator-equalities of (6.1.13) holds in ΛN , under the same notation of (6.1.10). ■ Similar to the above theorem, one obtains the following results. k Theorem 6.5. Let βN ∈ λN be a generating shift operator of ΛN on XN , for k ≥ 0 in {0, ±1, . . . , ±N }, and let −k k k −k Tk = βN βN , Sk = βN β N ∈ ΛN

be the corresponding shift operator (6.1.12). Then     c |XZk , Tk Sk = Sk Tk = IN |X]k+1,N −k[ ⋆ ON N

(6.1.16)

N

where Zk = {1, . . . , N } \ ]k + 1, N − k[ . Proof. Let Tk , Sk ∈ ΛN be the given projections, for k ∈ {0, ±1, . . . , ±N }. Then    Tk Sk = IN |X[1,N −k] IN |X[k+1,N ] N

N

= IN |X[1,N −k]∩[k+1,N ] = IN |X]k+1,N −k[ = Sk Tk , N

]k+1,N −k[

on XN

N

, since



c Tk Sk = ON |X[N −k+1,N ] N

[N −k+1,N ]∪[1,k]

on XN (6.1.16) holds.

  c c ON |X[1,k] = ON |X[N −k+1,N ]∪[1,k] = Sk Tk , N

N

, in ΛN . Therefore, the operator-equalities of ■

By the above two theorems, we obtain the following corollary. kl Corollary 6.6. Let βN ∈ λN be a generating shift operator of ΛN acting on XN , where kl ≥ 0 in {0, ±1, . . . , ±N }, and let kl −kl −kl kl β N ∈ ΛN Tkl = βN βN , Skl = βN

be shift operators, for all l = 1, 2. Then the finite products of {Tk1 , Tk2 , Sk1 , Sk2 } are projections in ΛN .

66 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. By (6.1.10),  Tkl =

IN

  c | [nl,1 nl,2 ] ⋆ ON | XN

 ,

[n ,n ]c XN l,1 l,2

and  Skl =

IN

  c | [ul,1 ,ul2 ] ⋆ ON | XN

 [u

XN l,1

,ul,2 ]c

,

for some nl,1 , nl,2 ul,1 , ul,2 ∈ {1, . . . , N }, for all l = 1, 2, where [n, k]c = {1, . . . , N } \ [n, k], ∀n < k ∈ {1, . . . , N } . i.e., all shift operators Tkl ,Skl ∈ ΛN , say A, have their forms, 

A = IN

  c |XJA ⋆ ON |



Ic XNA

N

in ΛN ,

where JA are the corresponding intervals of {1, . . . , N }, and c JA = {1, . . . , N } \ JA .

Moreover, by (6.1.13) and (6.1.16),        c c c c IN |XJB ⋆ ON | JB AB = IN |XJA ⋆ ON | JA XN XN N N     c | JAc ∪JBc = IN |XJA ∩JB ⋆ ON XN N     c = IN |XJA ∩JB ⋆ ON | (JA ∩JB )c , (6.1.17) 

N

XN

on XN , in ΛN , for all A, B ∈ {Tk1 , Tk2 , Sk1 , Sk2 }.

Adjointable Banach-Space Operators ■ 67 c Since the identity operator IN , and the operator ON of (5.2.3) are well-defined projections in ΛN , the operators formed by c (IN |Y1 ) ⋆ (ON |Y2 )

are projections in ΛN , whenever XN = Y1 ⋆ Y2 , because ∗

c c ((IN |Y1 ) ⋆ (ON |Y2 )) = (IN |Y1 ) ⋆ (ON |Y2 ) ,

and 2

c c ((IN |Y1 ) ⋆ (ON |Y2 )) = (IN |Y1 ) ⋆ (ON |Y2 ) ,

in ΛN , by (6.1.17). Therefore, inductively, all finite products of {Tk1 , Tk2 , Sk1 , Sk2 } are projections in ΛN . ■ The above corollary confirms that the products of the projections of (6.1.10), induced by the restricted-shift family λN , are projections in ΛN , too. Also, from the proof of Corollary 6.6, it is shown that: if [n ,n2 ]

XN 1

n2

= ⋆ C [{xj }] j=n1

is a ∗-subalgebra of XN , then the Banach-space operators Q of ΛN , formed by     c Q = IN |X[n1 ,n2 ] ⋆ ON |X[n1 ,n2 ]c N

N

are projections on XN , since     ∗ c ∗ Q∗ = IN |X[n1 ,n2 ] ⋆ (ON ) |X[n1 ,n2 ]c = Q, N

N

and     c Q2 = IN |X[n1 ,n2 ]∩[n1 ,n2 ] ⋆ ON |X([n1 ,n2 ]∩[n1 ,n2 ])c = Q, N

N

in ΛN , for all n1 ≤ n2 ∈ {1, . . . , N }. Notation. From below, we use the following notation;     denote c Q[n1 ,n2 ] = IN |X[n1 ,n2 ] ⋆ ON |X[n1 ,n2 ]c , N

(6.1.18)

N

in ΛN , for all n1 ≤ n2 ∈ {1, . . . , N }. For instance, by (6.1.10), −k k k −k βN βN = Q[1,N −k] , and βN βN = Q[k+1,N ] , k for βN ∈ λN ⊂ ΛN , for all k ∈ {0, 1, . . . , N }, in {0, ±1, . . . , ±N }.

□

68 ■ Banach-Space Operators On C ∗ -Probability Spaces

By the notation (6.1.18), one can summarize the above results as follows. Corollary 6.7. Let λN be the restricted-shift family generating the shift-operator algebra ΛN . If  k −k −k k k βN , βN βN ∈ ΛN : βN ∈ λN , Q = βN Q Q then the shift operators Q = T are projections in ΛN , where T ∈Q

means the finite product. In particular, for any such operator Q ∈ ΛN , there exists n1 ≤ n2 in {1, . . . , N }, such that     c Q = Q[n1 ,n2 ] = IN |X[n1 ,n2 ] ⋆ ON |X[n1 ,n2 ]c , (6.1.19) N

N

in ΛN , under the notation (6.1.18). Proof. The proof of (6.1.19) is done by Corollary 6.6 and (6.1.18). ■ Note that, if B is an arbitrary topological ∗-algebra, and if T, S ∈ B are projections in B, then the operator-product T S ∈ B is not a projection in B, in general. Lemma 6.8. Let B be a topological ∗-algebra, and let T, S ∈ B be projections in B. Then T S ∈ B is a projection, if and only if T S = ST, in B.

(6.1.20)

Proof. (⇐) Suppose two projections T and S are commuting in B, in the senes that: T S = ST, in B. Then ∗

(T S) = S ∗ T ∗ = ST = T S, and 2

(T S) = T (ST ) S = T (T S) S = T 2 S 2 = T S, in B, and hence, the operator-product T S is a projection in B. (⇒) Suppose that T S ∈ B is a projection, and suppose T S ̸= ST in B. Then ∗ (T S) = S ∗ T ∗ = ST ̸= T S, in B,

Adjointable Banach-Space Operators ■ 69

implying that T S cannot be a projection in B. It contradicts our assumption. Equivalently, if T S is a projection in B, then T and S are commuting in B. ■ Let Q be the subset of ΛN in the sense of Corollary 6.7. Then it is a “commuting family in ΛN ,” in the sense that: T S = ST, ∀T, S ∈ Q, in ΛN (not in Q).  −k k k −k k Theorem 6.9. Let Q = βN βN , βN βN ∈ ΛN : βN ∈ λN . Then Q is a commuting family of projections in ΛN .

(6.1.21)

Proof. By (6.1.19), if T ∈ Q in ΛN , then     c T = Q[n1 ,n2 ] = IN |X[n1 ,n2 ] ⋆ ON |X[n1 ,n2 ]c , N

N

in ΛN , for some n1 ≤ n2 ∈ {1, . . . , N }. Remark that, c c c IN W = W = W IN , and ON W = ON = W ON , c in ΛN , for all W ∈ ΛN . i.e., IN and ON are commuting with all shift operators of ΛN . So, for any

T1 = Q[n1 ,n2 ] , T2 = Q[m1 ,m2 ] ∈ Q, we have T1 T2 = Q[n1 ,n2 ]∩[m1 ,m2 ] = Q[m1 ,m2 ]∩[n1 ,n2 ] = T2 T1 , in ΛN . i.e., all elements of Q are commuting from each other in ΛN . Therefore, the statement (6.1.21) holds. ■ The above theorem characterizes the commuting property of the family Q in ΛN . i.e., all projections of Q are commuting from each other in ΛN by (6.1.21). Therefore, one can re-prove Corollaries 6.6 and 6.7, by Theorem 6.9.

70 ■ Banach-Space Operators On C ∗ -Probability Spaces

6.2

PARTIAL ISOMETRIES ON XN

∗ Throughout this section, we fix N ∈ N∞ >1 , and the C -probability space XN = (XN , φN ) generated by the free semicircular family N XN = {xj }j=1 . In Section 6.1, we considered certain projections in the shift-operator algebra ΛN induced by the restricted-shift family  k N λN = βN in the operator space B (XN ). Especially, it is shown k=−N that, for all k ≥ 0 in {0, ±1, . . . , ±N }, −k k k −k βN βN = Q[1,N −k] , and βN βN = Q[k+1,N ]

are projections in ΛN , where     c Q[n1 ,n2 ] = IN |X[n1 ,n2 ] ⋆ ON |X[n1 ,n2 ]c , N

(6.2.1)

(6.2.2)

N

in the sense of (6.1.10) and (6.1.18), for all n1 ≤ n2 ∈ {1, . . . , N }, with identification:     c C [{xj }] , Q[n,n] = Q{n} = IN |C[{xn }] ⋆ ON | ⋆ j∈{1,...,N }, j̸=n

in ΛN , for all n ∈ {1, . . . , N }. By (6.2.2), all finite products of such operators (6.2.1) are projections in ΛN , too (e.g., See Corollary 6.7). In this section, we study a new type of adjointable Banachspace operators of the shift-operator algebra ΛN . Such a new type is motivated by (6.1.11); by (6.2.1) and (6.2.2), one has
−k −k k −k k k −k k k ∗ = βN βN βN , (6.2.3) βN = βN βN βN , and βN = βN k in ΛN , for all βN ∈ λN (e.g., see (6.1.11)).

Definition 6.2. Let B be an arbitrary topological ∗-algebra over C. An operator y ∈ B is a partial isometry in B, if y ∗ y is a projection in B in the sense of Definition 6.1. By definition, it can be considered that an operator y of a topological ∗-algebra B is a partial isometry in B, if and only if y ∗ y is a projection in B (by definition), if and only if y = y (y ∗ y) = yy ∗ y in B, if and only if y ∗ = y ∗ (yy ∗ ) = y ∗ yy ∗ in B,

(6.2.4)

Adjointable Banach-Space Operators ■ 71

if and only if yy ∗ is a projection in B, if and only if y ∗ is a partial isometry in B. By (6.2.4), the readers can understand our partial isometries are generalized versions of partial isometries on Hilbert spaces (e.g., [14]). Remark 6.1. By definition, every projection in a topological ∗-algebra B is a partial isometry. Indeed, if y ∈ B is a projection, then ∗

(y ∗ y) = y ∗ y = y 2 = y = y 2


2

2

= (y ∗ y) ,

in B, implying that y ∗ y is a projection in B, and hence, it is a partial isometry in B. Clearly, not all partial isometries are projections in B (e.g., see Theorem below).  −k6.10 k k −k k Let Q = βN βN , βN βN : βN ∈ λN be the commuting family (6.1.22) of projections in the shift-operator algebra ΛN . Then the products of all projections in Q are projections in ΛN by (6.1.21), and hence, all such operator-products in Q are partial isometries in ΛN , too. □ k Theorem 6.10. Every generating shift operator βN ∈ λN is a partial isometry in ΛN , for all k ∈ {0, ±1, . . . , ±N }.

Proof. The proof is done by (6.2.3) and (6.2.4). Indeed, for any k ≥ 0, −k k k −k βN βN = Q[1,N −k] , and βN βN = Q[k+1,N ]

are projections in ΛN by (6.1.10), and hence, the restricted shifts k βN ∈ λN are partial isometries in ΛN by Definition 6.2, for all k ∈ {0, ±1, . . . , ±N }. ■ The above theorem implies that the shift-operator algebra ΛN is  k N generated by partial isometries βN = λN in B (XN ). Remark k=−N c that the operator ON , −N c N ON = βN = βN ∈ λN , if N < ∞,

is clearly a partial isometry in ΛN , since it is a projection. Also, Theorem 6.10 provides detailed information of products of restricted shifts of λN in ΛN , under the partial-isometry-property.

72 ■ Banach-Space Operators On C ∗ -Probability Spaces k1 k2 Theorem 6.11. Let βN , βN ∈ λN be generating shift operators of ΛN , for k1 , k2 ≥ 0 in {0, ±1, . . . , ±N }. Then k1 k2 k1 k2 βN βN = βN Q]k2 +1,N −k1 [ βN ,

(6.2.5)

and k2 k1 k2 k1 βN βN = βN Q]k1 +1,N −k2 [ βN ,

where ]n1 , n2 [ is in the sense of (6.1.16), for all n1 , n2 ∈ {1, . . . , N }. k1 k2 Proof. Let βN , βN ∈ λN be generating shift operators of ΛN , for

k1 , k2 ≥ 0 in {0, ±1, . . . , ±N } . Then, they are partial isometries in ΛN , satisfying (6.2.3). So,    k1 k2 k1 −k1 k1 k2 −k2 k2 βN βN = βN βN βN βN βN βN    k1 −k1 k1 k2 −k2 k2 = βN βN βN βN βN βN

k2 k1 = βN Q[1,N −k1 ] Q[k2 +1,N ] βN k1 k2 = βN Q[1,N −k1 ]∩[k2 +1,N ] βN k1 k2 = βN Q]k2 +1,N −k1 [ βN  k k2 1  βN Q[k2 +1,N −k1 ] βN k1 k2 = βN Q[N −k1 ,k2 +1] βN   c ON

if k2 + 1 ≤ N − k1 if k2 + 1 ≥ N − k1 if [1, N − k1 ] ∩ [k2 + 1, N ] = ¨ı¿œ, (6.2.6)

in ΛN , by (6.1.16). Similar to (6.2.6), we obtain that k2 k1 k2 k1 βN βN = βN Q]k1 +1,N −k2 [ βN ,

in ΛN , by (6.1.16). Therefore, the relation (6.2.5) holds by (6.2.6) and (6.2.7).

(6.2.7) ■

The above theorem re-confirms that, indeed, k1 k2 k2 k1 βN βN ̸= βN βN ∈ / λN , in ΛN ,

in general (whenever neither k1 = 0 nor k2 = 0). Similar to (6.2.5), one obtains the following cases.

Adjointable Banach-Space Operators ■ 73 k1 k2 Theorem 6.12. Let βN , βN ∈ λN be the generating partial isometries of ΛN . For all 0 ≤ k1 ≤ k2 ∈ {0, ±1, . . . , ±N } ,

we have that −k1 k2 −k1 k2 k2 −k1 k2 −k1 βN βN = βN Q[k2 +1,N ] βN , and βN βN = βN Q[k2 +1,N ] βN , (6.2.8) and ( k1 −k2 βN Q[k2 +1,N −k1 ] βN if k2 + 1 ≤ N − k1 k1 −k2 βN βN = (6.2.9) k1 −k2 βN Q[N −k1 ,k2 +1] βN if k2 + 1 ≥ N − k1 ,

and −k2 k1 βN βN

( k1 −k2 βN Q[k2 +1,N −k1 ] βN = k1 −k2 βN Q[N −k1 ,k2 +1] βN

if k2 + 1 ≤ N − k1 if k2 + 1 ≥ N − k1 ,

and −k1 −k2 −k1 −k2 βN βN = βN Q[k1 +1,k2 +1] βN ,

(6.2.10)

and −k2 −k1 −k2 −k1 βN βN = βN Q[k1 +1,k2 +1] βN ,

in ΛN ,. Proof. Observe that if 0 ≤ k1 ≤ k2 in {0, ±1, . . . , ±N }, then −k1 k2 −k1 k2 βN βN = βN Q[k1 +1,N ] Q[k2 +1,N ] βN −k1 k2 −k1 k2 = βN Q[k1 +1,N ]∩[k2 +1,N ] βN = βN Q[k2 +1,N ] βN ,

since k1 ≤ k2 ; similarly, k2 −k1 k2 −k1 k2 −k1 βN βN = βN Q[k2 +1,N ]∩[k1 +1,N ] βN = βN Q[k2 +1,N ] βN ,

in ΛN . Therefore, the operator-equalities of (6.2.8) hold. Also, one has k1 −k2 k1 −k2 βN βN = βN Q[1,N −k1 ] Q[k2 +1,N ] βN k1 −k2 k1 −k2 = βN Q[1,N −k1 ]∩[k2 +1,N ] βN = βN Q]k2 +1,N −k1 [ βN ( k1 −k2 βN Q[k2 +1,N −k1 ] βN if k2 + 1 ≤ N − k1 = k1 −k2 βN Q[N −k1 ,k2 +1] βN if k2 + 1 ≥ N − k1 ,

74 ■ Banach-Space Operators On C ∗ -Probability Spaces N

on XN = ⋆ C [{xj }], in ΛN . Similarly, j=1

−k2 k1 −k2 k1 βN βN = βN Q[k2 +1,N ]∩[1,N −k1 ] βN −k2 k1 = βN Q]k2 +1,N −k1 [ βN ( −k2 k1 Q[k2 +1,N −k1 ] βN βN = −k2 k1 βN Q[N −k1 ,k2 +1] βN

if k2 + 1 ≤ N − k1 if k2 + 1 ≥ N − k1 ,

in ΛN . Therefore, the operator-equalities of (6.2.9) holds, whenever 0 ≤ k1 ≤ k2 in {0, ±1, . . . , ±N }. Finally, consider that −k1 −k2 −k1 −k2 βN βN = βN Q[k1 +1,N ] Q[1,k2 +1] βN −k1 −k2 −k1 −k2 = βN Q[k1 +1,N ]∩[1,k2 +1] βN = βN Q[k1 +1,k2 +1] βN ,

since k1 ≤ k2 by assumption. Similarly, −k2 −k1 −k2 −k1 −k2 −k1 βN βN = βN Q[1,k2 +1]∩[k1 +1,N ] βN = βN Q[k1 +1,k2 +1] βN ,

in ΛN , whenever 0 ≤ k1 ≤ k2 . Therefore, the operator-equalities of (6.2.10) holds. ■ The operator-equalities (6.2.5), (6.2.8), (6.2.9) and (6.2.10) illustrate how the operator-product on ΛN in the generating set λN act on XN . Motivated by the above results, let’s focus on the partial isometries k βN of λN , more in detail. By (6.2.1), k k −k k k −k k −k k βN = βN βN βN = βN βN βN βN βN k k k −k k since βN is a partial isometry satisfying βN = βN βN βN in ΛN


k
−k k k −k k = βN βN βN βN βN = Q[nkfin ,mkfin ] βN Q[nkinit ,mkinit ] ,

(6.2.11)

in ΛN , where ( [1, N − |k|] [nkfin , mkfin ] = [1 + |k| , N ]

if k ≥ 0 if k < 0,

(6.2.12)

Adjointable Banach-Space Operators ■ 75

and [nkinit , mkinit ]

( [1 + |k| , N ] = [1, N − |k|]

if k ≥ 0 if k < 0,

in {1, . . . , N }, by (6.2.1), for all k ∈ {0, ±1, . . . , ±N }. k1 k2 So, if βN , βN ∈ λN are generating partial isometries of ΛN , for k1 , k2 ∈ {0, ±1, . . . , ±N }, then k1 k2 k1 βN βN = Q[nk1 ,mk1 ] βN Q[nk1 fin

k1 init ,minit ]

fin

k2 × Q[nk2 ,mk2 ] βN Q[nk2 fin

k2 init ,minit ]

fin

k1 h = Q[nk1 ,mk1 ] βN Q nk1 fin

k1 init ,minit

fin

k2 h × βN Q nk2

k2 init ,minit

i i h k2 k2 ∩ nfin ,mfin

i,

(6.2.13)

in ΛN , by (6.2.11), where h

l l nkinit , mkinit

i

, and

h

nkfinl , mkfinl

i

are in the sense of (6.2.12). i.e., the formula (6.2.13) generalizes (6.2.5), (6.2.8), (6.2.9) and (6.2.10), up to (6.2.12). k1 k2 Corollary 6.13. Let βN , βN ∈ λN be generating partial isometries of ΛN , for arbitrary k1 , k2 ∈ {0, ±1, . . . , ±N }. Then k1 k2 k1 k2 βN βN = βN Q[n,m] βN , in ΛN ,

(6.2.14)

where h i h i 1 1 [n, m] = nkinit , mkinit ∩ nkfin2 , mkfin2 , h i h i l l in {1, . . . , N }, where nkinit , mkinit , and nkfinl , mkfinl are in the sense of (6.2.12), for all l = 1, 2. Proof. The operator-equality (6.2.14) is proven by (6.2.5), (6.2.8), (6.2.9), (6.2.10) and (6.2.13). ■

76 ■ Banach-Space Operators On C ∗ -Probability Spaces

The operator-equality (6.2.14) implies the following result. k Theorem 6.14. Let βN ∈ λN be a generating partial isometry of ΛN , and xj ∈ XN , a generating semicircular element of XN . Then (     xj+k if j ∈ nkinit , mkinit ∩ nkfin − k, mkfin − k k βN (xj ) = (6.2.15) 0N otherwise,

in XN , where the quantities nkinit , mkinit , nkfin , mkfin ∈ {1, . . . , N } are in the sense of (6.2.12). Proof. By (6.2.11) and (6.2.14), one has that k k βN (xj ) = Q[nk ,mk ] βN Q[nk ,mk ] (xj ) init init fin fin    k Q k k β (xj ) if j ∈ nk , mk init init [nfin ,mfin ] N   = k Q nk ,mk βN (0N ) if j ∈ / nkinit , mkinit [ fin fin ] (   k Q[nk ,mk ] βN (xj ) if j ∈ nkinit , mkinit fin fin =   0N if j ∈ / nkinit , mkinit     if j ∈ nkinit , mkinit ,     xj+k    and j + k ∈ nkfin , mkfin    = if j ∈ nkinit , mkinit ,    0N   and j + k ∈ / nkfin , mkfin       0N if j ∈ / nkinit , mkinit (     xj+k if j ∈ nkinit , mkinit ∩ nkfin − k, mkinf − k = 0N otherwise,

in XN . Therefore, the formula (6.2.15) holds.

6.3

■

FREE-DISTRIBUTIONAL DATA AFFECTED BY PARTIAL ISOMETRIES

In this section, we consider how some shift operators of ΛN induced by the partial isometries of λN affect the free probability on XN . Recall that, by (6.2.15), we have that k (xj ) = χk (j)xj+k ∈ XN , βN

(6.3.1)

Adjointable Banach-Space Operators ■ 77

with ( χk (j) =

1 0

    if j ∈ nkinit , mkinit ∩ nkfin − k, mkfin − k otherwise,

for all j ∈ {1, . . . , N }, and k ∈ {0, ±1, . . . , ±N }, where the quantities nkinit , mkinit , nkfin , mkfin ∈ {1, . . . , N } are in the sense of (6.2.12), i.e., (  k  [1, N − |k|] if k ≥ 0 k ninit , minit = [1 + |k| , N ] if k < 0, and  k  nfin , mkfin =

( [1 + |k| , N ] [1, N − |k|]

if k ≥ 0 if k < 0,

in {1, . . . , N }, for k ∈ {0, ±1, . . . , ±N }. Note that, by (6.2.12), one can verify that (  k    [1, N − k] if k ≥ 0 ninit , mkinit ∩ nkfin − k, mkfin − k = [1 + |k| , N ] if k < 0, for all k ∈ {0, ±1, . . . , ±N }. Indeed, if k ≥ 0, then  k    ninit , mkinit ∩ nkfin − k, mkfin − k = [1, N − k] ∩ [1 + k − k, N − k], meanwhile, if k < 0, then  k    ninit , mkinit ∩ nkfin − k, mkfin − k = [1 + |k| , N ] ∩ [1 − k, N − |k| − k], implying that: if k ≥ 0, then  k    ninit , mkinit ∩ nkfin − k, mkfin − k = [1, N − k], meanwhile, if k < 0, then  k    ninit , mkinit ∩ nkfin − k, mkfin − k = [1 + |k| , N ]. Lemma 6.15. For all k ∈ {0, ±1, . . . , ±N },   k     ninit , mkinit ∩ nkfin − k, mkfin − k = nkinit , mkinit ,

(6.3.2)

in {1, . . . , N }, where the quantities nkinit , mkinit , nkfin , mkfin ∈ {1, . . . , N } are in the sense of (6.2.12).

78 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. The set-equality (6.3.2) is obtained by the straightforward computations of the very above paragraph. ■ By (6.3.2), the functions, χk (•) : {1, . . . , N } → {0, 1} of (6.3.1) can be re-defined by (   1 if j ∈ nkinit , mkinit χk (j) = 0 otherwise,

(6.3.3)

for all j ∈ {1, . . . , N }, for all k ∈ {0, ±1, . . . , ±N }. k Lemma 6.16. Let βN ∈ λN be a generating partial isometry of ΛN , and xj ∈ XN , a generating semicircular element of XN . Then

βN (xj ) = χk (j)xj+k , in XN ,

(6.3.4)

where χk (j) are in the sense of (6.3.3). Proof. The operator-equality (6.3.4) holds by (6.3.1) and (6.3.3).

■

By (6.3.4), we obtain the following free-distributional data. k Theorem 6.17. Let βN ∈ ΛN be a generating partial isometry, and xj ∈ XN , a generating semicircular element. If   j ∈ nkinit , mkinit in {1, . . . , N } , k then βN (xj ) is semicircular in XN , where nkinit , mkinit are in the sense of (6.2.12). Otherwise, it has the zero free distribution in XN .

Proof. By (6.3.4), one has that 
n  n k φN βN (xj ) = φN ((χk (j)xj+k ) ) , implying that φN



k βN (xj )


n 


= χk (j)n φN xnj+k ,

for all n ∈ N. Note that, by (6.3.3), χk (j)n = χk (j), ∀n ∈ N,

Adjointable Banach-Space Operators ■ 79

in {0, 1}, and hence, 
n 
k φN βN (xj ) = χk (j)φN xnj+k , therefore, φN



k βN


n  (xj ) =

( ωn c n2 0

if χk (j) = 1 if χk (j) = 0,

for all n ∈ N.   Recall that χk (j) = 1, if j ∈ nkinit , mkinit , meanwhile, χk (j) = 0, otherwise. Therefore, the semicircular law, or the zero free distribution k for βN (xj ) holds on XN . ■ k1 k2 By (6.2.14) and (6.3.4), if βN , βN ∈ λN are generating partial isometries of ΛN , and if xj ∈ XN is a generating semicircular element, then k2 k1 k2 βN βN (xj ) = βN (χk1 (j)xj+k1 ) = χk2 (j + k1 )χk1 (j)xj+k1 +k2 ,

i.e., k2 k1 βN βN (xj ) = (χk2 (j + k1 )χk1 (j)) xj+k1 +k2 ,

(6.3.5)

in XN , where χk (t) are in the sense of (6.3.3), for all k ∈ {0, ±1, . . . , ±N } , and t ∈ {1, . . . , N } . So, inductively, one can get the following result. kl Lemma 6.18. Let βN ∈ λN be generating partial isometries of ΛN (which are not necessarily distinct from each other), for l = 1, . . . , n, for n ∈ N, and xj ∈ XN , a generating semicircular element of XN . If

T =

n Y

k

kn k2 k1 βNn−l+1 = βN . . . βN βN ∈ ΛN

l=1

is a shift operator, then T (xj ) =

n Y l=1

χkl

j+

l−1 X t=1

!! kt

x

j+

n P

kl

l=1

∈ XN ,

(6.3.6)

80 ■ Banach-Space Operators On C ∗ -Probability Spaces

with axiomatization; 0 X

kt = 0,

t=1

where χk (t) are in the sense of (6.3.3). Proof. The formula (6.3.6) is obtained by the induction on (6.3.5). ■ For instance, k3 k2 k1 βN βN βN (xj ) = (χk1 (j)χk2 (j + k1 )χk3 (j + k1 + k2 )) xj+k1 +k2 +k3 ,

in XN , etc., where χk (t) are in the sense of (6.3.3). n Q

Theorem 6.19. Let T =

l=1

k

βNn−l+1 ∈ ΛN be a shift operator induced

kl by the partial isometries βN ∈ λN (which are not necessarily distinct from each other), for l = 1, . . . , n, for n ∈ N. If xj ∈ XN is a generating semicircular element of XN , then T (xj ) is semicircular in XN , if and only if " # l−1 l−1 X X n kl kl j ∈ ∩ ninit − kt , minit − kt , (6.3.7) l=1

t=1

t=1

in {1, . . . , N }. The free random variable T (xj ) is not semicircular, if and only if the free distribution of T (xj ) is the zero free distribution on XN . Proof. Let T ∈ ΛN be the given shift operator. Then, for an arbitrary semicircular element xj ∈ XN of XN , ! !! l−1 n Y X n T (xj ) = χkl j+ , kt x P l=1

t=1

j+

kl

l=1

in XN , by (6.3.6), where χk (t) are in the sense of (6.3.3), for all k ∈ {0, ±1, . . . , ±N } , and t ∈ {1, . . . , N } . So, this free random variable T (xj ) is nonzero in XN , if and only if ! l−1 n Y X χkl j+ kt ̸= 0, l=1

t=1

Adjointable Banach-Space Operators ■ 81

in {0, 1}, if and only if χkl

l−1 X

j+

! ̸= 0, ∀l ∈ {1, . . . , n} ,

kt

t=1

if and only if χkl

l−1 X

j+

! = 1, ∀l ∈ {1, . . . , n} ,

kt

t=1

in {0, 1}, if and only if j+

l−1 X

i h l l , ∀l ∈ {1, . . . , n} , , mkinit kt ∈ nkinit

t=1

if and only if " l j ∈ nkinit −

l−1 X

l kt , mkinit −

l−1 X

t=1

# kt , ∀l ∈ {1, . . . , n} ,

t=1

if and only if n

j∈∩

l=1

" l nkinit −

l−1 X

kt ,

t=1

l mkinit −

l−1 X

# kt ,

t=1

in {1, . . . , N }, if and only if T (xj ) = x

j+

n P

kl

∈ XN , in XN ,

l=1

and hence, it becomes a new generating semicircular element of XN . So, the free distribution of T (xj ) ∈ XN is the semicircular law, if and only if the condition (6.3.7) holds, for j ∈ {1, . . . , N }. Meanwhile, there exists at least one l0 ∈ {1, . . . , n} such that ! lX 0 −1 χkl0 j+ kt = 0, in {0, 1} , t=1

if and only if T (xj ) = 0N in XN , by (6.3.6), if and only if the free distribution of T (xj ) is the zero free distribution on XN . ■

82 ■ Banach-Space Operators On C ∗ -Probability Spaces

Now, let N X

U=

k tl βN ∈ ΛN , for tl ∈ C,

(6.3.8)

k=−N k where βN ∈ λN are the generating partial isometries of ΛN , for all k ∈ {0, ±1, . . . , ±N }. If U ∈ ΛN is a shift operator (6.3.8), then

U=

=

N X k=1 N X

−k t−k βN + t0 IN +

N X

k tk βN

k=1


−k t−k Q[1,N −k] βN Q[1+k,N ] + t0 IN

k=1 N X

+


k tk Q[1+k,N ] βN Q[1,N −k] ,

(6.3.9)

k=1

by (6.2.15). By (6.3.6) and (6.3.9), if xj ∈ XN is a generating semicircular element of XN , then N N X
X U xsj = t−k χ−k (j)xsj−k + t0 xj + tk χk (j)xsj+k , k=1

(6.3.10)

k=1

with ( 1 if j ∈ [1 + k, N ] χ−k (j) = 0 otherwise, and ( 1 χk (j) = 0

if j ∈ [1, N − k] otherwise,

by (6.3.2) and (6.3.3), for all k ∈ {1, . . . , N } in {0, ±1, . . . , ±N }, for all k s ∈ N, since the partial isometries βN ∈ λN are ∗-homomorphisms in Hom (XN ). So, we obtain the following free-distributional data.

Adjointable Banach-Space Operators ■ 83

Theorem 6.20. Let U ∈ ΛN be a shift operator (6.3.8), and let xj ∈ XN be a generating semicircular element of XN . Then   X


  φN U xsj = ωs c 2s  tk  , (6.3.11) k k k∈{0,±1,...,±N },j∈[ninit ,minit ] for all s ∈ N. Proof. By (6.3.10), for all s ∈ N, φN

N X



U xsj = φN

! tk χk (j)xsj+k

k=−N

=

N X

tk φN χk (j)xsj+k




k=−N

=

N X




tk χk (j) ωs c 2s = ωs c 2s




N X

! tk χk (j)

k=−N

k=−N


 = ωs c 2s 

 X

 t k · 1 k k∈{0,±1,...,±N },j∈[nk init ,minit ]

by (6.3.3) 
 = ωs c 2s 

 X

 tk  , k k∈{0,±1,...,±N },j∈[nk init ,minit ]

by (6.3.7). Therefore, the free-distributional data (6.3.11) holds.

■

The above theorem illustrate how the operator sum U ∈ ΛN of the partial isometries λN affect the original free-distributional data on XN by (6.3.11).

6.4

PARTIAL ISOMETRIES OF ΛN FOR N < ∞

In this section, we consider the above general results of Sections 6.2 and 6.3 to the cases where N < ∞ in N∞ >1 . In this section, we let N < ∞ in N∞ >1 ,

84 ■ Banach-Space Operators On C ∗ -Probability Spaces

and let XN be the corresponding C ∗ -probability space generated by the  k N N finite free semicircular family XN = {xj }j=1 , and λN = βN , k=−N the finite restricted-shift family, with ±N 0 c βN = IN , βN = ON ∈ λN in Hom (XN ) ,

generating the shift-operator algebra ΛN . k Recall that, if βN ∈ λN , then it is a partial isometry in ΛN , satisfying ( k Q[1+k,N ] βN Q[1,N −k] if k ≥ 0 k βN = k Q[1,N −|k|] βN Q[1+|k|,N ] if k < 0 ( k Q[1+k,N ] βN Q[1,N −k] if k ≥ 0 = (6.4.1) k Q[1,N +k] βN Q[1−k,N ] if k < 0 for all k ∈ {0, ±1, . . . , ±N }, expressed by k βN = Q[nk

k fin ,mfin

k ] βN Q[nkinit ,mkinit ] , in ΛN .

Assume that either k = 0, or k = N in {0, ±1, . . . , ±N }. Then
0 n n βN = IN = IN , (6.4.2) respectively, N βN


n

n

−N c c = (ON ) = ON = βN


n

,

for all n ∈ N. Suppose now that neither k = 0, nor k = ±N in {0, ±1, . . . , ±N }. Assume first that such a k is positive, i.e., k > 0. Then

n k n k βN = Q[1+k,N ] βN Q[1,N −k]
n−2 k = Q[1+k,N ] βN Q[1,N −k]
k k × Q[1+k,N ] βN Q[1,N −k] Q[1+k,N ] βN Q[1,N −k]
n−2 k = Q[1+k,N ] βN Q[1,1+k]
k k × Q[1+k,N ] βN Q]1+k,N −k[ βN Q[1,N −k] in ΛN , and hence, it goes to k k Q]1+k,N −k[ βN Q]1+k,N −k[ . . . = Q[1+k,N ] βN k k × Q]1+k,N −k[ βN Q]1+k,N −k[ βN Q[1,N −k] ,

Adjointable Banach-Space Operators ■ 85

i.e., k βN


n

k = Q[1+k,N ] βN Q]1+k,N −k[


n

k βN Q[1,N −k] ,

(6.4.3)

in ΛN , for all n ∈ N. Of course, if ]1 + k, N − k[ = ¨ı¿œ in {1, . . . , N } , then k βN


n

(6.4.4)

= ON , in ΛN .

Under the same condition k > 0 in {0, ±1, . . . , ±N }, if xj ∈ XN is a generating semicircular element of XN , then  
n k
k k n Q[1,N −k] (xj ) Q]1+k,N −k[ βN (xj ) = Q[1+k,N ] βN βN

n k k Q[1,N −k] (xj ) = Q[1+k,N ] βN Q]1+k,N −k[ βN
n k = Q[1+k,N ] βN Q]1+k,N −k[ (χk (j)xj+k ) where χk (j) is in the sense of (6.3.3)

n−1 k k βN Q]1+k,N −k[ (χk (j)xj+k ) = Q[1+k,N ] βN Q]1+k,N −k[ 
n−1 k k  Q[1+k,N ] βN Q[1+k,N −k] βN  
if χk (j) = 1   Q ]1+k,N −k[ (xj+k )
n−1 k = k  βN Q{1+k} Q[1+k,N ] βN  
if χk (j) = 0   Q ]1+k,N −k[ (0N ) 
n−1 k  Q[1+k,N ] βN Q[1+k,N −k] if χk (j) = 1 k = βN (xj+k )   0N if χk (j) = 0  k  βN (xj+k ) = xj+2k if χk (j) = 1, n = 2, j + 2k ≤ N    k   if χk (j) = 1, n = 2, j + 2k > N βN (xj+k ) = 0N
n−1 k = Q[1+k,N ] βN Q]1+k,N −k[  if χk (j) = 1, and n > 2  k  βN (xj+k )    0N if χk (j) = 0 (6.4.5) for all n ∈ N. By (6.4.5), one can verify the following result.

86 ■ Banach-Space Operators On C ∗ -Probability Spaces k Theorem 6.21. Let N < ∞ in N∞ >1 , and let βN ∈ λN be a generating partial isometry of ΛN , where k ̸= 0 in {0, ±1, . . . , ±N }. Then there exist n ∈ N, such that
k n c βN = ON , in ΛN . (6.4.6)

Proof. If k > 0 in {0, ±1, . . . , ±N } then, for any arbitrary generating semicircular element xj ∈ XN of XN ,
k n (xj ) = 0N , βN for some sufficiently big n > N in N (or as n → ∞ in N), by (6.4.5). Indeed, if χk (j) = 1, and if n → ∞ in N, then k Q[1+k,N ] βN Q[1+k,N −k]


n−1

k βN (xj+k ) = Q[1+k,N ] (xj+nk ) ,

and there do exist sufficiently big n ∈ N making j + nk > N, in {1, . . . , N } , by the Zorn’s lemma, and hence, Q[1+k,N ] (xj+nk ) = 0N , in XN , for such n ∈ N, by (6.4.5). If k < 0 in {0, ±1, . . . , ±N }, then ∗ n ∗n  n  
|k| −|k| |k| k n , (xj ) = βN (xj ) (xj ) = βN βN (xj ) = βN for all n ∈ N, for all j ∈ {1, . . . , N }. So, even though k < 0, there exists n ∈ N, such that  n |k| βN (xj ) = 0N , in XN , and hence, k βN


n

 n −|k| (xj ) = 0N , in XN (xj ) = βN

So, from below, k is taken from {±1, . . . , ±N } arbitrarily. Since j ∈ {1, . . . , N } is arbitrary, if N < ∞, then there do exist nxj ∈ N, such that
k nxj c , on XN , (6.4.7) βN = ON

Adjointable Banach-Space Operators ■ 87

for all j ∈ {1, . . . , N }. Thus, for any y=

n Y

xjl ∈ XN , for n ∈ N

l=1

(where xjl are not necessarily distinct in XN ), there do exist ny ∈ N, such that
k ny (y) = 0N , in XN , βN k since βN ∈ λN ⊂ Hom (XN ). In particular, such a quantity ny can be  ny = max nxj : j = 1, . . . , N ∈ N, (6.4.8)

where nxj ∈ N are the quantities satisfying (6.4.7), for each y ∈ XN . Since there are only finitely many generators {x1 , . . . , xN } of XN , for any arbitrary free reduced words W ∈ XN , there does exists nk ∈ N, such that
k nk βN (W ) = 0N , in XN , for all free reduced words W ∈ XN in XN , for all k ∈ {±1, . . . , ±N }, by (6.4.8). Note that the quantity nk can be taken to be  max nxj : j = 1, . . . , N ∈ N, (6.4.9) where nxj ∈ N are in the sense of (6.4.7), for all j ∈ {1, . . . , N }, where N < ∞. Therefore, since all free random variables of XN are the limits of linear combinations of free reduced words in XN by (5.6), there do exist n0 ∈ N, such that
k n0 βN (Y ) = 0N , in XN , for all Y ∈ XN ⊖ (C · 1N ), for “all” k ∈ {±1, . . . , ±N }. In particular, the quantity n0 ∈ N could be n0 = max {nk : k = ±1, . . . , ±N } ∈ N, where nk are in the sense of (6.4.9) (Remark that, in fact, nk = n−k !) It means that there do exist n ∈ N, such that
k n c , in ΛN . βN = ON Therefore, the relation (6.4.6) holds.

■

88 ■ Banach-Space Operators On C ∗ -Probability Spaces

The above theorem shows that, if N < ∞, then our generating partial isometries are conditional-nilpotent. Definition 6.3. Let B be a unital topological ∗-algebra with its unity 1B . A “nonzero” element y ∈ B is said to be (C)-conditional-nilpotent, if there exist n0 ∈ N, such that T n = 0cB , on B, whenever n ≥ n0 , where ( t · 1B 0cB (y) = 0B def

if y = t · 1B , for t ∈ C otherwise,

for all y ∈ B. In particular, the minimal quantity n0 ∈ N, satisfying the above operator-equality, is called the (C-)conditional-nilpotence of T . Corollary 6.22. If N < ∞ in N∞ >1 , then every nontrivial generating k partial isometry βN of λN is conditional-nilpotent on XN (or, in ΛN ). i.e., k N < ∞, k ̸= 0, k ̸= ±N =⇒ βN is conditional-nilpotent on XN . (6.4.10)

Proof. The conditional-nilpotent-property (6.4.10) on ΛN is proven by (6.4.6). ■ Theorem 6.23. If N < ∞ in N∞ >1 , then every finite operator sum NP −1 k k T = tk βN , of the nontrivial partial isometries βN of λN , for k=1

0 < k < N , is conditional-nilpotent in ΛN . Proof. Let T =

NP −1

k tk βN ∈ ΛN be a nonzero shift operator for tk ∈ C,

k=1

for k = 1, . . . , N − 1. k Since each “nonzero” summand tk βN ∈ λN is conditional-nilpotent k in ΛN (because βN are conditional-nilpotent by (6.4.10)), there exist the conditional-nilpotences nk ∈ N, such that

k nk k nk c βN = ON = tk βN , in ΛN , for k ∈ {1, . . . , N − 1}, whenever tk ̸= 0.

Adjointable Banach-Space Operators ■ 89

If we take n ∈ N, satisfying n > N max {nk : k = ±1, . . . , ± (N − 1)} , then c T m = ON , in ΛN ,

for all m ≥ n in N, by (6.4.10). i.e., the shift operator T ∈ ΛN is conditional-nilpotent on XN . ■ The above theorem also shows that the operator sum T ∗ of the given shift operator T ∈ ΛN in Theorem 6.23 is conditional-nilpotent, too. k Theorem 6.24. Let βN ∈ λN be a nontrivial partial isometry in ΛN , where neither k = 0, nor k = ±N , in {0, ±1, . . . , ±N }. Then, for any W ∈ XN \ {1N },  
k n (W ) = 0. (6.4.11) lim φN βN n→∞

Proof. For any W ∈ XN \ {1N }, observe that lim φN

n→∞



k βN


n

   
k n (xj ) (W ) = φN lim βN n→∞

since φN is bounded (equivalently, continuous) on XN , where the limit on the right-hand side is for the C ∗ -topology for XN  
 k n = φN lim βN (xj ) n→∞

where the limit here is for the operator-norm topology for ΛN inherited k from that for B (XN ) (since βN are bounded (or, continuous) on XN ) c = φN (ON (xj )) = φN (0N ) = 0, k by the conditional-nilpotent-property (6.4.10) of βN . Therefore, the free-distributional data (6.4.11) holds.

■

By (6.4.11), one can conclude the following asymptotic freedistributional information on XN , where N < ∞.

90 ■ Banach-Space Operators On C ∗ -Probability Spaces

Corollary 6.25. Let W ∈ XN be an arbitrary free random variable, where N < ∞ in N∞ >1 , and let n o
k n W = βN (W ) : n ∈ N ⊂ XN . Then the asymptotic free distribution of the family W is the zero free distribution on XN , as n → ∞. Proof. Let W ⊂ XN be the given family of free random variables. Then  
k n (W ) = 0, lim φN βN n→∞

by (6.4.11). Therefore, the asymptotic free distribution of W is the zero free distribution on XN . ■ By the above corollary, one can obtain the following generalized result. Theorem 6.26. Let W ∈ XN \ {1N } be an arbitrary free random variable, and let N −1 X k T = tl βN ∈ ΛN \ {ON } k=1

be a nonzero shift operator, for N < ∞ in N∞ >1 . Then a family, TW = {T n (W ) : n ∈ N} ⊂ XN , has the asymptotic zero free distribution on XN , as n → ∞. Proof. Let T ∈ ΛN be a given shift operator. Then, by Theorem 6.23, this shift operator T is conditional-nilpotent in ΛN . i.e., there exists the minimal quantity, say n0 ∈ N, such that c T n = ON on XN , whenever n ≥ n0 .

So, if n ≥ n0 in N, then T n (W ) = 0N , in XN , for all W ∈ XN \ {1N }, satisfying that lim φN (T n (W )) = 0,

n→∞

by (6.4.11). Equivalently, the asymptotic free distribution of the family TW is the zero free distribution on XN , as n → ∞. ■ n

The above theorem also shows that the family {(T ∗ ) (W )}n∈N has the asymptotic zero free distribution on XN , as n → ∞.

CHAPTER

7

Free-Probabilistic Information on XN Affected by Partial Isometries

I

N THIS SECTION, WE CONSIDER MORE ABOUT THE DEFORMED FREE-DISTRIBUTIONAL data on XN by the action of ΛN , for an arbitrarily fixed N ∈ N∞ >1 . In particular, we are interested in the operator-products of the generating partial isometries T of λN in ΛN , T =

n Y

k

kn k2 k1 βNn−l+1 = βN · · · βN βN ∈ ΛN ,

(7.1)

l=1 kl for l = 1, . . . , n (where βN are not necessarily distinct from each other in λN ), for n ∈ N. k Let βN ∈ λN be a generating partial isometry of ΛN , satisfying k βN = Q[nk

k ] βN Q[nkinit ,mkinit ] , in ΛN ,

(7.2)

( [1, N − k] if k ≥ 0 [1 + |k| , N ] if k < 0,

(7.3)

k fin ,mfin

where   k ninit , mkinit =

DOI: 10.1201/9781003263487-7

91

92 ■ Banach-Space Operators On C ∗ -Probability Spaces

and  k  nfin , mkfin =

( [1 + k, N ] [1, N − |k|]

if k ≥ 0 if k < 0,

for all k ∈ {0, ±1, . . . , ±N }. So, if T ∈ ΛN is a shift operator (7.1) and if xj ∈ XN is a generating semicircular element of XN , then !! n l−1 Y X n χkl j+ kt x P T (xj ) = ∈ XN , t=1

l=1

j+

kl

l=1

by (6.3.6), where χk (t) are in the sense of (6.3.3), and hence, the free distribution of T (xj ) is characterized by (6.3.7); i.e., the free distribution of it is the semicircular law on XN , if and only if " # l−1 l−1 X X n l l j ∈ ∩ nkinit kt , mkinit kt , − − l=1

t=1

t=1

in {1, . . . , N }, meanwhile, the free distribution is the zero free distribution on XN , if and only if " # l−1 l−1 X X n kl kl j∈ / ∩ ninit − kt , minit − kt , l=1

t=1

t=1

in {1, . . . , N }, with axiomatization: 0 X

kl = 0.

l=1

Now, let [1, N ] = {1, . . . , N }, for a fixed N ∈ N∞ >1 . For instance, [1, ∞] = N ∪ {∞} , and [1, 10] = {1, 2, . . . , 10} , etc.. Let 2[1,N ] be the power set of [1, N ] whose cardinality 2|[1,N ]| . i.e., 2[1,N ] = {K : K ⊆ [1, N ]}

(7.4)

is the set of all subsets of [1, N ]. For the set 2[1,N ] of (7.4), define a morphism, χ : 2[1,N ] × [1, N ] → {0, 1} ,

(7.5)

Free-Probabilistic Information on XN ■ 93

by a function, ( 1 if j ∈ K χ (K, j) = χK (j) = 0 if j ∈ / K, def

for all (K, j) ∈ 2[1,N ] ×[1, N ], where χK are the characteristic functions. By the definition (7.5), it is not difficult to check that, for any fixed j ∈ {1, . . . , N }, if K1 , K2 ∈ 2[1,N ] , then χ (K1 ∩ K2 , j) = χ (K1 , j) χ (K2 , j) , in {0, 1}. Indeed, j ∈ K1 ∩ K2 , equivalently, χ (K1 ∩ K2 , j) = 1, if and only if j ∈ K1 and j ∈ K2 , if and only if χ (K1 , j) = 1, and χ (K2 , j) = 1, satisfying the above equality. Similarly, χ (K1 ∩ K2 , j) = 0, if and only if j ∈ / K1 , or j ∈ / K2 , if and only if χ (K1 , j) = 0, or χ (K2 , j) = 0, satisfying the above equality, too. Inductively,   Y n n χ ∩ Kl , j = χ (Kl , j) , l=1

l=1

in {0, 1}. By (7.5), one of our main results, Theorem 6.19 of Section 6 can be re-characterized as follows. Theorem 7.1. Let T ∈ ΛN be a shift operator (7.1), and xj ∈ XN , a generating semicircular element of XN . Then the free random variable T (xj ) ∈ XN is semicircular, if and only if " # ! l−1 l−1 X X n l l χ ∩ nkinit − kt , mkinit − kt , j = 1; (7.6) l=1

t=1

t=1

94 ■ Banach-Space Operators On C ∗ -Probability Spaces

meanwhile, the free distribution of T (xj ) is the zero free distribution on XN , if and only if

χ

n

∩

l=1

" l nkinit −

l−1 X

l−1 X

l mkinit −

kt ,

t=1

#

!

kt , j

= 0,

(7.7)

t=1

where χ is the function (7.5). Proof. The semicircularity characterization (7.6), and the zero-freedistributed-ness (7.7), up to the function χ of (7.5), is shown by (6.3.6) and (6.3.7). Indeed, by (6.3.6), one has n Y

T (xj ) =

χkl

j+

!!

l−1 X

kt

x

j+

t=1

l=1

n P

kl

l=1

where χk (t) are in the sense of (6.3.3) n Y

=

χ

h

i

l l nkinit , mkinit , j+

l−1 X

!! kt

x

n P

j+

t=1

l=1

kl

l=1

by the definition (7.5) of χ

=

n Y

" l − nkinit

χ

=χ

∩

l=1

kt ,

l mkinit −

t=1

l=1 n

l−1 X

" l nkinit −

l−1 X

kt ,

l−1 X

#

!!

kt , j

x

l mkinit −

t=1

l−1 X

n P

j+

t=1

kl

l=1

#

!

kt , j

t=1

! x

j+

n P

kl

,

(7.8)

l=1

by (7.5), in XN . The computation (7.8) implies that, the condition (7.6) holds, if and only if T (xj ) =x

j+

n P

kl

l=1

and hence, it is semicircular in XN .

∈ XN , in XN ,

Free-Probabilistic Information on XN ■ 95

Meanwhile, the condition (7.7) holds, if and only if T (xj ) = 0N , in XN , whose free distribution is the zero free distribution on XN , by (7.8). ■ The above theorem shows that the semicircularity-preserving property for the generating semicircular elements of XN on the C ∗ probability space XN , under the action of the finite operator-products of ΛN in the generating partial isometries λN , is re-characterized by the functional values of the function χ of (7.5).

CHAPTER

8

Application I. Shift Operators Acting on X∞

N THIS SECTION, WE CONSIDER THE C ∗ -PROBABILITY ∞ space X∞ generated by the free semicircular family X∞ = {xj }j=1 , and the shift-operator algebra ∞ acting on X∞ , generated by the  kΛ restricted-shift family λ∞ = β∞ k∈Z , in the operator space B (XN ). Recall that, in Section 6.4, we concentrated on studying how the shift operator ΛN acting on the C ∗ -probability space XN , where N < ∞ in N∞ >1 . In particular, it is shown there that, if N < ∞, then the generating k partial isometries βN ∈ λN of ΛN with k ̸= 0 and k ̸= ±N , and their finite operator-products are conditional-nilpotent on XN (and hence, certain finite-sum shift operators in λN are conditional-nilpotent in ΛN ). However, if N = ∞, then the conditional-nilpotent-property of k β∞ ∈ λ∞ is not satisfied in Λ∞ , by (5.2.4). Thus, we here focus on the action α∞ of the partial isometries of λ∞ in Λ∞ , and the corresponding deformed free probability on X∞ . k Recall that, if β∞ ∈ λ∞ is a generating partial isometry of Λ∞ , then k k β∞ = Q[nk ,mk ] β∞ Q[nk ,mk ] , in ΛN , (8.1)

I

fin

with  k  ninit , mkinit =

fin

init

init

( [1, ∞ − k] = [1, ∞] if k ≥ 0 [1 + |k| , ∞] if k < 0,

DOI: 10.1201/9781003263487-8

97

98 ■ Banach-Space Operators On C ∗ -Probability Spaces

and  k  nfin , mkfin =

( [1 + k, ∞] if k ≥ 0 [1, ∞] = N if k < 0,

in N, for all k ∈ Z. k Lemma 8.1. Let β∞ ∈ λ∞ be a generating partial isometry of Λ∞ . If xj ∈ X∞ is a generating semicircular element of X∞ , then   xj+k if k < 0, j ≥ 1 + |k| k β∞ (xj ) = xj+k if k ≥ 0 (8.2)   0N otherwise,

in XN , for all j ∈ N, and k ∈ Z. Proof. By Theorem 7.1, one has  
k β∞ (xj ) = χ nkinit , mkinit , j xj+k , in X∞ , where 

nkinit , mkinit



(8.3)

( [1, ∞ − k] = [1, ∞] if k ≥ 0 = [1 + |k| , ∞] if k < 0,

in N for k ∈ Z, by (8.1), where the function χ : 2N × N → {0, 1} , is in the sense of (7.5), defined by ( 1 χ (K, j) = χK (j) = 0

if j ∈ K if j ∈ / K,

for all (K, j) ∈ 2N × N. By (8.3), for any generating semicircular element xj ∈ X∞ of X∞ , one can get that   xj+k if k < 0, j ≥ 1 + |k| k β∞ (xj ) = xj+k if k ≥ 0   0N otherwise, in XN . Therefore, the operator-equality (8.2) holds.

■

Shift Operators Acting on X∞ ■ 99

For example, if k = 1 in Z, then 1 β∞ (xj ) = xj+1 , in XN , ∀j ∈ N,

while
1 ∗ β∞

(xj ) =

−1 β∞

( xj−1 (xj ) = 0N

if j ≥ 2 otherwise,

in XN , by (8.2). kl Lemma 8.2. Let β∞ ∈ λ∞ be generating partial isometries of Λ∞ , n Q k for l = 1, . . . , n, for n ∈ N, and let T = β∞n−l+1 ∈ Λ∞ be a partial l=1

isometry. If xj ∈ X∞ is a generating semicircular element of X∞ , then      l−1 l−1 n P P  kl kl  n  x if χ ∩ n − k , m =1 − k t l , j  j+ P kl init init l=1 t=1 t=1 l=1 T (xj ) =     l−1 l−1  n P P  kl kl  − k = 0, − k , m 0 if χ ∩ n  ∞ l , j t init init l=1

t=1

t=1

(8.4) for all j ∈ N, where χ is the function (8.3). Proof. By Theorem 7.1 and (8.3), we obtain the formula (8.4) in X∞ . ■ By (8.4), we obtain the following free-distributional data. kl Corollary 8.3. Let β∞ ∈ λ∞ be generating partial isometries of the shift-operator algebra Λ∞ , for l = 1, . . . , n, for n ∈ N, and let n Q k T = β∞n−l+1 ∈ Λ∞ be the operator-product of them. If xj ∈ X∞ l=1

is a generating semicircular element of X∞ , then s

φ∞ ((T (xj )) )     l−1 l−1 n P P  k k l l  kt , minit − kl , ωs c 2s if χ ∩ ninit − l=1 t=1 t=1    = l−1 l−1 n P P  kl kl  0 if χ ∩ ninit − kt , minit − kl , l=1

for all s ∈ N.

t=1

t=1

 j

=1 (8.5)

 j

=0

100 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. The free-distributional data (8.5) is directly obtained by the operator-equality (8.4) on X∞ . ■ Now, consider a shift operator, n X

Un =

k β∞ ∈ ΛN , for n ∈ N.

(8.6)

k=1

For instance, 1 1 2 U1 = β ∞ , and U2 = β∞ + β∞ ,

in Λ∞ , etc.. Observe that, for any generating semicircular element xj ∈ X∞ of X∞ , one has Un (xj ) =

n X

k β∞ (xj ) =

k=1

n X

xj+k ∈ X∞ ,

(8.7)

k=1

for all n, j ∈ N. However, Un∗

(xj ) =

n X

−k β∞

(xj ) =

k=1

with

n X

χj,k ∈ X∞ ,

(8.8)

k=1

χj,k

( xj−k = 0∞

if j − k ≥ 1 otherwise,

χj,k

( xj−k = 0∞

if j ≥ 1 + k otherwise,

and hence,

in X∞ , for all k = 1, . . . , n. Note that j ≥ 1 + k ⇐⇒ k ≤ j − 1,

(8.9)

i.e., 1 ≤ k ≤ j − 1, in {1, . . . , n}, for n, j ∈ N, Note that the condition (8.9) holds only if j − 1 ≥ 1 in N, in (8.8)!

Shift Operators Acting on X∞ ■ 101

So, by (8.8) and (8.9), under the same conditions, one has that Un∗

(xj ) =

n X

−k β∞

(xj ) =

j−1 X

−k β∞

(xj ) =

xj−k ,

k=1

k=1

k=1

j−1 X

in X∞ , only if j − 1 ≥ 1 ⇐⇒ j ≥ 2, in N. In other words,  j−1  P x j−k ∗ Un (xj ) = k=1  0 ∞

if 1 ≤ j − 1 ≤ n

(8.10)

otherwise,

in X∞ , for all n, j ∈ N. Proposition 8.4. Let Un =

n P

k β∞ ∈ Λ∞ be a shift operator (8.5) for

k=1

n ∈ N, and let xj ∈ X∞ be a generating semicircular element of X∞ for j ∈ N. Then n X Un (xj ) = xj+k ∈ X∞ , (8.11) k=1

meanwhile

 j−1  P x j−k ∗ Un (xj ) = k=1  0 ∞

if 1 ≤ j − 1 ≤ n otherwise,

in X∞ . Proof. The operator-equalities of (8.11) are shown by (8.7) and (8.10), respectively. ■ By (8.11), one obtains the following free-distributional data on X∞ . Theorem 8.5. Let Un ∈ Λ∞ be a shift operator (8.5), and let xj ∈ X∞ in X∞ , for n, j ∈ N. Then, for any s ∈ N,


φ∞ Un xsj = n ωs c 2s , (8.12) and φ∞ Un∗



xsj =

(
(j − 1) ωs c 2s 0

if 1 ≤ j − 1 ≤ n otherwise.

102 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. For any arbitrary s ∈ N, if Un ∈ Λ∞ is a shift operator (8.5), then n
X s Un x j = xsj+k ∈ X∞ , k=1 l since β∞ ∈ λ∞ ⊂ Λ∞ are ∗-homomorphisms on X∞ , and similarly, j−1 X
xsj−k ∈ X∞ , Un∗ xsj = εj−1,n

(8.13)

k=1

where εj−1,n

( 1 if 1 ≤ j − 1 ≤ n = 0 otherwise,

def

for all j ∈ N, by (8.11). By (8.13), we have that φ∞ Un xsj





=

n X



ωs c 2s = n ωs c 2s ,

(8.14)

k=1

and φ∞ Un∗

xsj





= εj−1,n

j−1 X



ωs c 2s = εj−1,n (j − 1) ωs c 2s ,

k=1

for all s ∈ N. Therefore, the free-distributional data (8.12) holds by (8.14).

■

Let Un ∈ Λ∞ be a shift operator (8.6), and let xj ∈ X∞ be a generating semicircular element of X∞ . Then, by (8.11), one has that !s ! n s X X Y s xj+k = xj+kl , (8.15) (Un (xj )) = k=1

(k1 ,...,ks )∈{1,...,n}s

and s

(Un∗ (xj )) = εj−1,n

l=1

X

s Y

(k1 ,...,ks )∈{1,...,j−1}s

l=1

! xj−kl

,

in X∞ , where εj−1,n ∈ {0, 1} is in the sense of (8.13), for all j ∈ N.

Shift Operators Acting on X∞ ■ 103

Theorem 8.6. Let Un ∈ Λ∞ be a shift operator (8.6), and xj ∈ X∞ , a generating semicircular element of X∞ . Then ! s X Y s φ∞ ((Un (xj )) ) = φ∞ xj+kl , (8.16) (k1 ,...,ks )∈{1,...,n}s

l=1

and φ∞ ((Un∗

s

(xj )) ) = εj−1,n

X

φ∞

(k1 ,...,ks )∈{1,...,j−1}

s

s Y

! xj−kl

,

l=1

where the joint free moments, which are the summands, ! ! s s Y Y φ∞ xj+kl and φ∞ xj−kl l=1

l=1

in (8.16), are characterized by Corollary 5.4 (or (3.22), or (3.25)). Proof. The proof of (8.16) is done by (8.12), (8.15) and Corollary 5.4. ■ Now, let Un1 , Un2 ∈ Λ∞ be shift operators (8.6), for n1 , n2 ∈ N. Define a new shift operator U ∈ Λ∞ by def

U = Un∗1 + Un2 , in Λ∞ .

(8.17)

Lemma 8.7. Let U ∈ Λ∞ be a shift operator (8.17). If xj ∈ X∞ is a generating semicircular element of X∞ . then ! ! j−1 n2 X X U (xj ) = εj−1,n1 xj−l + xj+k , in X∞ , (8.18) l=1

k=1

where εj−1,n1 ∈ {0, 1} is in the sense of (8.13). Proof. If U ∈ Λ∞ is in the sense of (8.17), then U (xj ) = Un∗1 (xj ) + Un2 (xj ) , in X∞ . So, by (8.11) and (8.13), the operator-equality (8.18) holds. ■ By (8.18), we obtain the following free-distributional data.

104 ■ Banach-Space Operators On C ∗ -Probability Spaces

Theorem 8.8. Let U = Un∗1 + Un2 ∈ Λ∞ be a shift operator (8.17), and xj ∈ X∞ , a generating semicircular element of X∞ . Then


(8.19) φ∞ U xsj = (εj−1,n1 (j − 1) + n2 ) ωs c 2s , for all s ∈ N, where εj−1,n1 are in the sense of (8.13), for all j, n1 ∈ N. Proof. The free-distributional data (8.19) is obtained by (8.12) and (8.18). ■

CHAPTER

9

Application II. The Circular Law

HROUGHOUT THIS SECTION, we fix N ∈ N∞ >1 , and the ∗ C -probability space XN = (XN , φN ) generated by the free N semicircular family XN = {xj }j=1 , and the shift-operator algebra ΛN  k N induced by the restricted-shift family λN = βN of partial k=−N isometries. In this section, we consider how our partial isometries of ΛN affect the circular law on XN .

T 9.1

CIRCULAR ELEMENTS

Let (B, ψ) be a topological ∗-probability space, and assume that x1 and x2 are two, free, semicircular elements of (B, ψ). Then one can construct a new free random variable y ∈ (B, ψ), √ def 1 y = √ (xi + ix2 ) , with i = −1 ∈ C. 2

(9.1.1)

Then, it has its adjoint, 1 y ∗ = √ (x1 − ix2 ) ∈ (B, ψ) , 2 since the semicircular elements x1 and x2 are self-adjoint in B. It shows that the free random variable y of (9.1.1) is not self-adjoint in (B, ψ). So, the free distribution of y is characterized by the joint free moments, or equivalently, by the joint free cumulants of {y, y ∗ }, by (2.1.1).

DOI: 10.1201/9781003263487-9

105

106 ■ Banach-Space Operators On C ∗ -Probability Spaces

Observe that, if y ∈ (B, ψ) is a free random variable (9.1.1), and if k• (. . .) is the free cumulant on B in terms of the linear functional ψ on B (e.g., [17,22,23]), then     1 1 √ (x1 + ix2 ) , . . . , √ (x1 + ix2 ) kn y, y, . . . . . . , y  = kn | {z } 2 2 n-times n  1 kn ((x1 + ix2 ) , . . . , (x1 + ix2 )) = √ 2 by the bimodule-map property of free cumulant (e.g., [17,23]) n  1 (kn (x1 , . . . , x1 ) + kn (ix2 , . . . , ix2 )) = √ 2 since x1 and x2 are free, and hence, x1 and ix2 are free in (B, ψ) (Recall that if multi free random variables are free, then all “mixed” free cumulants of them vanish. e.g., [17])  n 1 = √ (δn,2 + in kn (x2 , . . . , x2 )) 2 by the semicircularity (2.2.3), or (2.2.5) of x1 , where δ is the Kronecker delta  n  n 1 1 n = √ (δn,2 + i δn,2 ) = √ δn,2 (1 + in ) 2 2 by the semicircularity of x2 ( 0 if n ̸= 2
= 1 2 = 0 if n = 2 2 1+i = 0,

(9.1.2)

for all n ∈ N. Similar to (9.1.2), one can get that 



kn y ∗ , y ∗ , . . . .., y ∗  = 0, | {z } n-times

for all n ∈ N.

(9.1.3)

The Circular Law ■ 107

The above free-cumulant computations (9.1.2) and (9.1.3) show that all n-th free cumulants of y and those of y ∗ vanish in (B, ψ). So, to consider the free distribution of the free random variable y ∈ (B, ψ), it suffices to study “mixed” free cumulants of {y, y ∗ }, kn (y r1 , y r2 , . . . , y rn ) , for all mixed n-tuples (r1 , . . . , rn ) of {1, ∗}, for n ∈ N \ {1}. However, as it is discussed in [24,25], all mixed free cumulants of {y, y ∗ } vanish, except for the possible nonzero cases where kn (y, y ∗ , y, y ∗ , . . . , y, y ∗ ) ,

(9.1.4)

and kn (y ∗ , y, y ∗ , y, . . . , y ∗ , y) , by (9.1.2) and (9.1.3). i.e., the R-diagonality of y is satisfied in (B, ψ) by (9.1.4). So, let’s concentrate on “possible” non-vanishing mixed free cumulants of {y, y ∗ } of (9.1.4). First, consider the case where n = 2; if n = 2, then   1 1 ∗ k2 (y, y ) = k2 √ (x1 + ix2 ) , √ (x1 − ix2 ) 2 2 1 = (k2 (x1 , x1 ) + k2 (ix2 , −ix2 )) 2 by the freeness of {x1 } and {±ix2 } in (B, ψ) =


1 1 − i2 k2 (x2 , x2 ) 2

by the bi-module-map property of free cumulant =

1 (1 + 1) = 1, 2

(9.1.5)

and similarly, k2 (y ∗ , y) = 1.

(9.1.6)

If n ≥ 3, then, by the semicircularity (2.2.3) and (2.2.5), kn (y, y ∗ , y, y ∗ , . . . , y, y ∗ ) = 0,

(9.1.7)

108 ■ Banach-Space Operators On C ∗ -Probability Spaces

and kn (y ∗ , y, y ∗ , y, . . . , y ∗ , y) = 0, in (B, ψ). Therefore, the free distribution of a free random variable y ∈ (B, ψ) of (9.1.1) (or that of y ∗ ) is characterized by (9.1.2), (9.1.3), (9.1.5), (9.1.6) and (9.1.7). Theorem 9.1. If y ∈ (B, ψ) is a free random variable (9.1.1), then the only non-vanishing joint free cumulants of {y, y ∗ } are k2 (y, y ∗ ) = 1 = k2 (y ∗ , y) .

(9.1.8)

Proof. The proof of (9.1.8) is done by (9.1.2), (9.1.3), (9.1.5), (9.1.6) and (9.1.7) (e.g., see [24,25]). ■ The above theorem fully characterizes the free distribution of an element y ∈ (B, ψ) by the only non-vanishing joint free cumulants (9.1.8). Now, let’s study equivalent free-distributional data of a free random variable y ∈ (B, ψ) of (9.1.1). Observe that, for any n ∈ N, ! X Y n ψ (y ) = k|V | (y, . . . , y) π∈N C(n)

V ∈π

X

Y

π∈N C(n)

V ∈π

by the M¨ obius inversion ! =

0

= 0,

(9.1.9)

by (9.1.2). Similarly, n

ψ ((y ∗ ) ) = 0,

(9.1.10)

for all n ∈ N, by (9.1.3). By (9.1.8), the only possible non-vanishing mixed free moments of {y, y ∗ } are n n ψ ((yy ∗ ) ) and ψ ((y ∗ y) ) , for all n ∈ N.

The Circular Law ■ 109

Observe that ! X

Y

π∈N C(2n)

V ∈π

n

ψ ((yy ∗ ) ) =

kV

where kV are the block-depending free cumulants of {y, y ∗ }, by the M¨ obius inversion of [17] ! X Y = χB kB θ∈N C2 (2n)

B∈θ

by (9.1.8), where N C2 (2n) = {π ∈ N C(2n) : V ∈ π ⇔ |V | = 2}, as in Section 3, and   if a block B induces either    1 k =k (y, y ∗ ) or k = k (y ∗ , y) B 2 B 2 χB =     0 otherwise, for all B ∈ θ, for all θ ∈ N C2 (2n), and hence, it goes to ! ! X Y X Y = kB = 1 = |X2 (2n)| , θ∈X2 (2n)

B∈θ

θ∈X2 (2n)

by (9.1.8), where  X2 (2n) = π ∈ N C2 (2n)

(9.1.11)

B∈θ

∀V ∈ π induces either kV = k2 (y, y ∗ ), or kV = k2 (y ∗ , y)

 .

Note that, since N C2 (2n) is defined over {1, . . . , 2n}, the subset X2 (2n) of (9.1.11) is not empty for all n ∈ N. For example, X2 (2) = {{(1, 2)}} = {12 } , X2 (4) = {{(1, 2), (3, 4)}, {(1, 4), (2, 3)}} , and   {(1, 2), (3, 4), (5, 6)}, {(1, 6), (2, 3), (4, 5)}, X2 (6) = {(1, 6), (2, 5), (3, 4)}, {(1, 2), (3, 6), (4, 5)}, ,   {(1, 4), (2, 3), (5, 6)}

(9.1.12)

110 ■ Banach-Space Operators On C ∗ -Probability Spaces

etc., and hence,     2 3 ψ (yy ∗ ) = 1, ψ (yy ∗ ) = 2, ψ (yy ∗ ) = 5, respectively, by (9.1.11) and (9.1.12). Theorem 9.2. Let y = √12 (x1 + ix2 ) ∈ (B, ψ) be a free random variable (9.1.1) generated by two, free semicircular elements x1 , x2 ∈ (B, ψ). Then the only non-vanishing joint free moments are ψ ((yy ∗ )n ) = |X2 (2n)| = ψ ((y ∗ y)n ) ,

(9.1.13)

for all n ∈ N, where X2 (2n) is the subset (9.1.11) of the noncrossingpartition lattice N C(2n). Proof. By (9.1.11), one has n

ψ ((yy ∗ ) ) = |X2 (2n)| , for all n ∈ N. Similar to (9.1.11), n

ψ ((y ∗ y) ) = |X2 (2n)| , ∀n ∈ N. Thus, the only non-vanishing free-distributional data (9.1.13) are obtained. ■ The above theorem re-characterizes the free distribution of a free random variable y ∈ (B, ψ) of (9.1.1), in terms of the joint free moments of {y, y ∗ }, which is equivalent to (9.1.8). Definition 9.1. A free random variable y = √12 (x1 + ix2 ) ∈ (B, ψ) of (9.1.1) is said to be the circular element generated by two, free semicircular elements x1 and x2 . Theorems 9.1 and 9.2 characterize the free distribution of a circular element y ∈ (B, ψ) by the only non-vanishing joint free cumulants, k2 (y, y ∗ ) = 1 = k2 (y ∗ , y) , by (9.1.8); and equivalently, by the only non-vanishing joint free moments, n

n

ψ ((yy ∗ ) ) = |X2 (2n)| = ψ ((y ∗ y) ) , ∀n ∈ N, by (9.1.13).

The Circular Law ■ 111

By the universality (2.2.4) and (2.2.5) of “the” semicircular law, the free distributions of all circular elements are universal, by the very definition (9.1.1). So, we call the free distributions of all circular elements, “the” circular law. The circular law is characterized by the only non-vanishing joint free cumulants (9.1.8), or by the only non-vanishing joint free moments (9.1.13).

9.2

DEFORMED CIRCULAR LAWS ON XN BY PARTIAL ISOMETRIES OF λN

 k N Let λN = βN be the restricted-shift family generating the shiftk=−N operator algebra ΛN in the operator space B (XN ) of the C ∗ -probability space XN . Corollary 9.3. Let j1 ̸= j2 in {1, . . . , N }, and xjl ∈ XN , the corresponding generating semicircular elements of XN , for l = 1, 2. Then a free random variable def 1 y = √ (xj1 + ixj2 ) ∈ XN (9.2.1) 2 is a circular element. Proof. If j1 ̸= j2 in {1, . . . , N }, then two generating free random variables xj1 and xj2 are distinct in the free semicircular family XN . It implies that they are free in XN . So, by Definition 9.1, the free random variable y of (9.2.1) is a circular element of XN . ■ By the circularity of the free random variable y ∈ XN of (9.2.1), the free distribution of y is characterized by the universal free-distributional data (9.1.8), or (9.1.13). k Let βN ∈ λN be a generating partial isometry of ΛN , with k ≥ 0 in {0, ±1, . . . , ±N }. Then k k βN = Q[1+k,N ] βN Q[1,N −k] , in ΛN .

If y = (9.2.2), k βN

√1 2

(y) =

(9.2.2)

(xj1 + ixj2 ) ∈ XN is a circular element (9.2.1), then, by k Q[1+k,N ] βN Q[1,N −k]






1 √ (xj1 + ixj2 ) 2



1 = √ (χ ([1, N − k], j1 ) xj1 +k + iχ ([1, N − k], j2 ) xj2 +k ) , 2 (9.2.3) where χ is the function (7.5).

112 ■ Banach-Space Operators On C ∗ -Probability Spaces k Lemma 9.4. Let βN ∈ λN be a generating partial isometry of ΛN , with k ≥ 0. If y ∈ XN is a circular element (9.2.1), then   if χ ([1, N − k], j1 ) = 1,   √1 (xj +k + ixj +k )  1 2  2  and if χ ([1, N − k], j2 ) = 1      1 if χ ([1, N − k], j1 ) = 1,     √2 xj1 +k and χ ([1, N − k], j2 ) = 0 k βN (y) =  if χ ([1, N − k], j1 ) = 0,   √i xj +k  2  2  and χ ([1, N − k], j2 ) = 1      if χ ([1, N − k], j1 ) = 0,    0N and χ ([1, N − k], j2 ) = 0, (9.2.4) and   if χ ([1 + |k| , N ], j1 ) = 1,   √1 (xj −k + ixj −k )  1 2  2  and if χ ([1 + |k| , N ], j2 ) = 1      1 if χ ([1 + |k| , N ], j1 ) = 1,     √2 xj1 −k and χ ([1 + |k| , N ], j2 ) = 0 −k βN (y) =  if χ ([1 + |k| , N ], j1 ) = 0,   √i xj −k  2  2  and χ ([1 + |k| , N ], j2 ) = 1      if χ ([1 + |k| , N ], j1 ) = 0,    0N and χ ([1 + |k| , N ], j2 ) = 0,

in XN . Proof. The first formula in (9.2.4) is shown by (9.2.3).
−k k ∗ Observe now that, if k ≥ 0, and βN = βN ∈ λN is a partial isometry of ΛN , then 1 −k βN (y) = √ (χ ([1 + |k| , N ], j1 ) xj1 +k + iχ ([1 + |k| , N ], j2 ) xj2 +k ) , 2 (9.2.5) as in (9.2.3), by (7.3). Thus, the second formula in (9.2.4) holds by (9.2.5). ■ By (9.2.4), we obtain the following free-distributional data.

The Circular Law ■ 113 k Theorem 9.5. Let βN ∈ λN be a generating partial isometry of ΛN for k ∈ {0, ±1, . . . , ±N }, and let y ∈ XN be a circular element k (9.2.1). Then the free distribution of βN (y) ∈ XN is the circular law, characterized by the only nonzero joint free cumulants (9.1.8), or the only nonzero joint free moments (9.1.3), if and only if  
χ nkinit , mkinit , jl = 1, ∀l = 1, 2, (9.2.6)

where   k ninit , mkinit =

( [1, N − k] [1 + |k| , N ]

if k ≥ 0 if k < 0,

where the function χ is in the sense of (7.5). Proof. By (9.2.4), the condition (9.2.6) holds, if and only if 1 k βN (y) = √ (xj1 +k + ixj2 +k ) ∈ XN , 2 induced by two distinct semicircular elements, xj1 +k ̸= xj2 +k ∈ XN , and hence, free in XN , if and only if the free random variable βN (y) is circular in XN , if and only if the free distribution of it is the circular law, characterized by the nonzero free-distributional data (9.1.8), or (9.1.13). ■ Consider the following free-probabilistic concept (e.g., [5–7]). Definition 9.2. Let (B, ψ) be a topological ∗-probability space. Then a self-adjoint free random variable u ∈ (B, ψ) is said to be weighted-semicircular with its weight t0 ∈ C× = C \ {0} (in shift, t0 -semicircular), if n

ψ (un ) = ωn t02 c n2 , for all n ∈ N.

(9.2.7)

By the M¨ obius inversion of [17], the above t0 -semicircularity definition (9.2.7) is re-characterized by that: a self-adjoint free random variable u ∈ (B, ψ) is t0 -semicircular, if and only if   knψ u, u, . . . , u = δn,2 t0 , ∀n ∈ N, | {z } n-times

(9.2.8)

114 ■ Banach-Space Operators On C ∗ -Probability Spaces

where k•ψ (. . .) is the free cumulant on B in terms of ψ, where δ is the Kronecker delta. k Theorem 9.6. Let βN ∈ λN be a generating partial isometry of ΛN . k If y ∈ XN is a circular element (9.2.1), then βN (y) is 12 -semicircular in XN , if and only if

χ

 k 
 
ninit , mkinit , j1 = 1, χ nkinit , mkinit , j2 = 0,

(9.2.9)

  where χ is the function (7.5), and nkinit , mkinit is in the subset (9.2.6) of {1, . . . , N }. Proof. Under hypothesis, the condition (9.2.9) is satisfied, if and only if 1 k βN (y) = √ xj1 +k ∈ XN , 2 by (9.2.4). It is easy to verify that this free random variable is selfadjoint in XN , since 

1 √ xj1 +k 2

∗

1 1 = √ x∗j1 +k = √ xj1 +k , in XN . 2 2

Observe that, for any n ∈ N, one can get that φN



k βN


n  (xj ) = =



1 √ 2

n



1 √ 2

n

φN xnj1 +k ωn c n2







  n2 1 c n2 , = ωn 2

(9.2.10)

by the semicircularity of xj1 +k ∈ XN in XN . So, by (9.2.7) and (9.2.10), this self-adjoint free random variable
√1 xj +k is 1 -semicircular in XN . ■ 2 2 1 Equivalently, the condition (9.2.9) holds, if and only if knN

k βN

by (9.2.8) and (9.2.10).

k (y) , . . . , βN

  1 (y) = δn,2 , 2


The Circular Law ■ 115 k Now, suppose βN ∈ λN is a generating partial isometry of ΛN , for k ∈ {0, ±1, . . . , ±N }, and y ∈ XN is a circular element (9.2.1), and assume that  
 
χ nkinit , mkinit , j1 = 0, χ nkinit , mkinit , j2 = 1, (9.2.11)

Then, by (9.2.4), we obtain that i k βN (y) = √ xj2 +k , in XN . 2 This free random variable 

i √ xj2 +k 2

√i xj +k 2 2

∗

(9.2.12)

satisfies that

−i = √ xj2 +k , in XN , 2

implying that it is not self-adjoint in XN . So, the free distribution of this free random variable is characterized by the joint free moments of   i −i √ xj2 +k , √ xj2 +k , 2 2 on XN . denote Let wj2 +k = √i2 xj2 +k ∈ XN be the above non-self-adjoint element. For any n

(r1 , . . . , rn ) ∈ {1, ∗} , ∀n ∈ N, one has that

n Y l=1

wjr2l +k =

r l n  Y i √ 2 l=1

! xnj2 +k ,

in XN , satisfying that #(1)  #(∗)  n Y −i i rl √ xnj2 +k ∈ XN , wj2 +k = √ 2 2 l=1 where #(1) = the number of 1′ s, and #(∗) = the number of ∗′ s, n

in an n-tuple (r1 , . . . , rn ) ∈ {1, ∗} .

(9.2.13)

116 ■ Banach-Space Operators On C ∗ -Probability Spaces k Theorem 9.7. Let βN ∈ λN be a generating partial isometry of ΛN , and y ∈ XN , a circular element of (9.2.1). Then the condition (9.2.11) is satisfied, if and only if the free distribution of the free random k variable is characterized by the following joint free moments  k βN (y) ∗ k of βN (y) , βN (y) ,

φN

n Y

! k βN

(y)

rl

 = ωn

l=1

i √ 2

#(1) 

−i √ 2

#(∗) c n2 ,

(9.2.14)

n

for all (r1 , . . . , rn ) ∈ {1, ∗} , for all n ∈ N, where the quantities #(1) and #(∗) are in the sense of (9.2.13). Proof. The condition (9.2.11) is satisfied, if and only if i k βN (y) = √ xj2 +k ∈ XN , 2 n

by (9.2.4). So, for any (r1 , . . . , rn ) ∈ {1, ∗} , for all n ∈ N, φN

n Y

! k βN

(y)

rl

 = φN

l=1

i √ 2

#(1) 

−i √ 2

#(∗)

! xnj2 +k

by (9.2.13)  =

i √ 2

#(1) 

−i √ 2

#(∗) ωn c n2




by the semicircularity of xj2 +k ∈ XN in XN , where #(1) and #(∗) are in the sense of (9.2.13). Therefore, the joint-free-moment formula (9.2.14), characterizing k the free distribution of βN (y), holds on XN . ■ Finally, one has the following free-probabilistic information. k Theorem 9.8. Let βN ∈ λN be a generating partial isometry of ΛN , and y ∈ XN be a circular element (9.2.1). Then the free distribution of k βN (y) is the zero free distribution on XN , if and only if

χ

 k 
ninit , mkinit , jl = 0, ∀l = 1, 2.

(9.2.15)

The Circular Law ■ 117

Proof. By (9.2.4), the condition (9.2.15) is satisfied, if and only if k βN (y) = 0N , in XN .

Therefore, the free distribution of it is the zero free distribution on XN . ■ Theorems 9.5, 9.6, 9.7 and 9.8 characterize how our partial isometries of λN deform the circular law on XN .

CHAPTER

10

Application III. Free Poisson Distributions

HROUGHOUT THIS SECTION, WE FIX N ∈ N∞ >1 . In Section 9, we studied how our generating partial isometries  k N βN k=−N = λN of the shift-operator algebra ΛN deform the circular law on the C ∗ -probability space XN generated by the free semicircular N family XN = {xj }j=1 . Here, we are interested in how they deform certain free Poisson distributions (induced by XN ) on XN .

T

10.1

FREE POISSON ELEMENTS

Let (B, ψ) be a topological ∗-probability space, and let x ∈ (B, ψ) be a semicircular element. Assume that a self-adjoint free random variable a ∈ (B, ψ) is free from x in (B, ψ). i.e., all their “mixed” free cumulants of {a, x} vanish. By the self-adjointness of a ∈ (B, ψ), its free distribution is completely characterized by the free-moment sequence, ∞

(ψ (an ))n=1 , or, by the free-cumulant sequence,  

∞

kn a, a, . . . . . . ., a | {z } n-times

,

n=1

where k• (. . .) is the free cumulant on B in terms of the linear functional ψ on B. DOI: 10.1201/9781003263487-10

119

120 ■ Banach-Space Operators On C ∗ -Probability Spaces

Definition 10.1. Let x ∈ (B, ψ) be a semicircular element, and let a ∈ (B, ψ) be a self-adjoint free random variable, which is free from x. Define a free random variable Wxa ∈ (B, ψ) by def

Wxa = xax in (B, ψ).

(10.1.1)

Then this free random variable Wxa ∈ (B, ψ) is called the free Poisson element generated by x and a. The free distribution of Wxa is called the free Poisson distribution of Wxa . By definition, if Wxa = xax ∈ (B, ψ) is a free Poisson element (10.1.1), then ∗ (Wxa ) = x∗ a∗ x∗ = xax = Wxa , in (B, ψ), and hence, it is a self-adjoint free random variable. So, the free Poisson distribution of it is fully characterized either by the free moments or by the free cumulants of Wxa . Under the partial ordering (≤), π ≤ θ ⇐⇒ ∀B ∈ π, ∃V ∈ θ, s.t. B ⊆ V, the noncrossing-partition lattice N C(n) over {1, . . . , n}, for n ∈ N, has its maximal element, 1n = {(1, 2, . . . , n)} , the one-block partition, and its minimal element, 0n = {(1), (2), . . . , (n)} , the n-block partition (e.g., [17,22,23]). Suppose {1, . . . , n} = Ω1 ⊔ Ω2 , where Ωl are the subsets of {1, . . . , n}, for l = 1, 2, and ⊔ is the disjoint union, and let N C (Ωl ) be the noncrossing-patition lattice over Ωl , for l = 1, 2. Then, for θl ∈ N C(Ωl ), for l = 1, 2, one can construct a noncrossing partition, θ = θ1 ∨ θ2 ∈ N C(n),

Free Poisson Distributions ■ 121

as the join of θ1 and θ2 , as in [17,22,23]. For example, if θ1 = {(2, 5), (3)} ∈ N C ({2, 3, 5}) , and θ2 = {(1, 6, 7) , (4)} ∈ N C ({1, 4, 6, 7}) , then θ1 ∨ θ2 = {(1, 6, 7), (2, 5), (3), (4)} ∈ N C(7). Now, fix n ∈ N, and let N C(3n) be the noncrossing-partition lattice over {1, . . . , 3n}, and let Ω3n,1 = {1, 3, 4, 6, 7, 9, 10, . . . , 3n − 3, 3n − 2, 3n} ,

(10.1.2)

and Ω3n,2 = {2, 5, 8, 11, . . . , 3n − 1} , satisfying {1, . . . , 3n} = Ω3n,1 ⊔ Ω3n,2 . And then, take θ0 = {(1, 3n), (3, 4), (6, 7), . . . , (3n − 3, 3n − 2)} ∈ N C (Ω3n,1 ) , (10.1.3) where Ω3n,1 is in the sense of (10.1.2). Theorem 10.1. Let Wxa ∈ (B, ψ) be a free Poisson element (10.1.1). Then   kn Wxa , Wxa , . . . .., Wxa  = ψ (an ) , | {z } n-times

for all n ∈ N. Proof. By the semicircularity (2.2.5) of x ∈ (B, ψ), kn (x, . . . , x) = δn,2 , ∀n ∈ N. So, one has that kn (Wxa , . . . , Wxa ) = kn (xax, . . . , xax) =

X π∈N C(Ω3n,2 ), θ∈N C(Ω3n,1 ) π∨θ≤π0

kπ (xax, . . . xax)

(10.1.4)

122 ■ Banach-Space Operators On C ∗ -Probability Spaces

where kπ (. . .) =

Q

kV are the partition-depending free cumulants,

V ∈π

and π0 = {(1, 2, 3), (4, 5, 6), . . . , (3n − 2, 3n − 1, 3n)} ∈ N C(3n) (e.g., [17,22,23]), and hence, it goes to X

=

kπ∨θ0 (xax, . . . , xax)

π∈N C(Ω3n,2 ), π∨θ0 ∈N C(3n)

where Ω3n,2 is in the sense of (10.1.2), and θ0 ∈ N C(Ω3n,1 ) is in the sense of (10.1.3)     X = kπ a, a, . . . .., a kθ0 x, x, x, . . . . . . , x {z } {z } | | π∈N C(Ω3n,2 ), π∨θ0 ∈N C(3n)

n-times

2n-times

by the freeness of a and x (e.g., [17,22,23])   X n = kπ a, a, . . . .., a (k2 (x, x)) {z } | π∈N C(Ω3n,2 ), π∨θ0 ∈N C(3n) n-times X X = kπ (a, . . . , a) = kθ (a, . . . , a) π∈N C(Ω3n,2 )

θ∈N C(n)

since two lattices N C (Ω3n,2 ) and N C(n) are isomorphic (or, equivalent), because |Ω3n,2 | = n = |{1, . . . , n}| , and hence, it goes to = ψ (an ) , by the M¨ obius inversion. Therefore, the free-distributional data (10.1.4) holds.

■

The above theorem characterizes the free Poisson distribution of ∈ (B, ψ) by (10.1.4). By the M¨ obius inversion of [17] and (10.1.4), one obtains the following equivalent result.

Wxa

Free Poisson Distributions ■ 123

Theorem 10.2. Let Wxa = xax ∈ (B, ψ) be a free Poisson element (10.1.1). Then !  X Y  a n |V | ψ ((Wx ) ) = ψ a , (10.1.5) π∈N C(n)

V ∈π

for all n ∈ N. Proof. For any n ∈ N, we have that    X Y    n ψ ((Wxa ) ) = k|V | Wxa , Wxa , . . . ., Wxa   | {z } π∈N C(n)

V ∈π

|V |-times

by the M¨ obius inversion of [17] =

X

Y

π∈N C(n)

V ∈π



ψ a|V |



! ,

by (10.1.4), for all n ∈ N. Therefore, the free-moment computations (10.1.5) hold.

■

The above two theorems fully characterize the free Poisson distribution of the free Poisson element Wxa ∈ (B, ψ) of (10.1.1). They illustrate that the free Poisson distribution of Wxa is determined by the free distribution of a in (B, ψ), either by (10.1.4) or by (10.1.5). They also illustrate that, different from the semicircularity and the circularity, free Poisson distributions are not universal. They depend on the free distributions of self-adjoint free random variables a ∈ (B, ψ), which are free from a fixed semicircular element x ∈ (B, ψ), inducing the free Poisson elements Wxa ∈ (B, ψ) of (10.1.1).

10.2

DEFORMED FREE POISSON DISTRIBUTIONS ON XN

In this section, we concentrate on free Poisson elements of XN induced N by the free semicircular family XN = {xj }j=1 . Suppose j1 ̸= j2 in {1, . . . , N }. Then the corresponding semicircular elements xj1 and xj2 are distinct in XN , implying that they are free in XN . Define a free random variable Wj1 ,j2 ∈ XN by Wj1 ,j2 = xj1 xj2 xj1 ∈ XN .

(10.2.1)

124 ■ Banach-Space Operators On C ∗ -Probability Spaces

Since xj1 and xj2 are free in XN , and since xj2 ∈ XN is self-adjoint, this free random variable Wj1 ,j2 is a free Poisson element of XN in the sense of (10.1.1). Corollary 10.3. Let xj1 , xj2 ∈ XN be generating semicircular elements for j1 ̸= j2 in {1, . . . , N }, and let Wj1 ,j2 = xj1 xj2 xj1 ∈ XN be a free Poisson element (10.2.1). Then the free Poisson distribution of Wj1 ,j2 is characterized by knN (Wj1 ,j2 , . . . , Wj1 ,j2 ) = ωn c n2 ,

(10.2.2)

or, equivalently,  φN Wjn1 ,j2 = ωn 


Y

X

c |V | 2

V ∈π

π∈N Ce (n)



! ,

for all n ∈ N, where N Ce (n) = {θ ∈ N C(n) : ∀B ∈ θ, |B| ∈ 2N} . Proof. Let Wj1 ,j2 ∈ XN be a free Poisson element (10.2.1). Then, by (10.1.4),
knN (Wj1 ,j2 , . . . , Wj1 ,j2 ) = φN xnj2 = ωn c n2 , by the semicircularity of xj2 ∈ XN in XN . Equivalently, for all n ∈ N, one has that !   X Y
|V | n φN Wj1 ,j2 = φN xj2 , π∈N C(n)

V ∈π

by (10.1.5). More precisely, one can have φN Wjn1 ,j2 =


=

X

Y

π∈N C(n)

V ∈π

  |V | φN xj2

X

Y

π∈N Ce (n)

V ∈π

φN



|V | xj2

!



!

Free Poisson Distributions ■ 125

where N Ce (n) = {θ ∈ N C(n) : ∀B ∈ θ, |B| is even} is a subset of N C(n) consisting of all even noncrossing partitions !  X Y , = c |V | π∈N Ce (n)

2

V ∈π

for all n ∈
N. It is clear that if m is odd in N, then the free moments φN Wjm vanish, i.e., 1 ,j2
φN Wjm = 0, 1 ,j2 since the subsets N Ce (m) are empty in N C(m) whenever m is odd. Therefore, the free Poisson distribution of Wj1 ,j2 is characterized by the free-distributional data (10.2.2). ■ More generally, one may take a self-adjoint free random variable,   T ∈ ⋆ C [{xj }] , j̸=j1 in {1,...,N }

satisfying T ̸= t · IN for t ∈ C, in XN , and then construct a self-adjoint free random variable, Wj1 ,T = xj1 T xj2 , in XN , which is a free Poisson element of XN , such that knN (Wj1 ,T , . . . , Wj1 ,T ) = φN (T n ) , and φN

Wjn1 ,T




=

X

Y

π∈N C(n)

V ∈π



φN T

|V |



! ,

for all n ∈ N, by (5.3). However, for convenience, we here focus on free Poisson elements Wj1 ,j2 = xj1 xj2 xj1 of (10.2.1) in XN . k If βN ∈ λN is a generating partial isometry of ΛN , and if Wj1 ,j2 ∈ XN is a free Poisson element (10.2.1), then k βN (Wj1 ,j2 ) = χk (j1 )χk (j2 )xj1 +k xj2 +k xj1 +k ∈ XN ,

(10.2.3)

126 ■ Banach-Space Operators On C ∗ -Probability Spaces

with χk (jl ) = χ

 k 
ninit , mkinit , jl , ∀l = 1, 2,

in XN , where χ is the function (7.5), and (  k  [1, N − k] if k ≥ 0 ninit , mkinit = [1 + |k| , N ] if k < 0, in {1, . . . , N }, for all k ∈ {0, ±1, . . . , ±N }. Lemma 10.4. Let Wj1 ,j2 ∈ XN be a free Poisson element (10.2.1), and k let βN ∈ λN be a generating partial isometry of ΛN . Then ( xj1 +k xj2 +k xj1 +k if χk (jl ) = 1, ∀l = 1, 2 k βN (Wj1 ,j2 ) = (10.2.4) 0N otherwise, k in XN . Equivalently, βN (Wj1 ,j2 ) ̸= 0N in XN , if and only if it is a free Poisson element of XN .

Proof. The operator-equality (10.2.4) holds by (10.2.3). So, if k βN (Wj1 ,j2 ) is nonzero in XN , it becomes a free Poisson element of XN , too, since xj1 +k and xj2 +k are free in XN . Indeed, since   j1 ̸= j2 , in nkinit , mkinit ,   the quantities j1 +k and j2 + k are distinct in nkfin , mkfin in {1, . . . , N }, by (10.2.3). So, the freeness of them, and the self-adjointness of xj2 +k ∈ XN in XN , guarantee the free-Poisson-ness (10.1.1) of the nonzero free random variable xj1 +k xj2 +k xj1 +k in XN . k Conversely, by (10.2.3), if βN (Wj1 ,j2 ) becomes a free Poisson element xj1 +k xj2 +k xj1 +k in XN , then, definitely, it is a nonzero element of XN . ■ The above lemma proves the following result. Theorem 10.5. Let Wj1 ,j2 ∈ XN be a free Poisson element (10.2.1), k and let βN ∈ ΛN be a partial isometry (10.2.3). The quantities j1 , j2 ∈ {1, . . . , N } satisfy   
 jl ∈ nkinit , mkinit ⇐⇒ χ nkinit , mkinit , jl = 1, (10.2.5)

Free Poisson Distributions ■ 127

for all l = 1, 2, if and only if the free distribution of W is a free Poisson distribution, characterized by

denote

=

k βN (Wj1 ,j2 )

knN (W, . . . , W ) = ωn c n2 , ∀n ∈ N,

(10.2.6)

equivalently, by  φN (W n ) = ωn 

Y

X

π∈N Ce (n)

V ∈π

c |V |



!  , ∀n ∈ N.

2

Meanwhile, the quantities j1 , j2 ∈ {1, . . . , N } satisfy either     j1 ∈ / nkinit , mkinit , or j2 ∈ / nkinit , mkinit , (10.2.7) equivalently, either  
 
χ nkinit , mkinit , j1 = 0, or χ nkinit , mkinit , j2 = 0, if and only if the free distribution of W is the zero free distribution on XN . Proof. By (10.2.4), the condition (10.2.5) holds, if and only if W = xj1 +k xj2 +k xj1 +k ̸= 0N , is a free Poisson element of XN , where xj1 +k , xj2 +k ∈ XN . So, we have that
knN (W, . . . , W ) = φN xnj2 +k = ωn c n2 , and

 φN (W n ) = ωn 

X

π∈N Ce (n)

Y V ∈π

c |V | 2



! ,

by (10.2.2), for all n ∈ N. So, the free-distributional data (10.2.6) holds, and hence, the free distribution of W is the free Poisson distribution characterized by (10.2.6). Therefore, the condition (10.2.5) is satisfied, if and only if the free-Poisson-distributional data (10.2.6) for W holds. Meanwhile, the condition (10.2.7) is satisfied, if and only if k W = βN (Wj1 ,j2 ) = 0N , in XN ,

128 ■ Banach-Space Operators On C ∗ -Probability Spaces

by (10.2.4). Therefore, the free distribution of W is the zero free distribution on XN . ■ The above theorem characterizes how our generating partial N isometries {λk }k=−N = λ of the shift-operator algebra ΛN deform the free Poisson distributions induced by the free semicircular family XN on XN .

CHAPTER

11

Examples

I

N THIS SECTION, WE CONSIDER SOME EXAMPLES supporting/summarizing our main results of Sections 6–10.

11.1

PARTIAL ISOMETRIES OF λ5 ⊂ Λ5 ON X5

In this section, we fix 5 ∈ N∞ >1 , and consider how our main results are parallel to this special case where the C ∗ -probability space, X5 = (X5 , φ5 ) , generated by the free semicircular family, X5 = {x1 , x2 , x3 , x4 , x5 } , and the shift-operator algebra Λ5 , generated by the restricted-shift family,   λ5 = β5−4 , . . . , β5−1 , β50 = I5 , β51 , . . . , β54 ∪ β5−5 = O5 = β55 , in the operator space B (X5 ). Let’s take a partial isometry β53 ∈ λ5 in Λ5 , and take a generating semicircular element x2 ∈ X5 in X5 . Then β5−3 β53 = Q[1,5−3] , and β53 β5−3 = Q[1+3,5] , i.e., β5−3 β53 = Q[1,2] , and β53 β5−3 = Q[4,5] ,

(11.1.1)

β53 = β53 β5−3 β53 = Q[4,5] β53 = β53 Q[1,2] ,

(11.1.2)

satisfying that

DOI: 10.1201/9781003263487-11

129

130 ■ Banach-Space Operators On C ∗ -Probability Spaces

and hence, 3 β53 = Q[4,5] βN Q[1,2] , in Λ5 ,

by (11.1.1). Similarly, if a partial isometry β5−2 ∈ λ5 is taken in Λ5 , then −2 −2 βN = Q[1,3] βN Q[3,5] , in Λ5 ,

(11.1.3)

as in (11.1.2). So, the operator-product T3,−2 = β53 β5−2 ∈ Λ5 satisfies that −2 3 T3,−2 (xj ) = β53 β5−2 (xj ) = (Q[4,5] βN Q[1,2] Q[1,3] βN Q[3,5] ) (xj )
−2 3 = Q[4,5] βN (Q[1,2] Q[1,3] )βN Q[3,5] (xj )
−2 3 = Q[4,5] βN Q[1,2]∩[1,3] βN Q[3,5] (xj )
−2 3 = Q[4,5] βN Q[1,2] βN Q[3,5] (xj )

= χ ([1 − (−2), 2 − (−2)] ∩ [3, 5], j) xj+3+(−2) = χ ([3, 4] ∩ [3, 5], j) xj+1 = χ ([3, 4], j) xj+1 ,

(11.1.4)

in X5 , for all j ∈ {1, . . . , 5}, by (7.8), (11.1.2) and (11.1.3). By (11.1.4), we obtain that: if j ∈ {1, 2, 5}, then T3,−2 (xj ) = 0N ,

(11.1.5)

meanwhile, T3,−2 (x3 ) = x3+3−2 = x4 , and T3,−2 (x4 ) = x4+3−2 = x5 , in X5 . The relation (11.1.5) shows that     T3,−2 = β51 |X[3,4] ⋆ O5 |X{1,2,5} , in Λ5 . 5

(11.1.6)

5

This operator-equality (11.1.6) provides an example Theorem 7.1. Observe now that


2 β53 = β53 Q[1,2] Q[4,5] β53 = β53 Q[1,2]∩[4,5] β53 = ON ,

for

Examples ■ 131

in Λ5 . It shows that β53


n

= ON , for all n ≥ 2,

i.e., the partial isometry β53 ∈ λ5 is conditional-nilpotent with its conditional-nilpotence 2 in Λ5 . Similarly,
2

β5−2 = β5−2 Q[3,5] Q[1,3] β5−2 = β5−2 Q{3} β5−2 , in Λ5 , where Q{3} = Q[3,3] in Λ5 . So, β5−2


3

= β5−2 Q{3} β5−2 Q[1,3] β5−2 , in Λ5 ,

implying that
3 β5−2 (xj ) = χ ({3 − (−2)} ∩ [1, 3]) xj−2 = χ ({5} ∩ [1, 3], j) xj−2 , in XN , with {5} ∩ [1, 3] = ¨ı¿œ, in {1, . . . , 5} . Thus, β5−2


3

(xj ) = 0N , for all j ∈ {1, . . . , 5} .

It shows that β5−2


n

= ON , in Λ5 , ∀n ≥ 3 in N.

i.e., the partial isometry β5−2 ∈ λ5 is conditional-nilpotent with its conditional-nilpotence 3 in Λ5 . Let 1 y = √ (x1 + ix5 ) ∈ X5 2 be a circular element. Then 1 β53 (y) = √ x4 , in X5 . 2 So,
one can verify that the self-adjoint free random variable β53 (y) is 1 2 -semicircular in X5 , by (9.2.9). Similarly, one has that i β5−2 (y) = √ x3 , in X5 , 2

132 ■ Banach-Space Operators On C ∗ -Probability Spaces

and hence, the free distribution of β5−2 (y) is characterized by the joint free moments obtained by (9.2.14). Now, let W = x3 x4 x3 ∈ X5 be a free Poisson element. Then β53 (W ) = 0N , in X5 , whose free distribution is the zero free distribution on X5 , by (10.2.7). Similarly, we have β5−2 (W ) = x1 x2 x1 , in X5 , whose free distribution is the free Poisson distribution characterized by
kn5 β5−2 (W ) , . . . , β5−2 (W ) = φ5 (xn2 ) = ωn c n2 , and

 φ5



β5−2 (W )


n 

= ωn 

X

π∈N Ce (n)

Y

c |V |

V ∈π

2



! ,

for all n ∈ N, by (10.2.5) and (10.2.6).

11.2

PARTIAL ISOMETRIES OF λ∞ ⊂ Λ∞ ON X∞

Some general results showing how the generating partial isometries of λ∞ , and some Banach-space operators of Λ∞ induced by them are already considered in Section 8. Here, we simply focus on basic examples as in Section 11.1. 2 Let β∞ ∈ λ∞ be a generating partial isometry of Λ∞ . Then 2 2 β∞ = Q[3,∞] β∞ Q[1,∞] ,

and 2 β∞


∗

−2 −2 = β∞ = Q[1,∞] β∞ Q[3,∞] ,

in Λ∞ , where Q[1,∞] = I∞ , the identity map on X∞ .

(11.2.1)

(11.2.2)

Examples ■ 133

By (11.2.2), one can simply rewrite (11.2.1) by 2 2 −2 −2 β∞ = Q[3,∞] β∞ , and β∞ = β∞ Q[3,∞] ,

(11.2.3)

in Λ∞ . As in (11.2.3), one can have −3 −3 1 1 β∞ = β∞ Q[4,∞] , and β∞ = Q[2,∞] β∞ ,

in Λ∞ . Then the shift operator 2 −3 T2,−3 = β∞ β∞ ∈ Λ∞

is well-defined, identical to 2 −3 2 −3 T2,−3 = Q[3,∞] β∞ Q[1,∞]∩[1,∞] β∞ Q[4,∞] = Q[3,∞] β∞ β∞ Q[4,∞] ,

in Λ∞ . So, T2,−3 (xj ) = 0N , ∀j = 1, 2, 3, having their free distributions, the zero free distribution on X∞ , and T2,−3 (xj ) = xj+2+(−3) = xj−1 , ∀j ≥ 4, in N, which are the semicircular elements of X∞ , contained in the free semicircular family X∞ . Similarly, let −3 2 −3 2 −3 2 T−3,2 = β∞ β∞ = β∞ Q[4,∞] Q[3,∞] β∞ = β∞ Q[4,∞] β∞ ,

in Λ∞ . So, it satisfies that: if xj ∈ X∞ is a generating semicircular element of X∞ , then T−3,2 (xj ) = χ ([4 − 2, ∞ − 2] ∩ [1, ∞], j) xj+(−3)+2 = χ ([2, ∞], j) xj−1 , in X∞ . So, one can have that −3 −3 T−3,2 (x1 ) = β∞ Q[4,∞] (x1+2 ) = β∞ Q[4,∞] (x3 ) = 0N ,

having its zero free distribution on X∞ , and T−3,2 (xj ) = xj+(−3)+2 = xj−1 , ∀j ≥ 2, in N,

134 ■ Banach-Space Operators On C ∗ -Probability Spaces

which are the semicircular elements of χ∞ . Now, let 2 1 −3 T2,1,−3 = β∞ β ∞ β ∞ ∈ Λ∞ . Then it is identified with 3 −3 T2,1,−3 = β∞ β∞ , in Λ∞ ,

since 2 1 2 1 2 1 2 1 β∞ β∞ = β∞ Q[1,∞]∩[2,∞] β∞ = β∞ Q[2,∞] β∞ = β∞ β∞ ,

in ΛN , moreover, for any j ∈ N, one has that 2 1 2 3 β∞ β∞ (xj ) = β∞ (xj+1 ) = xj+1+2 = xj+3 = β∞ (xj ) ,

in X∞ . Since X∞ generates X∞ by (5.3), the above equalities imply that 2 1 3 β∞ β∞ = β∞ , on Λ∞ . So, this shift operator T2,1,−3 becomes a projection in Λ∞ , i.e., 3 −3 T2,1,−3 = β∞ β∞ = Q[1+|−3|,∞] = Q[4,∞] ,

(11.2.4)

in ΛN . So, T2,1,−3 (xj ) = 0N , ∀j = 1, 2, 3, and T2,1,−3 (xj ) = xj+2+1−3 = xj , ∀j ≥ 4, in N, by (11.2.4). Let −3 1 2 T−3,1,2 = β∞ β∞ β∞ ∈ Λ∞

be a shift operator. Similar to (11.2.4), it becomes a projection, −3 3 T−3,1,2 = β∞ β∞ = Q[1,∞] = I∞ ,

the identity operator on X∞ . So, T−3,1,2 (xj ) = xj , ∀j ∈ N, implying that T−3,1,2 (W ) = W, ∀W ∈ X∞ , by (11.2.5).

(11.2.5)

Examples ■ 135

Suppose 1 −3 2 T1,−3,2 = β∞ β∞ β∞ ∈ Λ∞

is a shift operator. Then, −3 2 1 T1,−3,2 = β∞ Q[1,∞] Q[1,∞] β∞ Q[4,∞] Q[3,∞] β∞ 1 −3 2 = β∞ Q[1,∞] β∞ Q[4,∞] β∞ ,

in Λ∞ . Thus, for any j ∈ N, one can have that T1,−3,2 (xj ) = χ ([1 − (−3 + 2), ∞ − (−3 + 2)] ∩ [4 − 2, ∞ − 2], j) × xj+1+(−3)+2 , satisfying T1,−3,2 (xj ) = χ ([2, ∞] ∩ [2, ∞], j) xj ,

(11.2.6)

if and only if T1,−3,2 (xj ) = χ ([2, ∞], j) xj , in X∞ , and hence, T1,−3,2 (x1 ) = 0N , whose free distribution is the zero free distribution on X∞ , while T1,−3,2 (xj ) = xj , ∀j ≥ 2, in N, are the semicircular elements of X∞ , contained in X∞ , by (11.2.6). Now, let √ 1 y = √ (x2 + ix5 ) ∈ X∞ , with i = −1 ∈ C, 2

(11.2.7)

be a circular element of X∞ . If y ∈ X∞ is a circular element (11.2.7), then i −2 β∞ (y) = √ x3 ∈ X∞ , 2 whose free distribution is characterized by the joint-free-moment formula (9.2.14).

136 ■ Banach-Space Operators On C ∗ -Probability Spaces

For the circular element y ∈ X∞ of (11.2.7), and for all k ≥ 0 in Z, the free random variables 1 k β∞ (y) = √ (xk+2 + ixk+5 ) ∈ X∞ 2 are circular in X∞ , too. Note that, if k < 0 in Z, one cannot guarantee k the circularity of β∞ (y) in X∞ , by the very above paragraph. However, under non-negativity, the circularity of y is preserved by our partial  k ∞ ⊂ λ∞ . isometries β∞ k=0 If T−3,1,2 , T1,2,−3 ∈ Λ∞ are as above, then T−3,1,2 (y) is circular in X∞ , meanwhile the free distribution of T1,2,−3 (y) = √i2 x5 is characterized by (9.2.14) because T−3,1,2 = I∞ , and T1,2,−3 = Q[4,∞] , in Λ∞ . For a circular element y ∈ X∞ of (11.2.7), the free random variable 1 T1,−3,2 (y) = √ (x2 + ix5 ) = y 2 is circular in X∞ , by (11.2.6). Now, let W = x2 x5 x2 ∈ X∞ be a free Poisson element, whose free distribution is characterized by φ∞ (W n ) = kn∞ (x5 , . . . , x5 ) = δn,2 ,

(11.2.8)

or, by  kn∞ (W, . . . , W ) = ωn 

X

π∈N Ce (n)

for all n ∈ N, by (10.2.6).

Y V ∈π

c |V | 2



! ,

Examples ■ 137

The free random variable −2 β∞ (W ) = 0N x3 0N = 0N , in X∞

has its zero free distribution on X∞ , by (10.2.7). For all non-negative k ≥ 0 of Z, k β∞ (W ) = xk+2 xk+5 xk+2 ∈ X∞

are free Poisson elements, too. Moreover, all these free Poisson elements have the same free distributions on X∞ , characterized by the freedistributional data (11.2.8). The free random variable T−3,1,2 (W ) = I∞ (W ) = W is free Poisson, having its free distribution characterized by (11.2.8), however, the free random variable T1,2,−3 (W ) = Q[4,∞] (W ) = 0N , having its zero free distribution on X∞ . Now, observe that T1,−3,2 (W ) = x2 x5 x2 = W, by (11.2.6). Therefore, it becomes a free Poisson element of XN , characterized by the free-distributional data (11.2.8).

CHAPTER

12

The Group-Dynamical System Γ

I

N SECTION 4, WE STUDIED HOW THE INTEGER-SHIFT group λ ⊂ Aut (Xφ ) acts on the C ∗ -probability space Xφ = (X, φ), generated by the free semicircular family X = {xj }j∈Z . In particular, every group-element β n ∈ λ is a free-isomorphism on Xφ by (4.19). i.e., the group-action α of λ on Xφ preserves the free probability on Xφ , where α : λ → Aut (Xφ ) satisfying α (β n ) = β n , for all n ∈ Z. However, the group algebra Λ, the shift-operator algebra generated by λ, deforms the free probability on Xφ . In particular, if a shift operator T ∈ Λ has more than one summand, then it distorts the free-distributional data on Xφ , characterized by (4.32). Similar to Section 4, we considered, in Section 5, how the restrictedshift families λN , and the corresponding restricted-shift-operator algebras ΛN affect the free probability on XN , for all N ∈ N∞ >1 . From below, we concentrate on the C ∗ -probability space Xφ (having its free-probabilistic sub-structures {XN }N ∈N∞ ), and the integer-shift >1 group λ (inducing restricted-shift families {λN }N ∈N∞ ), and study >1 certain dynamical systems. Since our integer-shift group λ acts on Xφ through the group-action α, one can have the group dynamical system, def

Γ = (Xφ , λ, α) ,

DOI: 10.1201/9781003263487-12

(12.1)

139

140 ■ Banach-Space Operators On C ∗ -Probability Spaces

in the sense of (2.3.1). So, for the dynamical system Γ of (12.1), the corresponding crossed product Banach ∗-algebra Xφ is constructed to be def Xφ = Xφ ⋊α λ, (12.2) as in (2.3.2), satisfying the α-relation; (T1 , β n1 ) (T2 , β n2 ) = (T1 β n1 (T2 ) , β n1 β n2 ) ,

(12.3)

and
∗ (T, β n ) = β n (T ∗ ) , β −n , in Xφ , by (2.3.3) and (2.3.4), for all T, T1 , T2 ∈ Xφ , and β n , β n1 , β n2 ∈ λ. In this section, we study the dynamical system Γ of (12.1), and the free probability on the crossed product algebra Xφ of (12.2). At the end of this section, we will discuss that such a dynamical-systematic structure is “not” applicable (or, “not” restricted) to our restrictedshift families λN acting on XN , for all N ∈ N∞ >1 .

12.1

DYNAMICS ON Xφ

Let Xφ = Xφ ⋊α λ be the crossed product Banach ∗-algebra (12.2) of the group dynamical system Γ = (Xφ , λ, α) of (6.1.1), satisfying the α-relation (6.1.3). Since Xφ = (X, φ) is a C ∗ -probability space, one can define a well-defined linear functional, τ : Xφ → C,

(12.1.1)

by a morphism satisfying





τ T, β k = φ α β k (T ) = φ β k (T ) , ∀ T, β k ∈ Xφ , by (2.3.5), inducing the Banach ∗-probability space, X

denote

=

(Xφ , τ ) .

(12.1.2)

Definition 12.1. The Banach ∗-probability space X = (Xφ , τ ) of (12.1.2) is called the Γ(-dynamical)-semicircular (Banach-) ∗probability space, where τ is the linear functional (12.1.1) on Xφ . From the definition, it is verified that the free probability on the Γ-semicircular ∗-probability space X is determined by the free probability on Xφ under the action of λ, equivalently, under the dynamics of Γ.

The Group-Dynamical System Γ ■ 141

12.2

THE Γ-SEMICIRCULAR ∗-PROBABILITY SPACE X

Let X = (Xφ , τ ) be the Γ-semicircular ∗-probability space of Definition 12.1, and let Λ be the shift-operator algebra (4.22), a Banach ∗-algebra embedded in the operator space B (Xφ ), generated by the integer-shift group λ. Define a conditional tensor product Banach ∗-algebra, def

X = Xφ ⊗α Λ,

(12.2.1)

by the Banach ∗-subalgebra of the usual tensor product Banach ∗-algebra, Xφ ⊗C Λ, where ⊗C is the usual tensor product of Banach ∗-algebras. In the definition (12.2.1), the conditional tensor product ⊗α satisfies its αcondition: (T1 ⊗ β n1 ) (T2 ⊗ β n2 ) = (T1 β n1 (T2 )) ⊗ β n1 β n2 ,

(12.2.2)

and ∗

∗

(T ⊗ β n ) = (T ∗ ) ⊗ (β n ) , for all T, T1 , T2 ∈ Xφ , and β n1 , β n1 , β n2 ∈ λ. Note that, since Xφ is a C ∗ -algebra, and Λ is a Banach ∗-algebra, the usual tensor product algebra Xφ ⊗C Λ is well-defined as a Banach ∗-algebra, and hence, the conditional tensor product Banach ∗-algebra X of (12.2.1) is well-defined as its Banach ∗-subalgebra. Note also that the α-condition (12.2.2) can be re-expressed by (T1 ⊗ β n1 ) (T2 ⊗ β n2 ) = T1 β n1 (T2 ) ⊗ β n1 +n2 , respectively, ∗

(T ⊗ β n ) = β n (T ∗ ) ⊗ β −n . Theorem 12.1. Let Xφ be the crossed product algebra (12.0.2) of the dynamical system Γ, and let X = Xφ ⊗α Λ be the conditional tensor product Banach ∗-algebra (12.2.1) satisfying the α-condition (12.2.2). Then ∗-iso Xφ = X , (12.2.3) ∗-iso

where “ = ” means “being ∗-isomorphic as Banach ∗-algebras.”

142 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. By the definition (12.2.1), the Banach ∗-algebra X is generated by the operators,  s xj ⊗ β k |j, k ∈ Z, s ∈ N , and, by the definition (12.2) of Xφ , it is generated by the elements,  s k
xj , β |j, k ∈ Z, s ∈ N . So, one can define a linear morphism, Φ : Xφ → X ,

(12.2.4)

satisfying Φ

xsj , β k





= xsj ⊗ β k in X ,


as a generating operator of X , for all generating operators xsj , β k ∈ Xφ . By definition, this linear transformation Φ of (12.2.4) is generatorpreserving, and hence, bijective and bounded. Also, it satisfies that  


Φ xsj11 , β k1 xsj22 , β k2 = ΦN xsj11 xsj22+k1 , β k1 +k2
= xsj11 xsj22+k1 ⊗ β k1 +k2 = xsj11 β k1 xsj22 ⊗ β k1 β k2

= xsj11 ⊗ β k1 xsj22 ⊗ β k2 ,
in X , for all xsjll , β kl ∈ X, for l = 1, 2, by (12.2) and (12.2.2). It implies that, for all T1 , T2 ∈ X, Φ (T1 T2 ) = Φ (T1 ) Φ (T2 ) in X .

(12.2.5)

And the morphism Φ satisfies that 
∗ 
Φ xsj , β k = Φ xsj+k , β −k 

∗
∗ ∗ = xsj+k ⊗ β −k = β k xsj ⊗ β k = xsj ⊗ β k ,
in X , for all xj , β k ∈ X, by (12.2.2), implying that ∗

Φ (T ∗ ) = (Φ (T )) , in X ,

(12.2.6)

for all T ∈ X. By (12.2.5) and (12.2.6), the bijective and bounded linear morphism Φ of (12.2.4) is a ∗-homomorphism, equivalently, it is a ∗-isomorphism. Therefore, the ∗-isomorphic relation (12.2.3) holds. ■

The Group-Dynamical System Γ ■ 143

By the above theorem, one can regard our crossed product algebra Xφ and the conditional tensor product algebra X as the same Banach ∗-algebra. Remark that if ! X tn β n ∈ X , for W ∈ Xφ , U =W ⊗ k∈Z

then ! Φ

−1

−1

(U ) = Φ

X

n

tn (W ⊗ β )

n∈Z

=

X


X tn Φ−1 (W ⊗ β n ) = tn (W, β n ) ,

n∈Z

n∈Z

in Xφ , where Φ−1 is the inverse of the ∗-isomorphism Φ of (12.2.4). Now, define a linear functional τ ′ : X → C, by a morphism satisfying τ ′ (T ⊗ S) = φ (S(T )) , ∀T ⊗ S ∈ X . (12.2.7) By (12.2.7), one has that,




τ xsj , β k = φ β k xsj = φ xsj+k



= τ ′ xsj ⊗ β k = τ ′ Φ xsj , β k , (12.2.8)
k for all generating operators xnj , βN ∈ Xφ , where Φ is the ∗-isomorphism (12.2.4). By (12.2.3) and (12.2.8), we obtain the following result. Theorem 12.2. The Γ-semicircular ∗-probability space X = (Xφ , τ ) denote

and the Banach ∗-probability space X = (X , τ ′ ) are freeisomorphic, where τ ′ is the linear functional (12.2.7) on X . i.e., free-iso

X

=

(X , τ ′ ) .

(12.2.9)

Proof. Let Φ be the ∗-isomorphism (12.2.4) from Xφ onto X . By (12.2.8), it also satisfies that τ ′ (Φ (T )) = τ (T ) , ∀T ∈ X. i.e., it is a free-isomorphism from X onto X . Therefore, the freeisomorphic relation (12.2.9) holds. ■

144 ■ Banach-Space Operators On C ∗ -Probability Spaces

By the above free-isomorphic relation (12.2.9), we regard two Banach ∗-probability spaces X and X , as the same free-probabilistic structure, and hence, we use them alternatively for our purposes. These two identified Banach ∗-probability spaces X and X would be denoted by X.

12.3

DISCUSSION

In this section, we briefly consider the dynamics of the integer-shift group λ on Xφ cannot be restricted to those on XN , canonically, for all N ∈ N∞ >1 . For a fixed N ∈ N∞ >1 , assume that there were a certain system, ΓN = (XN , λN , αN ) , induced by
def k αN β k = βN , ∀k ∈ Z,  k N where λN = βN is the restricted-shift family of λ. Then, from k=−N this mathematical triple, there is no “nice” way to define the αN relation like (12.3), since the restricted-shift families λN do not form algebraic structures in the homomorphism semigroups Hom (XN ), i.e., the structures ΓN do not have “suitable” dynamics on XN , for all N ∈ N∞ >1 . However, if we consider Banach ∗-algebras ΛN , the (restricted-) shift-operator algebras generated by λN , one can consider a canonical dynamics “of ΛN ” on XN , on the “usual” tensor products, XN ⊗C ΛN , for N ∈ N∞ >1 . We study such tensor product Banach ∗-algebras in Sections 15 and 16.

CHAPTER

13

On The Γ-Semicircular ∗-Probability Space X

I

N THIS SECTION, WE CONSIDER FREEDISTRIBUTIONAL data on the Γ-semicircular ∗-probability space (12.1.2), X = (Xφ , τ ) , identified with the Banach ∗-probability space X = (X , τ ′ ) of Theorem 12.2, by the free-isomorphic relation (12.2.9).

13.1

GENERATING FREE RANDOM VARIABLES OF X

In this section, we concentrate on the generating free random variables, 


xsj , β k ∈ X |j, k ∈ Z, s ∈ N ,

of the Γ-semicircular ∗-probability space X. By the α-relation (12.3) (or, the α-condition (12.2.2)), n Y



xsjll , β kl = xsj11 xsj22+k1 , β k1 +k2


l=1

n  Y

! xsjll , β kl




l=3

=



xsj11 xsj22+k1 xsj33+(k1 +k2 ) , β k1 +k2 +k3

n  Y

! xsjll , β kl




l=4

= ···   = xsj11 xsj22+k1 xsj33+(k1 +k2 ) · · · xsjnn+(k1 +···+kn−1 ) , β k1 +···+kn DOI: 10.1201/9781003263487-13

145

146 ■ Banach-Space Operators On C ∗ -Probability Spaces

 n Y xsl = l=1

jl +

n P l−1 P

 ki

, β i=1  ,

(13.1.1)

ki = 0

(13.1.2)

ki

i=1

in X, with axiomatization: 0 X

in Z.

i=1


Proposition 13.1. Let xsjll , β kl ∈ X be generating free random variables, for l = 1, . . . , n, for n ∈ N. Then   n P n n Y Y ki
(13.1.3) , β i=1  , xsjll , β kl =  xsl l−1 P l=1

l=1

ki

jl +

i=1

in X, under the axiomatization (13.1.2). Proof. The relation (13.1.3) is shown by (13.1.1), under (13.1.2).

■

By (13.1.3), we obtain the following special case.
Corollary 13.3. If xsj , β k is a generating operator of X, then xsj , β


k n

=

n Y

! xsj+(l−1)k , β nk

,

(13.1.4)

l=1

in X, for all n ∈ N.
Proof. Let xsj , β k be a given generating free random variable of X. Then, for any n ∈ N, we have 
n  xsj , β k = xsj xsj+k xsj+2k · · · xsj+(n−1)k , β nk ! n Y s nk = xj+(l−1)k , β , (13.1.5) l=1

in XN , by (13.1.3). Therefore, the computation (13.1.4) is obtained by (13.1.5). ■

Γ-Semicircular ∗-Probability Space X ■ 147

By (12.3), (12.2.2) and (13.1.4), it is trivial that  
∗ n  s xsj , β k = xj+k xsj xsj−k . . . xsj−(n−2)k , β −nk ,
k for all generating operators xsj , βN ∈ X, since
∗
xsj , β k = xsj+k , β −k in X,

(13.1.6)

for all n ∈ N. s1 ,s2 s1 ,s2 Now, define new free random variables Tj,k and Sj,k of X by def


∗

def




s1 ,s2 Tj,k = xsj 1 , β k

and

s1 ,s2 Sj,k = xsj 1 , β k


xsj 2 , β k , xsj 2 , β k


∗

(13.1.7)

,

in X. Then, by the definition (13.1.7),   
 1 s2 s1 ,s2 1 Tj,k = xsj+k , β −k xsj 2 , β k = xsj+k xj−k , β 0 ,

(13.1.8)

respectively s1 ,s2 Sj,k = xsj 1 , β k




   2 2 xsj+k , β −k = xsj 1 xsj+2k , β0 ,

in X, where β 0 = 1Xφ ∈ λ is the unity of the shift-operator algebra Λ, identified with the identity ∗-isomorphism of Aut (Xφ ). s1 ,s2 s1 ,s2 Proposition 13.4. Let Tj,k , Sj,k ∈ XN be the free random variables (13.1.7). Then  n  n  s1 ,s2 s1 s2 0 Tj,k = xj+k xj−k , β , (13.1.9)

and



s1 ,s2 Sj,k

n

=



2 xsj 1 xsj+2k

n

 , β0 ,

(13.1.10)

in X, for all n ∈ N. Proof. For any n ∈ N,  n s1 ,s2 Tj,k        1 2 1 2 1 2 = xsj+k xsj−k β 0 xsj+k xsj−k · · · β (n−1)0 xsj+k xsj−k , β n0 in X, by (13.1.2) and (13.1.8). So, the formula (13.1.9) holds.

148 ■ Banach-Space Operators On C ∗ -Probability Spaces

Similar to (13.1.9), 

s1 ,s2 Sj,k

n

=



2 xsj 1 xsj+2k

n

 , β0 ,

in X, by (13.1.2) and (13.1.8). So, the formula (13.1.10) holds.

■

s1 ,s2 s1 ,s2 Let Tj,k , and Sj,k be the free random variables (13.1.7) of the ΓN -semicircular ∗-probability space X. Define new free random variables of X by

n  n
def s1 ,s2 s1 ,s2 Tj,k = Tj,k xsj 3 , β k , s ,1, n n3  
∗ def s1 ,s2 s1 ,s2 = Tj,k xsj 3 , β k , Tj,k s ,∗  n3  n
def s1 ,s2 s1 ,s2 Sj,k = Sj,k xsj 3 , β k ,



(13.1.11)

s3 ,1

and

 n
∗ s1 ,s2 Sj,k xsj 3 , β k ,

in X. Then, one can get the following computations. Corollary 13.5. The free random variables of (13.1.11) of X are simplified to be 

s1 ,s2 Tj,k

n s ,1

n3



s1 ,s2 Tj,k s ,∗  n3 s1 ,s2 Sj,k

s3 ,1

and 

s1 ,s2 Sj,k



n

 xsj 3 , β k , n   2 1 xsj−k xsj 3 , β −k , = xsj+k  n  2 = xsj 1 xsj+2k xsj 3 , β k , =

n s3 ,∗

1 2 xsj+k xsj−k

=



2 xsj 1 xsj+2k

n

(13.1.12)

 xsj 3 , β −k ,

in X. Proof. By (13.1.3), (13.1.9), (13.1.10) and (13.1.11), one obtains (13.1.12). ■

Γ-Semicircular ∗-Probability Space X ■ 149

13.2

FREE-DISTRIBUTIONAL DATA ON X

Let X be our Γ-semicircular ∗-probability space, X = (Xφ ⋊α λ, τ ) , induced by G=




xsj , β k xj ∈ X, β k ∈ λ, s ∈ N .

In this section, we study free-distributional data on X by computing free moments of some basic free random variables induced by the generator set G of (7.2.1).
Lemma 13.6. The free random variables xj , β 0 ∈ G are semicircular in X, for all j ∈ Z.
Proof. Let Wj = xj , β 0 ∈ G in X. Observe that
∗

Wj∗ = xj , β 0 = x∗j+0 , β −0 = xj , β 0 = Wj , in X, showing the self-adjointness of Wj ∈ G in X. Such a self-adjoint operator Wj ∈ X satisfies that
n
Wjn = xj , β 0 = xnj , β 0 in X, by (13.1.2), for all n ∈ N. So, one has that





τ Wjn = τ xnj , β 0 = φ β 0 xnj = φ xnj = ωn c n2 , by the semicircularity of xj ∈ X in Xφ , for all n ∈ N. Therefore, the generating free random variable Wj ∈ G is semicircular in X, for all j ∈ Z. ■ By the above lemma, one obtains the following generalized result. Theorem 13.7. Let Is = (i1 , . . . , is ) be an s-tuple in Z (without n Q considering repetitions), and let X[Is ] = xil be the corresponding l=1

operator of Xφ , for xi1 , . . . , xis ∈ X are the generating semicircular elements of Xφ . If
W [Is ] = X[Is ], β k is a free random variable of X, for k ∈ Z, then τ (W [Is ]) = φ (X[Is ]) characterized by (3.22) or (3.25).

in C,

150 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. Observe that
τ (W [Is ]) = φ β k (X[Is ]) = φ (X[Is ]) , since β k ∈ λ ⊂ Λ is a free-isomorphism on Xφ , by (4.19). Moreover, the free moment φ (X[Is ]) is determined by (3.22). ■ Similarly, we have the following joint free-momental data. Theorem 13.8. Let W =

n Q l=1

induced by G. Then


xsj1l , β kl be a free random variable of X

  n Y  sl τ (W ) = φ  x



j+

l=1

l−1 P

!  ,



ki

(13.2.1)

i=1

where the right-hand side of (13.2.1) is characterized by Theorem 13.7 (or, (3.22)). Proof. Let W ∈ X be given as above. Then     n P n kl   Y  sl !  , β l=1  in X, W = x l−1 P l=1

ki

j+

i=1

by (13.1.2). Thus, one can get that    n P n kl Y   s τ (W ) = φ β l=1  x l

j+

l=1

  n Y   sl = φ x l=1

j+

l=1

j+

l−1 P

ki



! 

i=1

 

!    n P ki + kl

l−1 P i=1

  n Y  sl = φ x



l=1

 l−1 P

ki

!  ,



i=1

by (4.19). This C-quantity is determined by (3.22) or (3.25).

■

Γ-Semicircular ∗-Probability Space X ■ 151

In the rest of this section, let’s consider some specific cases of (13.2.1).
Theorem 13.9. If W = xsj , β k ∈ G is a generating free random variable of X for j, k ∈ Z, then
n τ (W n ) = ωs c 2s , ∀n ∈ N. (13.2.2) Proof. If W ∈ G is given as above in X, then   W n = xsj xsj+k xsj+2k · · · xsj+(n−1)k , β nk in X, by (7.1.4), for all n ∈ N. So, for any n ∈ N, one has   τ (W n ) = τ xsj xsj+k xsj+2k · · · xsj+(n−1)k , β nk    = φ β nk xsj xsj+k xsj+2k · · · xsj+(n−1)k   = φ xsj+nk xsj+k+nk xsj+2k+nk · · · xsj+(n−1)k+nk   = φ xsj+nk xsj+(n+1)k xsj+(n+2)k · · · xsj+(2n−1)k , for n ∈ N. Thus, there exists an n-tuple, J = (j + nk, j + (n + 1)k, . . . , j + (2n − 1)k) in Z, such that xsj+nk xsj+(n+1)k xsj+(n+2)k · · · xsj+(2n−1)k = X[J], in Xφ , where X[J] ∈ Xφ is in the sense of (3.7). Therefore, in this case,   2n−1 Y
n
φ xsj+nk xsj+(n+1)k · · · xsj+(2n−1)k = φ xsj+lk = ωs c 2s , l=n

by (3.25), since the corresponding noncrossing partitions π (j + mk) of (3.6) for all entries j + mk, for all m = n, n + 1, . . . , 2n − 1, are identical to the n-block partition, πJ = {(j + nk) , (j + (n + 1)k) , . . . , (j + (2n − 1)k)} ,

152 ■ Banach-Space Operators On C ∗ -Probability Spaces

inducing Π ([J]) = {πJ } , showing that ( {πJ } = Π ([J]) if all blocks have even elements Πe ([J]) = ¨ı¿œ otherwise. Therefore, the relation (13.2.2) holds.

■

Similar to (13.2.2), one obtains the following result.
Theorem 13.10. Let W = xsj , β k ∈ G be a free random variable of X. Then
n n (13.2.3) τ ((W ∗ ) ) = ωs c 2s , ∀n ∈ N. Proof. Observe that, for any n ∈ N, n

(W ∗ ) = xsj+k , β −k


n

by (12.3)   = xsj+k xsj xsj−k · · · xsj−(n−2)k , β −nk , in X, by (13.1.4). So, similar to the proof of (13.2.2), one obtains that
n n τ ((W ∗ ) ) = ωs c 2s , for all n ∈ N.

■

Remark that the relation (13.2.3) can be proven directly by (13.2.2) because τ (T ∗ ) = τ (T ), ∀T ∈ X. However, we here emphasize (13.2.2) and (13.2.3), independently, in terms of direct computations from (13.1.4). Now, let   s1 ,s2 1 2 T = Tj,k = xsj+k xsj−k , β0 , (13.2.4) and

  s1 ,s2 2 S = Sj,k = xsj 1 xsj+2k , β0 ,

be the operators (13.1.8) of X, for arbitrarily fixed j, k ∈ Z, and s1 , s2 ∈ N.

Γ-Semicircular ∗-Probability Space X ■ 153

Theorem 13.11. Let T and S be the free random variables (13.2.4) of X. Then, for any n ∈ N,   n   n τ (T n ) = ωns1 c ns21 ωs2 c s22 + ωns2 c ns22 ωs1 c s21 , (13.2.5) and   n   n τ (S n ) = ωns1 c ns21 ωs2 c s22 + ωns2 c ns22 ωs1 c s21 .

(13.2.6)

Proof. If T, S ∈ X are free random variables of (13.2.4), then  n   n  s1 s2 s1 s2 n 0 n 0 T = xj+k xj−k , β and S = xj xj+2k , β , in X, by (13.1.10) and (13.1.11), respectively, for all n ∈ N. So,   1 2 1 2 1 2 τ (T n ) = φ xsj+k xsj−k xsj+k xsj−k · · · xsj+k xsj−k  n      n  s2 1 ns1 s2 ω c , = φ xns φ x = ω c s2 2 ns1 2 j+k j−k by Corollary 4.2 (or (3.22), or (3.25)). Indeed, the operator 1 2 1 2 1 2 xsj+k xsj−k xsj+k xsj−k · · · xsj+k xsj−k ∈ Xφ

is identical to the operator X[J] ∈ Xφ of (3.6) (ns1 + ns2 )-tuple J,  j + k, . . . j + k , j − k, . . . j − k , · · · {z } | {z } |  s1 −times s2 −times  J = j + k, . . . j + k , j − k, . . . j − k | {z } | {z } s1 −times

induced by the  ,  , 

s2 −times

in Z. Note that J has the corresponding noncrossing partition π (j + k) , and π (j − k) of (3.6), inducing Π ([J]) = {π (j + k) , π (j − k)}, where π (j + k) = {U1 , U2 , . . . , Un+1 } , with





U1 = j + k, j + k, . . . , j + k  , | {z } ns1 −times

154 ■ Banach-Space Operators On C ∗ -Probability Spaces

and 



U2 = · · · = Un+1 = j − k, j − k, . . . , j − k  , | {z } s2 −times

and π (j − k) = {V1 , V2 , . . . , Vn+1 } , with 



V1 = j − k, j − k, . . . , j − k  , | {z } ns2 −times

and 



U2 = · · · = Un+1 = j + k, j + k, . . . , j + k  . | {z } s1 −times

So, by (3.22), or (3.25), n+1 Y

τ (T n ) = φ (X[J]) =

l=1



 n+1  Y ω|Ul | c |Ul | + ω|Vl | c |Vl | , 2

l=1

2

for all n ∈ N. Therefore, the formula (13.2.5) holds. Similarly, one can get that   n   n ωs2 c s22 + ωns2 c ns22 ωs1 c s21 , τ (S n ) = ωns1 c ns21 for all n ∈ N. So, the formula (13.2.6) holds.

■

By (13.2.5) and (13.2.6), one can get the following corollary. Corollary 13.12. Let T and S be free random variables (13.2.4) of X. Then τ (T n ) = τ (S n ) , ∀n ∈ N. Proof. The proof is done by (13.2.5) and (13.2.6).

■

Γ-Semicircular ∗-Probability Space X ■ 155

13.3

MORE ABOUT FREE-DISTRIBUTIONAL DATA ON X

Let X = (Xφ ⋊α λ, τ ) be the Γ-semicircular ∗-probability space, generated by the generator set G of (13.2.1). Recall the definition (12.2.7) of the linear functional τ ′ , which is also denoted by τ below, on the conditional tensor product algebra, def

X = Xφ ⊗α Λ of (12.2.1),

where Λ is the shift-operator algebra generated by λ. Let X T = tl β l ∈ Λ, with tl ∈ C, l∈Z

be an arbitrary shift operator, and xj ∈ X, a generating semicircular element of Xφ , for j ∈ Z, and let def

xsj,T = xsj ⊗ T ∈ X = X, for s ∈ N.

(13.3.1)

Then, one can have that ! τ xsj,T




X



= φ T xsj = φ tl β l xsj l∈Z

! =φ

X

X
X
= tl φ xsj+l = tl φ xsj

tl xsj+l

l∈Z

l∈Z

l∈Z

by (4.19) ! =

X




tl ωs c 2s = ωs c 2s


X

l∈Z

tl

.

(13.3.2)

l∈Z

Lemma 13.16. Let xsj,T ∈ X be a free random variable (13.3.1). Then ! X

τ xsj,T = ωs c 2s tl . (13.3.3) l∈Z

Proof. The formula (13.3.3) is shown by (13.3.2).

■

The above lemma illustrates how our shift-operator algebra Λ affects the free probability on Xφ , and hence, how it determines freedistributional data on the Γ-semicircular ∗-probability space X = X.

156 ■ Banach-Space Operators On C ∗ -Probability Spaces

Now, let
W = xsj11 xsj22 . . . xsjnn ⊗ T ∈ X = X,

(13.3.4)

where xsj11 xsj22 · · · xsjnn is a free reduced word of Xφ with its length n ∈ N, and X T = tn β n ∈ Λ n∈Z

is a shift operator. Theorem 13.17. Let W ∈ X be a free random variable (13.3.4), and assume
φ xsj11 xsj22 . . . xsjnn = w0 in C, characterized by Corollary 4.2 (or, (3.22), or (3.25)). Then ! X τ (W ) = w0 tl .

(13.3.5)

l∈Z

Proof. If U = xjs11 xsj22 . . . xsjnn ∈ Xφ , and W = U ⊗ T ∈ X = X, then ! τ (W ) = φ (T (U )) = φ

X

tl β n (U )

l∈Z

=

X

n

tl φ (β (U ))

l∈Z

! =

X l∈Z

X tl φ (U ) = φ (U ) tl

,

(13.3.6)

l∈Z

by (4.19) and (5.3.2). So, the formula (13.3.5) holds by (3.22), (3.25) and (13.3.6). ■

CHAPTER

14

Operator-Theoretic Properties on X

L

ET X = (Xφ ⋊α λ, τ ) BE THE Γ-SEMICIRCULAR ∗probability space of the group dynamical system, Γ = (Xφ , α, λ) ,

where λ is the integer-shift group acting on the C ∗ -probability space Xφ = (X, φ) generated by the free semicircular family X = {xj }j∈Z . In this section, we study certain operator-theoretic properties of basic free random variables of X induced by the generator set, 
G = xsj , β k xj ∈ X, β k ∈ λ, s ∈ N of X.

14.1

OPERATOR-THEORETIC PROPERTIES ON G

Let B be an arbitrary Banach ∗-algebra over C. (One may consider it as a C ∗ -algebra, or, a von Neumann algebra, however, generally, we consider B as a Banach ∗-algebra.) In this text, we automatically assume that B is a unital algebra with its unity (or the multiplicationidentity) 1B , satisfying T · 1B = T = 1B · T in B, for all T ∈ B.

DOI: 10.1201/9781003263487-14

157

158 ■ Banach-Space Operators On C ∗ -Probability Spaces

Definition 14.1. Let T ∈ B be an arbitrary element. (1) T is self-adjoint in B, if T ∗ = T in B. (2) T is a projection in B, if T ∗ = T = T 2 in B. (3) T is normal in B, if T ∗ T = T T ∗ in B. (4) T is an isometry in B, if T ∗ T = 1B in B. (5) T is a unitary in B, if T ∗ T = 1B = T T ∗ in B. Readers can recognize that the above definition is a generalization of the (operator-theoretic-)spectral properties in the Hilbert-space operator theory, or the C ∗ -algebra theory (e.g., see [14]). Up to the adjoint (∗) on the Banach ∗-algebra B, we extend the spectral properties (on Hilbert spaces, or in C ∗ -algebras) to those in B as above. Throughout this section, we let X = (Xφ , τ ) be our Γ-semicircular ∗-probability space. Then, by (5.3) and (5.6), X T = tW W ∈ X, with tW ∈ C. (14.1.1) W :free reduced word in G

Notation 14.1. In (14.1.1), the condition: “free reduced words W =
U, β k ∈ X with their length-n in G” means that “U ∈ Xφ is a free reduced words with their length-n in the free semicircular family X,” for all n ∈ N. □
Recall that if W = xsj , β k ∈ G, then
W ∗ = xsj+k , β −k ∈ G, in X, by the α-relation (12.3). It shows that a generating operator W ∈ G is not self-adjoint in X, in general.
Proposition 14.1. Let W = xsj , β k ∈ G. Then W is self-adjoint in X, ⇐⇒ k = 0.

(14.1.2)
0

Proof. (⇐) Assume that k = 0 in Z, and hence, W = xsj , β in G ⊂ X, where β 0 = 1Xφ is the group-identity of the integer-shift group λ. Then

W ∗ = xsj+0 , β −0 = xsj , β 0 = W, in X, by (12.3), implying the self-adjointness of W .

Operator-Theoretic Properties on X ■ 159


(⇒) Let W = xsj , β k ∈ G in X. If k ̸= 0, then

W ∗ = xsj+k , β −k ̸= xsj , β k = W, in X. i.e., if k ̸= 0, then W is not self-adjoint in X. Equivalently, if W is self-adjoint in X, then k = 0 in Z. Therefore, the self-adjointness characterization (14.1.2) on G holds in X. ■ The relation (14.1.2) characterizes the self-adjointness on the generator set G in X.
Proposition 14.2. Every operator W = xsj , β k ∈ G is not a projection in X. Proof. By the projection-property, if there exists a projection W ∈ G of X, then it should be self-adjoint first of
all. So, let’s focus on selfadjoint generating operators W 0 = xsj , β 0 ∈ G, by (14.1.2). Observe that
2

0 W 0 = xsj xsj+0 , β 0+0 = x2s ∈ G, j ,β in X by (13.1.2), implying that W0


2

̸= W 0 in X,

since s ∈ N, and hence, 2s ̸= s in N. It shows that all self-adjoint generating operators W of G are not idempotent in the sense that: W 2 = W in X, equivalently, all generating operators of G cannot be projections in X. ■ The above proposition characterizes the projection-property on the generator set G of X. i.e., all generating operators of G are not projections in X. By the definition of self-adjointness, if T ∈ X is self-adjoint, then it is automatically normal in X, since T ∗ T = T T = T 2 = T T = T T ∗ in X, and hence, one can get the following proposition by (14.1.2).
Corollary 14.3. A generating operator W = xsj , β 0 ∈ G is normal in X.

160 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. A given operator W is self-adjoint in X by (14.1.2). So it is automatically normal in X. Indeed,
0 W ∗ W = x2s = W W ∗, j ,β in X, implying the normality.

■

By Corollary 14.3, we are now interested in normal generating operators of G, which is not self-adjoint in X. Equivalently, we consider
the normality of the generating operators xsj , β k ∈ G with k ̸= 0, in X.
Lemma 14.4. A generating operator W = xsj , β k ∈ G, with k ̸= 0, is not normal in X. i.e., W ∈ G is not normal ⇐⇒ k ̸= 0 in Z.

(14.1.3)

Proof. Let W ∈ G be given as above in X. Then


W ∗ W = xsj+k , β −k xsj , β k = xsj+k xsj−k , β 0 , and W W ∗ = xsj , β k






xsj+k , β −k = xsj xsj+2k , β 0 ,

in X, by (13.1.2). It shows that: W ∗ W ̸= W W ∗ in X, whenever k ̸= 0, because xsj+k xsj−k ̸= xsj xsj+2k in Xφ , as nonzero free reduced words with their length-2, whenever k ̸= 0 in Z. Therefore, the non-normality characterization (14.1.3) on G holds. ■ The above lemmas characterize the normality on the generator set G in X, by (14.1.2) and (14.1.3); the self-adjointness on G and the normality on G are identical in X.
Proposition 14.5. A generating operator W = xsj , β k ∈ G is normal in X, if and only if it is self-adjoint in X. i.e., W ∈ G is normal, ⇐⇒ k = 0 in Z.

(14.1.4)

Operator-Theoretic Properties on X ■ 161

Proof. By (14.1.2), if k = 0 in Z, then W ∈ G is self-adjoint, and hence, normal in X. By (14.1.3), if k ̸= 0 in Z, then W is not normal in X. Therefore, the normality (14.1.4) holds. It implies that W ∈ G is normal ⇐⇒ k = 0 ⇐⇒ W is self-adjoint, in X.

■

By (14.1.4), one can get the following unitarity of G in X. Proposition 14.6. Every generating operator of G is not unitary in X. Proof. By definition, if W ∈ G were a unitary in X, then W ∗ W = I = W W ∗ , in X, (14.1.5)

where I = x0j , β 0 = 1Xφ , β 0 is the unity of X, where 1Xφ is the unity of the C ∗ -algebra Xφ , and β 0 is the identity of the integer-shift group λ (or, the unity of the shift-operator algebra Λ). However, by (14.1.3) and (14.1.4), if k ̸= 0, then W ∗ W ̸= W W ∗ in X, showing that W cannot be a unitary in X. If k = 0, then


0 W ∗ W = xsj+0 xsj−0 , β 0 = x2s = xsj xsj+0 , β 0 = W W ∗ , j ,β but W ∗ W = W W ∗ ̸= I in X, disobeying (14.1.5), and hence, it contradicts our assumption that W is a unitary in X. Therefore, every generating operator W ∈ G is not a unitary in X. ■ The above proposition proves that there
are no generating unitaries of G in X. Recall now that, if W = xsj , β k ∈ G, then


W ∗ W = xsj+k , β −k xsj , β k = xsj+k xsj−k , β 0 , (14.1.6) and W W ∗ = xsj , β k






xsj+k , β −k = xsj xsj+2k , β 0 ,

in X (as in the proof of (14.1.3)).

162 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proposition 14.7. Every generating operator of G is not an isometry in X.
Proof. It is shown directly by (14.1.6). i.e., if W = xsj , β k ∈ G were an isometry in X, then

W ∗ W = xsj+k xsj−k , β 0 = 1Xφ , β 0 = I, in X, if and only if xsj+k xsj−k = 1Xφ , in Xφ , which does not hold in Xφ , since xsj+k xsj−k is a free reduced word of Xφ with its length-2 in the free semicircular family X. ■ In this section, we characterized the operator-theoretic properties on the generating set G of X. In the following sections, we concentrate on general cases.

14.2

OPERATOR-THEORETIC PROPERTIES OF FREE REDUCED WORDS OF X IN G

Let X = (Xφ , τ ) be the Γ-semicircular ∗-probability space. In this section, operator-theoretic properties of free reduced words of X in the generator set G are studied. Consider a free reduced word U ∈ X in the generator set G (in the sense of N otation 14.1). i.e., U=

n Y


xsjll , β kl ∈ X.

(14.2.1)

l=1

Indeed, the operator U of (14.2.1) is a well-defined free reduced words of X in G (in the sense of N otation 14.1), since   U = xsj11 xsj22+k1 xsj33+k1 +k2 . . . xsjnn+k1 +···+kn−1 , β k1 +···+kn−1 +kn , (14.2.2) in X, by (13.1.2). However, note here that the element, T = xsj11 xsj22+k1 xsj33+k1 +k2 . . . xsjnn+k1 +···+kn−1 ∈ Xφ ,

(14.2.3)

in (14.2.2), is not a free “reduced” word of Xφ in X, in general. It is simply a “free word” in X, which can be simplified to be a free

Operator-Theoretic Properties on X ■ 163

“reduced” word under the operator-multiplication on Xφ , by (5.6). So, the statement “an element U ∈ X of (14.2.1) is a free reduced word in G” means “the element U is regarded as a free reduced word of X in the sense of N otation 14.1, by simplifying T of (14.2.3) to its identified free reduced word of Xφ in X.” Notation 14.2. As we discussed in the very above paragraph, from below, we simply say “ an element U of (14.2.1) is a free reduced word of X in G,” in the above sense. □ Indeed, the following result well characterizes the terminology of N otation 14.2. Theorem 14.8. Let X be our Γ-semicircular ∗-probability space. Then   free-homo X −→ ⋆ C [{xj }] ⊗C Λ , j∈Z

where (⋆) is the free-probabilistic free product, and Λ is the shiftoperator algebra generated by λ, where ⊗C is the usual tensor product of Banach ∗-algebras. Proof. By the structure theorem (4.5),   ∗−iso Xφ = ⋆ C [{xj }] , j∈Z

one has X

free-iso

def

X = Xφ ⊗α Λ   ∗-homo −→ ⋆ C [{xj }] ⊗C Λ j∈Z   ∗-homo → ⋆ C [{xj }] ⊗C Λ ,

=

j∈Z

since the conditional tensor product Banach ∗-algebra X = Xφ ⊗α Λ is a Banach ∗-subalgebra of the usual tensor product Banach ∗-algebra ∗-homo Xφ ⊗C Λ, where “ −→ ” means “being ∗-homomorphic to.” The free probability on X is preserved to be that on X , by the freeisomorphism Φ of (12.2.4). Therefore, the above homomorphic relation implies the free-homomorphic relation. ■

164 ■ Banach-Space Operators On C ∗ -Probability Spaces

Let U ∈ X be a free reduced word (14.2.1). Then  ∗ U ∗ = xsj11 xsj22+k1 xsj33+k1 +k2 . . . xsjnn+k1 +···+kn−1 , β kU where kU =

n P

kl ∈ Z

l=1

  = xsjnn+k1 +···+kn−1 +kU . . . xsj33+k1 +k2 +kU xsj22+k1 +kU xsj11+kU , β −kU , (14.2.4) in X, by (12.3). So, one can verify that U = U ∗ in X,

(14.2.5)

if and only if   xsj11 xsj22+k1 . . . xsjnn+k1 +···+kn−1 =       sn xjn +k1 +···+kn−1 +kU . . . xsj22+k1 +kU xsj11+kU   and     kU −kU β =β , in Xφ , and in λ, respectively. The above relation (14.2.5) is equivalent to ! n n X X kU = kl = − kl = −kU , l=1

(14.2.6)

l=1

in Z, and j1 = jn + k1 + · · · + kn−1 + kU , j2 + k1 = jn−1 + k1 + · · · + kn−2 + kU , j3 + k1 + k2 = jn−2 + k1 + · · · + kn−3 + kU , .. . j[ n ]−1 + k1 + · · · + k[ n ]−2 = j[ n ]+1 + k1 + · · · + k[ n ] + kU , 2 2 2 2 in Z, where hni 2

def

= the maximal integer, less than or equal to

n , 2

Operator-Theoretic Properties on X ■ 165

for instance,     4 5 = 2, and = 2. 2 2 The above conditions (14.2.6) can be re-written by U ∗ = U in X, if and only if n n−i X X kU = kl = 0, ⇐⇒ kl = − l=1

l=1

n X

! kl

,

(14.2.7)

l=n−i+1

in Z, and ji +

i−1 X

kl = jn+1−i +

l=1

n−i X

kl + kU = jn+1−i +

l=1

n−i X

kl ,

l=1

in Z, because kU = 0, for all i = 1, . . . , n − 1, with axiomatization: 0 X

let

kl = 0, in Z.

l=1

Theorem 14.9. A free reduced word U of (14.2.1) in G is self-adjoint in X, if and only if ! n−i n n X X X kl = − kl ⇐⇒ kl = 0, (14.2.8) l=1

l=n−i+1

and ji +

i X

l=1

kl = jn+1−i +

l=1

n−i X

kl ,

l=1

in Z, for all i = 1, . . . , n − 1. Proof. The proof of the relation (14.2.8) is done by (14.2.7).

■

The above theorem characterizes the self-adjointness of free reduced words of X in G by (14.2.8). Suppose k1 = 1, k2 = 0, k3 = −1,

166 ■ Banach-Space Operators On C ∗ -Probability Spaces

and j1 = 3, j2 = 2, j3 = 2, in Z, then k1 = − (k2 + k3 ) , and k1 + k2 = − (k3 ) ,

(14.2.9)

and j1 = 3 = 2 + 1 = j3 + (k1 + k2 ), and j2 + k1 = 2 + 1 = j2 + k1 , Then we have


U0 = xs31 , β 1 xs22 , β 0 xs23 , β −1   2 3 = xs31 xs2+1 xs2+1+0 , β 1+0+(−1)

= xs31 xs32 xs33 , β 0 = xs31 +s2 +s3 , β 0 , having its adjoint, U0∗ =



xs31 +s2 +s3


∗

 , β −0 = U0 ,

in X. Indeed, the above condition (14.2.9) satisfies the relation (14.2.8), and hence, Theorem 14.9 guarantees the self-adjointness of this free reduced word U0 in X. Also, independently, one can check that, as a free reduced word of X with its length-1, this operator U0 ∈ G is self-adjoint in X, by (14.1.2).
Corollary 14.10. Let U = W, β k ∈ X be a free reduced word (14.2.1) in G, where W ∈ Xφ is a free reduced word in the free semicircular family X. If W has its length 1 in Xφ , and k = 0, then U is self-adjoint in X. Proof. Suppose W ∈ Xφ
is a free-reduced word with its length-1 in X, and assume U = W, β 0 in X. Then it is contained in the generator set G of X by (14.2.1). So, the self-adjointness of U is determined by (14.1.2) in X. ■

Operator-Theoretic Properties on X ■ 167

The above corollary illustrates
that if a free reduced word U of (14.2.1) is simplified to be xsj , β 0 ∈ G, for some j ∈ Z, and s ∈ N, then it is self-adjoint in X, by (14.1.2), without considering (14.2.8). Conjecture 14.2.1. A free reduced word U of (14.2.1) is self-adjoint in X, if and only if
U = xsj , β 0 ∈ X, for some j ∈ Z,
and s ∈ N. By Corollary 14.10, we already showed that: if U = xsj , β 0 ∈ X, then it is self-adjoint in X. So, one may focus on the converse. □ Let U ∈ X be a self-adjoint free random variable (14.2.1), satisfying the self-adjointness condition (14.2.8). Observe that  2 U 2 = xsj11 xsj22+k1 xsj33+k1 +k2 . . . xsjnn+k1 +···+kn−1 , β 0   2 s1 s2 sn 0+0 = xj1 xj2 +k1 . . . xjn +k1 +···+kn−1 , β   = xsj11 xsj22+k1 . . . xsjnn+k1 +···+kn−1 xsj11 xsj22+k1 . . . xsjnn+k1 +···+kn−1 , β 0 , in X. So, this self-adjoint free random variable U were a projection in X, if and only if U 2 = U in X, (14.2.10) if and only if xsj11 xsj22+k1 . . . xsjnn+k1 +···+kn−1 xsj11 xsj22+k1 . . . xsjnn+k1 +···+kn−1 let

= xsj11 xsj22+k1 . . . xsjnn+k1 +···+kn−1 = W, in Xφ , as free reduced words in X. By the relation (14.2.10), if W is a free reduced word of Xφ with its lengh-n in X, and if n > 1, then W 2 ̸= W in Xφ , implying that U is not idempotent in X, i.e., U 2 ̸= U in X, and hence, such a self-adjoint free reduced word U cannot be a projection in X.

168 ■ Banach-Space Operators On C ∗ -Probability Spaces

Theorem 14.11. Every free reduced word U of (14.2.1) is not a projection in X.
Proof. Let U = W, β k ∈ X be a free reduced word (14.2.1) with a free reduced word W ∈ Xφ . Let’s assume that it is self-adjoint in X. Then, by the self-adjointness characterization (14.2.8) of U , k = 0 in Z. i.e.,
0 U = W, β in X, for a free reduced word W ∈ Xφ .
As we discussed in the very above paragraph, if U = W, β 0 ∈ X, and if W has its length-n, with n > 1, in Xφ , then the free reduced word U cannot be a projection in X. Assume now that W = xsj ∈ Xφ is a free reduced word with its length-1, equivalently, U ∈ G in X. Then, by Proposition 14.2, this element U cannot be a projection in X, either. Therefore, all free reduced words (14.2.1) in G are not projections in X. ■ The above theorem shows that all free
reduced words of X in G are not projections. Recall that if U = xsj , β k ∈ X is a free reduced word with its length-1 in G, then U is normal in X, if and only if k = 0 in Z,

(14.2.11)

if and only if it is self-adjoint in X, by (14.1.4). By (14.2.11), we here focus on the cases where a given free random variable U ∈ X of (14.2.1) has its length-n, where n > 1 in N. Observe that: if U has its length-n, with n > 1, then   U ∗ U = xjn +k1 +,,,+kn−1 +kU . . . xsj22+k1 +kU xsj11+kU , β −kU   × xsj11 xsj22+k1 . . . xsjnn+k1 +···+kn−1 , β kU ! xsjnn+k1 +···+kn−1 +kU . . . xsj11+kU xsj11−kU . . . = , (14.2.12) xsjnn+k1 +···+kn−1 −kU , β 0 and   U U ∗ = xsj11 xsj22+k1 . . . xsjnn+k1 +···+kn−1 , β kU   × xsjnn+k1 +···+kn−1 +kU . . . xsj22+k1 +kU xsj11+kU , β −kU   = xsj11 . . . xsjnn+k1 +···+kn−1 xsjnn+k1 +,,,+kn−1 +kU . . . xsj11+kU , β 0 , (14.2.13)

Operator-Theoretic Properties on X ■ 169

where kU =

n P

kl in Z.

l=1

From below, we denote the right-hand sides of (14.2.12) and (14.2.13) simply by U ∗U

denote

=


W∗ , β 0 , and U U ∗

denote

=


W ∗, β0 ,

(14.2.14)

respectively, with W∗ , W ∗ ∈ Xφ . Theorem 14.13. Every free reduced word U ∈ X with its length-n in G, with n > 1, is not normal in X. Proof. Suppose U ∈ X is a free reduced word in G, with its lengthn, where n > 1. Then the free random variables U ∗ U and U U ∗ are determined to be the operators of (14.2.14). So, they cannot be identical in X, because W∗ ̸= W ∗ in Xφ , as free reduced words (with their length-2n) in the free semicircular family X, by (14.2.12) and (14.2.13). It implies that U ∗ U ̸= U U ∗ in X, whenever n > 1.

■

The above theorem reduced words (14.2.1) In conclusion, we have normality of free reduced

characterizes the (non-)normality of free of X, with their length-(n > 1), in G. the following corollary, characterizing the words (14.2.1) of X in G.
Corollary 14.14. Let U = W, β k ∈ X be a free reduced word (14.2.2) in G. Then it is normal in X, if and only if |W | = 1 in Xφ , and k = 0 in Z,

(14.2.15)

where |W | is the (reduced-)length of W ∈ Xφ . Proof. Let U ∈ X be a free reduced word (14.2.2) in G. (⇐) If the condition (14.2.15) holds, then the operator U is normal in X, by (14.2.11) (or, by (14.1.4)). (⇒) Assume that the condition (14.2.15) does not hold, i.e., suppose either |W | > 1 in Xφ , or k ̸= 0 in Z. First of all, if |W | > 1, then, by

170 ■ Banach-Space Operators On C ∗ -Probability Spaces

Theorem 14.13, this operator U is not normal in X. Second, if k ̸= 0, and |W | = 1, then U is not normal in X, by (14.2.11), while, if k ̸= 0, and |W | > 1, then it is not normal either, by Theorem 14.13. Therefore, the normality characterization (14.2.15) holds for all free reduced words U ∈ X of (14.2.2) in G. ■ The above corollary shows that the normality (14.2.15) of free reduced words of X in G implies the self-adjointness (14.1.2) of free reduced words of X with their length-1. Also, by the normality (14.2.15), the following unitarity of free reduced words of X in G is obtained. Theorem 14.15. Every free reduced word U of (14.2.1) in G is not a unitary in X. Proof. If a free reduced word U of (14.2.1) were a unitary in X. Then, by definition, this operator U should be normal in X. So, by (14.2.15), the operator U should be of the form,
U = xsj , β 0 ∈ X, for some j ∈ Z, and s ∈ N, where xj ∈ X is a semicircular element of Xφ . However, such a free reduced word U ∈ X with its length-1 cannot be a unitary in X, by Proposition 14.6. Therefore, all free reduced words (14.2.1) in G are not unitary in X. ■ Similar to the proof of Theorem 14.15, one obtains the following isometry-property of the free reduced words (14.2.1) of X.
Theorem 14.16. Every free reduced word U = W, β k of (14.2.2) in G is not an isometry in X.
Proof. Suppose |W | = 1 in Xφ , and hence, U = xsj , β k in X. Then it is not an isometry in X, by Proposition 14.7. Assume now that |W | > 1. If U were an isometry, then

U ∗ U = W∗ , β 0 = 1Xφ , β 0 = I in X, by (14.2.14), if and only if W∗ = 1Xφ in Xφ ,

Operator-Theoretic Properties on X ■ 171

where W∗ is the free reduced word of Xφ in X, obtained in (14.2.12). However, W∗ ̸= 1Xφ in Xφ , implying that U ∗ U ̸= I in X. It contradicts our assumption.

■

The main results of this section, characterizing operator-theoretic properties of free reduced words of the Γ-semicircular ∗-probability space X in the generator set G generalize the main results of Section 8.1.

14.3

CERTAIN OPERATOR SUMS OF X

In this section, different from Section 14.2, the operator-theoretic properties of certain operator sums of X induced by the generating operators of G, are studied. In particular, we are interested in the operators formed by T = t0 I+

n X


tl xsjll , β kl ∈ X with tl ∈ C,

(14.3.1)

l=1

in G ∪ {I}, where Wl

denote

=


xsjll , β kl are mutually distinct,

in G, for all l = 1, . . . , n, for n ∈ N∞ = N ∪ {∞}, and sl ∈ N. In the definition (14.3.1), we need to keep in mind that our operator sums (14.3.1) can contain the nonzero unity-summand t0 I, whenever t0 ̸= 0 in C. Note that the identity element I of X is understood as

I = x0jl , β 0 = 1Xφ , β 0 in X. By (14.3.1), it is verified that such an operator T is naturally decomposed by T = T− + t0 I + T+ in X, (14.3.2) with

n+ X
T+ = tl xsjll , β kl with kl > 0 and sl ̸= 0, l=1

and T− =

n− X i=1


ti xsjii , β ki with ki < 0,

172 ■ Banach-Space Operators On C ∗ -Probability Spaces

where

( n+ + n− + 1 n= n+ + n−

if t0 ̸= 0 in C if t0 = 0 in C,

in N∞ . Assumption. From below, if there is no confusion, the operators T of (14.3.1) are automatically assumed to be “non-zero” operators in X. Equivalently, T+ ̸= O, or T− ̸= O in X, or t0 ̸= 0 in C, by (14.3.2). Also, we denote Wl

denote

=


xsjll , β kl ∈ G, ∀l = 1, . . . , n,

for convenience.

□

Observe that T ∗ = t0 I+

n X

tl Wl∗ = t0 I+

l=1

n X

  tl xsjll+kl , β −kl in X,

l=1

implying that T ∗ = T−∗ + t0 I + T+∗ in X,

(14.3.3)

where T− and T+ are in the sense of (14.3.2). Theorem 14.17. Let T = T− + t0 I + T+ ∈ X be an operator (14.3.1) with its decomposition (14.3.2). Then it is self-adjoint, if and only if t0 ∈ R, and T+∗ = T− in X.

(14.3.4)

Equivalently, there exists an operator T0 =

n0 X

tl Wl ∈ XN , for some n0 ∈ N,

l=1

such that T = T0∗ + t0 I + T0 ∈ X, with t0 ∈ R, where

in N∞ .

( 2n0 + 1 n= 2n0

if t0 ̸= 0 in C if t0 = 0 in C,

(14.3.5)

Operator-Theoretic Properties on X ■ 173

Proof. Suppose T = T− + t0 I + T+ ∈ X is an operator (14.3.1), decomposed by (14.3.2). Then, by (14.3.3), T ∗ = T ⇐⇒ t0 = t0 , and T+∗ = T− , in X. Therefore, the self-adjointness characterization (14.3.4) holds. n0 P tl Wl ∈ X, having its adjoint, Suppose there is an operator T0 = l=1

T0∗ =

n0 X

tl Wl∗ ∈ X,

l=1

where Wl∗ = xsjll , β kl


∗

  = xsjll+kl , β −kl .

If T = T0∗ + t0 I + T0 ∈ X with t0 ∈ R, then T ∗ = T0∗∗ + t0 I ∗ + T0∗ = T0 + t0 I + T0∗ = T, in X, by (14.3.3), implying the self-adjointness (14.3.4) of T in X. Assume now that T ∈ X is an operator (14.3.1), and suppose T is not of the form of (14.3.5). Then T cannot be self-adjoint in X by (14.3.4). Therefore, the relation (14.3.5) is equivalent to (14.3.4). ■ The above theorem characterizes the self-adjointness of operators T of (14.3.1) in X, by (14.3.4), or by (14.3.5). Corollary 14.18. An operator T of (14.3.1) is self-adjoint in X, if and only if m m   X X
sl −kl T = tl xjl +kl , β tl xsjll , β kl , + t0 I+ l=1

(14.3.6)

l=1

in X, with t0 ∈ R, for some m ∈ N∞ . Proof. The relation (14.3.6) is shown by (14.3.5) (or, by (14.3.4)). ■ Let T = T− + t0 I + T+ ∈ X be an operator (14.3.1), decomposed by (14.3.2). Consider that T 2 = (T− + t0 I + T+ ) (T− + t0 I + T+ )
= T−2 + T+2 + 2t0 (T− + T+ ) + (T− T+ + T+ T− ) + t20 I,

174 ■ Banach-Space Operators On C ∗ -Probability Spaces

in X, implying that: if T is self-adjoint in X, then


T 2 = T+∗2 + T+2 + 2t0 T+∗ + T+ + T+∗ T+ + T+ T+∗ + t20 I, (14.3.7) by (14.3.4) and (14.3.6), where t0 ∈ R. Theorem 14.19. An operator T = T− + t0 I + T+ of (14.3.1) is a projection in X, if and only if t0 = 0, or t0 = 1 in R,

(14.3.8)

and


T+∗2 + T+2 + 2t0 T+∗ + T+ + T+∗ T+ + T+ T+∗ = T+ + T+∗ in X. Proof. By definition, if an operator T ∈ X is a projection, it is selfadjoint in X. So, it is sufficient to consider the case where T is selfadjoint in X. Then, by (14.3.4), if T is self-adjoint in X, then t0 ∈ R, and T+∗ = T− in X, i.e., T = T+∗ + t0 I + T+ in X, with t0 ∈ R. Such a self-adjoint operator T is a projection in X, if and only if it is idempotent in X, if and only if t20 I = t0 I, and


T+∗2 + T+2 + 2t0 T+∗ + T+ + T+∗ T+ + T+ T+∗ = T+ + T+∗ , in X, by (14.3.7), if and only if t20 = t0 in R, and


T+∗2 + T+2 + 2t0 T+∗ + T+ + T+∗ T+ + T+ T+∗ = T+ + T+∗ , in X, if and only if t0 = 0, or t0 = 1 in R, and


T+∗2 + T+2 + 2t0 T+∗ + T+ + T+∗ T+ + T+ T+∗ = T+ + T+∗ , in X. Therefore, the projection-property (14.3.8) holds.

■

Operator-Theoretic Properties on X ■ 175

Also, one obtains a following refined case. Theorem 14.20. Let T = T− + t0 I + T+ ∈ X be a “non-zero” operator (14.3.1), where n < ∞ in N∞ . Then it is a projection in X, if and only if t0 = 1 in C, and T+ = O in X, (14.3.9) if and only if T =I

in X.

Proof. Assume that T ∈ X is an operator (14.3.1), with n < ∞ in N∞ . i.e., suppose a given operator T ∈ X is a “finite” sum over the generator set G. This operator T of (14.3.1) is a projection in X, if and only if t0 = 0, or t0 = 1 in R, and


T+∗2 + T+2 + 2t0 T+∗ + T+ + T+∗ T+ + T+ T+∗ = T+ + T+∗ , in X, by (14.3.8). So, a “finite-sum” self-adjoint operator T is a “non-zero” projection in X, if and only if t0 = 1 in R, and T+∗2 + T+2 = O = T+∗ T+ + T+ T+∗ and

2t0 T+∗ + T+ = 2 T+∗ + T+ = T+ + T+∗ , in X, if and only if t0 = 1 in R, and T+ + T+∗ = O in X, if and only if t0 = 1 in R, and T+ = O in X, whenever n < ∞. i.e., this self-adjoint “finite-sum” operator T can be a projection in X, if and only if there exists t0 ∈ R, such that T = O + I + O = I in X. It proves the relation (14.3.9).

■

176 ■ Banach-Space Operators On C ∗ -Probability Spaces

The above theorem shows that the only non-zero “finite-sum” projection T of (14.3.1) is the identity operator I of X, by (14.3.9). Now, let T ∈ X be an operator (14.3.1), decomposed by (14.3.2), with its adjoint T ∗ of (14.3.3). Then
T ∗ T = T−∗ + t0 I + T+∗ (T− + t0 I + T+ ) 2

= T−∗ T− + t0 T−∗ + T−∗ T+ + t0 T− + |t0 | I + t0 T+ + T+∗ T− + t0 T+∗ + T+∗ T+
= T−∗ T− + T−∗ T+ + T+∗ T− + T+∗ T+
2 + t0 T−∗ + T+∗ + t0 (T− + T+ ) + |t0 | I,

(14.3.10)

and T T ∗ = (T− + t0 I + T+ ) T−∗ + t0 I + T+∗


2

= T− T−∗ + t0 T− + T− T+∗ + t0 T−∗ + |t0 | I + t0 T+∗ + T+ T−∗ + t0 T+ + T+ T+∗
= T− T−∗ + T− T+∗ + T+ T−∗ + T+ T+∗
2 + t0 T−∗ + T+∗ + t0 (T− + T+ ) + |t0 | I, in X. By (14.3.10), one can get that an operator T is normal in X, if and only if T−∗ T− + T−∗ T+ + T+∗ T− + T+∗ T+ = T− T−∗ + T− T+∗ + T+ T−∗ + T+ T+∗ ,

(14.3.11)

in X. Theorem 14.21. Let T ∈ X be an operator (14.3.1), decomposed by (14.3.2). Then it is normal in X, if and only if T−∗ T− + T−∗ T+ + T+∗ T− + T+∗ T+ = T− T−∗ + T− T+∗ + T+ T−∗ + T+ T+∗ ,

(14.3.12)

in X. Proof. The proof of the normality (14.3.12) of T in X is shown by (14.3.11). ■

Operator-Theoretic Properties on X ■ 177

By the normality (14.3.12), one obtains the following corollary. Corollary 14.22. Let T = T− + t0 I + T+ ∈ X be an operator (14.3.1). (14.3.13) If T− = O, then T is normal, if and only if T+ is normal in X. (14.3.14) If T+ = O, then T is normal, if and only if T− is normal in X. (14.3.15) If T+ = O = T− , then T is normal in X. Proof. The proofs of the above three statements are done by the normality (14.3.12). Suppose first that T− = O in X. Then such an operator T of (14.3.2) is normal in X, if and only if T+∗ T+ = T+ T+∗ in X, by (14.3.12), if and only if T+ is normal in X. Thus, the statement (14.3.13) holds. Similarly, the statement (14.3.14) also holds true. Assume that T− = O = T+ in X. Then 2

T ∗ T = |t0 | I = T T ∗ in X, implying the normality of T in X. It proves the statement (14.3.15). ■ n P tl Wl ∈ X be an operator (14.3.1), where Now, let T = l=1
Wl = xsjll , β kl ∈ G, with “kl ≥ 0 in Z.” i.e.,

T = T+ ,

in X,

(14.3.16)

in the sense of (14.3.2). Then n   X T∗ = tl Wl∗ ∈ X, with Wl∗ = xsjll+kl , β −kl ∈ G, l=1

for all l = 1, . . . , n.

(14.3.17)

178 ■ Banach-Space Operators On C ∗ -Probability Spaces

By (14.3.16) and (14.3.17), we have   X s s T ∗T = tl1 tl2 xjll1 +kl xjll2 −kl , β −kl1 +kl2 , 1

(l1 ,l2

1

2

(14.3.18)

1

)∈{1,...,n}2

and  s s ti1 ti2 xjii1 xjii2 +ki

X

TT∗ =

1

2

2 +ki1

 , β ki1 −ki2 ,

(i1 i2 )∈{1,...,n}2

in X. By (14.3.13) and (14.3.18), we obtain the following normality condition. n P Theorem 14.23. Let T = tl Wl ∈ X be an operator (14.3.16). l=1

Assume that k1 = k2 = . . . = kn = 0 in Z. Then T is normal in X, if and only if tl1 tl2 = tl1 tl2 in C,

(14.3.19)

2

for all (l1 , l2 ) ∈ {1, . . . , n} . n P

Proof. Suppose T = T+ =

tl Wl ∈ X is an operator (14.3.16) of X.

l=1

Assume that k1 = k2 = . . . = kn = 0, in Z, i.e., T =

n X


tl xsjll , β 0 ∈ X.

l=1

Then T∗ =

n X

n
X
tl xsjll+0 , β −0 = tl xsjll , β 0 ∈ X.

l=1

l=1

So, by (14.3.18), we have that   X s s tl1 tl2 xjll1 +0 xjll2 −0 , β −0+0 , T ∗T = 1

(l1 ,l2

)∈{1,...,n}2

2

(14.3.20)

Operator-Theoretic Properties on X ■ 179

and

  s s ti1 ti2 xjii1 xjii2 +0+0 , β 0−0 ,

X

TT∗ =

1

2

(i1 i2 )∈{1,...,n}2

in X. Thus, such an operator T is normal in X, if and only if     s s s s tl1 tl2 xjll1 xjll2 , β 0 = tl1 tl2 xjll1 xjll2 , β 0 in X, 1

2

1

2

2

for all (l1 , l2 ) ∈ {1, . . . , n} , by (14.3.20), if and only if tl1 tl2 = tl1 tl2 in C, 2

for all (l1 , l2 ) ∈ {1, . . . , n} . Therefore, under the condition that k1 = k2 = . . . = kn = 0 in Z, the normality characterization (14.3.19) holds.

■

The above special case (14.3.19) of (14.3.12) refines (14.3.13). Like the normality (14.3.12), the computation (14.3.10) gives the following isometry-characterization. Theorem 8.24. An operator T = T− + t0 I + T+ of (14.3.1), decomposed by (14.3.2), is an isometry in X, if and only if t0 ∈ T ⊂ C,

(14.3.21)

and

T−∗ T− + T−∗ T+ + T+∗ T− + T+∗ T+ + t0 T−∗ + T+∗ + t0 (T− + T+ ) = O, in X, where T is the unit circle of C. Proof. By (14.2.10), one has that
T ∗ T = T−∗ + t0 I + T+∗ (T− + t0 I + T+ )
= T−∗ T− + T−∗ T+ + T+∗ T− + T+∗ T+
2 + t0 T−∗ + T+∗ + t0 (T− + T+ ) + |t0 | I,

(14.3.22)

in X. So, such an operator T is an isometry in X, if and only if the resulted computation (14.3.22) of T ∗ T is identified with the identity operator I of X, if and only if the condition (14.3.21) holds in X. Recall 2 that t0 ∈ T in C, if and only if |t0 | = 1. Thus, the isometry-characterization (14.3.21) holds. ■

180 ■ Banach-Space Operators On C ∗ -Probability Spaces

By the unitarity, one can get the following unitarity characterization with help of (14.3.21). Theorem 8.25. An operator T = T− + t0 I + T+ of (14.3.1), decomposed by (14.3.2), is a unitary in X, if and only if T−∗ T− + T−∗ T+ + T+∗ T− + T+∗ T+ = T− T−∗ + T− T+∗ + T+ T−∗ + T+ T+∗ ,

(14.3.23)

and t0 ∈ T ⊂ C,

(14.3.24)

and
T−∗ T− + T−∗ T+ + T+∗ T− + T+∗ T+
+ t0 T−∗ + T+∗ + t0 (T− + T+ ) = O, in X. Proof. By definition, a given operator T is unitarty in X, if and only if it is both a normal operator, and an isometry in X. Thus, it is a unitary in X, if and only if it satisfies both the normality (14.3.23) and the isometry-property (14.3.24), by (14.3.12) and (14.3.21), respectively. ■

14.4

FREE DISTRIBUTIONS OF SOME SELF-ADJOINT OPERATORS

In this section, as applications of Sections 13 and 14.3, we study free distributions of a certain self-adjoint free random variable of our Γsemicircular ∗-probability space X = (X ⋊α λ, τ ). Throughout this section, for j, k ∈ Z, let

T = xsj+k , β −k + xsj , β k ∈ X. (14.4.1) It is not hard to check that this operator T is self-adjoint in X, by (14.3.6). Indeed,  
∗ s −(−k) T = xj+k−k , β + xsj+k , β −k = T, and hence, it is self-adjoint in X.

Operator-Theoretic Properties on X ■ 181

If we let xs,k j

denote

=


xsj , β k ∈ X,

then the operator T of (14.4.1) is re-written by ∗  s,k s,−k + xs,k T = xs,k j = xj+k + xj ∈ X. j ∗ If W = xs,k j ∈ X, inducing T = W + W in X, then

τ (W ∗n1 W n2 )   s s s s −n1 k = τ xj+k xj xj−k . . . xj−(n1 −2)k , β   × xsj xsj+k . . . xsj+(n2 −1)k , β n2 k =τ

xsj+k xsj . . . xsj−(n1 −2)k xsj−n1 k xsj+(1−n1 )k . . . xsj+(n2 −n1 −1)k , β (−n1 +n2 )k

=φ β

(n2 −n1 )k

!

xsj+k xsj . . . xsj−(n1 −2)k xsj−n1 k xsj+(1−n1 )k . . . xsj+(n2 −n1 −1)k

!!

  = φ xsj+k xsj . . . xsj−(n1 −2)k xsj−n1 k xsj+(1−n1 )k . . . xsj+(n2 −n1 −1)k , (14.4.2) by (4.19) and (14.2.2), and τ (W n2 W ∗n1 )   = τ xsj xsj+k . . . xsj+(n2 −1)k , β n2 k   × xsj+k xsj xsj−k . . . xsj−(n1 −2)k , β −n1 k =τ

xsj xsj+k . . . xsj+(n2 −1)k xsj+(n2 +1)k xsj+n2 k . . . xsj+(n2 −n1 +2)k , β (n2 −n1 )k

= φ β (n2 −n1 )k

!

xsj xsj+k . . . xsj+(n2 −1)k xsj+(n2 +1)k xsj+n2 k . . . xsj+(n2 −n1 −2)k

!!

  = φ xsj xsj+k . . . xsj+(n2 −1)k xsj+n2 k xsj+(n2 −1)k . . . xsj+(n2 −n1 −2)k , (14.4.3) by (4.19) and (14.2.2).

182 ■ Banach-Space Operators On C ∗ -Probability Spaces


s k Lemma 14.25. Let W = xs,k ∈ G be a generating operator j = xj , β of X. Then
n1 +n2 = τ (W n2 W ∗n1 ) in C, (14.4.4) τ (W ∗n1 W n2 ) = ωs c 2s for all n1 , n2 ∈ N0 = N ∪ {0}. Proof. By (3.22), (3.25), (13.2.1), (14.4.2) and (14.4.3), we have that τ (W ∗n1 W n2 ) = ωs c 2s


n1 +n2

= τ (W n2 W ∗n1 ) ,

for all n1 , n2 ∈ N. Clearly, if either n1 = 0, or n2 = 0, the above equality holds. Therefore, the free-distributional data (14.4.4) holds, for all n1 , n2 ∈ N0 . ■ Inductive to (14.4.4), we obtain the following generalized result. Proposition 14.26. Let W = xs,k j ∈ G be a generating operator of X. Then ! ! n n Y Y
n1 +n2 , τ W il = τ W iσ(l) = τ (W ∗n1 W n2 ) = ωs c 2s l=1

l=1

(14.4.5) n for (i1 , . . . , in ) ∈ {1, ∗} , and for all permutations σ ∈ Sn , where Sn is the symmetric group over {1, . . . , n}, for all n ∈ N, where, in the last equality of (14.4.5), n1 , n2 ∈ N0 , with n1 + n2 = n, in N. Proof. The proof of the first equality in (14.4.5) is done by the induction on (14.4.4), with help of (3.25) and (13.2.1). The second equality in (14.4.5) is a special case of the first equality; and the third equality of (14.4.5) is shown by (14.4.4). ■ Observe now that, for any n ∈ N, if T ∈ X is an operator (14.4.1), then ! n X Y W il ∈ X, (14.4.6) Tn = (i1 ,...,in )∈{1,∗}n

l=1


s k for all n ∈ N, where W = xs,k ∈ G. j = xj , β

Operator-Theoretic Properties on X ■ 183

Theorem 14.27. Let T = W ∗ + W ∈ X be a self-adjoint operator (14.4.1), where W = xs,k j ∈ G. Then τ (T n ) = ωs c 2s


n

n   X n

k

k=0

! ,

(14.4.7)

for all n ∈ N, where   n! n = , ∀k ≤ n ∈ N0 . k k!(n − k)! Proof. For any n ∈ N, we have that  X τ (T n ) = τ 

n Y

(i1 ,...,in )∈{1,∗}n

! W il 

l=1

by (14.4.6) =

n   X n k=0

k

τ W ∗k W n−k




by (14.4.5), where   n! n = , ∀k ≤ n ∈ N0 , k k!(n − k)! and hence, it goes to = ωs c 2s


n

n   X n k=0

k

! ,

by (14.4.5). Therefore, the free-distributional data (14.4.7) holds.

■

The above theorem characterizes the free distributions of the selfadjoint free random variables T of (14.4.1) in the Γ-semicircular ∗probability space X, by (14.4.7).

CHAPTER

15

Free Probability on XN = (XN ⊗C ΛN , τN )

N SECTIONS 12, 13 AND 14, we studied free probability, and Banach-space-operator theory on the Γ-semicircular ∗-probability space, X = (Xφ ⋊α λ, τ ) ,

I

induced by the group dynamical system, Γ = (Xφ , λ, α) , where λ ∈ Aut (Xφ ) is the integer-shift group acting on the C ∗ probability space Xφ generated by the free semicircular family X = {xj }j∈Z , via the group-action α. In Section 5, we considered the restricted-shift families λN =  k N βN k=−N ⊂ Hom (XN ) acting on the free-probabilistic substructures XN of Xφ , generated by the free semicircular families N XN = {xj }j=1 , by acting the (restricted-)shift-operator algebras ΛN ⊂ B (XN ) generated by λN , for all N ∈ N∞ >1 . In Section 12.3, we discussed that, the dynamics of Section 12 cannot be restricted to suitable dynamical systems of λN on XN , for each N ∈ N∞ >1 . However, one may have suitable free-probabilistic structures induced by {XN , λN }N ∈N∞ , >1 containing informations how {λN }N ∈N∞ (or {ΛN }N ∈N∞ ) act on >1 >1 {XN }N ∈N∞ , under the tensor product of XN and ΛN , for N ∈ N∞ >1 . >1 The main purpose of this section is to consider the free probabilities on such tensor products.

DOI: 10.1201/9781003263487-15

185

186 ■ Banach-Space Operators On C ∗ -Probability Spaces

15.1

BANACH ∗-PROBABILITY SPACES XN

Throughout this section, we fix N ∈ N∞ >1 , and let XN = (XN , φN ) be the corresponding C ∗ -probability space (5.6), generated by the free  k N N semicircular family XN = {xj }j=1 , and let λN = βN be the k=−N restricted-shift family (5.2.6), which is a subset of the homomorphism semigroup Hom (XN ), acting on XN under the action of the shiftoperator algebra ΛN of (5.2.10). Even though there does not exist a “good” crossed product Banach ∗-probability space generated by {XN , λN } (compared with the settings of Sections 12, 13 and 14), one can naturally consider a Banach ∗probabilistic structure induced by {XN , ΛN } under the usual tensor product of XN and ΛN , as we discussed in Section 12.3. Define the tensor product Banach ∗-algebra XN by def

X N = X N ⊗C ΛN .

(15.1.1)

Since XN is a C ∗ -algebra, and ΛN is a Banach ∗-algebra, the tensor product Banach ∗-algebra XN of (15.1.1) is a well-defined Banach ∗algebra, satisfying the operator-multiplication, (T1 ⊗ S1 ) (T2 ⊗ S2 ) = T1 T2 ⊗ S1 S2 , and the adjoint, ∗

(T ⊗ S) = T ∗ ⊗ S ∗ , for all T, T1 , T2 ∈ XN , and S, S1 , S2 ∈ ΛN . Define a linear functional τN on XN by the linear morphism satisfying τN (T ⊗ S) = φN (S (T )) , (15.1.2) for all T ∈ XN , and S ∈ ΛN , where φN is in the sense of (5.6). Definition 15.1. The Banach ∗-probability space, XN

denote

=

(XN , τN )

(15.1.3)

is called the λN -semicircular (Banach-)∗-probability space, where τN is the linear functional (15.1.2) on the Banach ∗-algebra XN of (15.1.1). By definition, one obtains the following structure theorem of XN up to τN .

Free Probability on XN = (XN ⊗C ΛN , τN ) ■ 187

Theorem 15.1. Let XN = (XN , τN ) be the λN -semicircular ∗probability space. As a Banach ∗-algebra (15.1.1), XN N

Proof. Since XN = ⋆

j=1

∗−iso N

=

⋆

j=1



 C [{xj }] ⊗C ΛN .

(15.1.4)

Bj , with Bj = C [{xj }] for j = 1, . . . , N , by

(5.3), we have X N = X N ⊗C ΛN =



N

⋆ Bj

j=1



∗-iso N

⊗C ΛN = ⋆ (Bj ⊗C ΛN ) j=1

(e.g., [22,23,28,30]). So, the relation (15.1.4) holds.

■

By (15.1.4), it is easily verified that the λN -semicircular ∗probability space XN of (15.1.3) is generated by the generator set,

GN

  xj ∈ XN , ∀j = 1, . . . , N       k s k βN ∈ λN , ∀k = 0, ±1, . . . , ±N = x j ⊗ βN . and       for all s ∈ N

(15.1.5)

Notation. From below, if there are no confusions, we denote the k operators xsj ⊗ βN of the generator set GN of (15.1.5), simply by xs,k j .□ Lemma 15.2. If xs,k j ∈ GN be a generating operator of XN , then τN

 n  ̸= 0, xs,k = ωsn c sn j 2

(15.1.6)

if and only if −N ≤ kn ≤ N, and 1 ≤ j + kn ≤ N,  n  for all n ∈ N. Equivalently, τN xs,k = 0, if and only if the j condition of (15.1.6) does not hold.

188 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. For xs,k j ∈ GN ⊂ XN , observe that 

xs,k j

n

k = xsj ⊗ βN


n

sn,kn kn = xsn , j ⊗ βN = x j

in XN , satisfying that xsn,kn ̸= ON in XN , if and only if j −N ≤ kn ≤ N, for n ∈ N, by (5.2.5), where ON is the zero element of XN . If xsn,kn ̸= ON in XN , equivalently, if −N ≤ kn ≤ N , then j  


kn sn τN xsn,kn = φN βN xsn = φN xsn j j+kn = ωsn c 2 ̸= 0, j if and only if 1 ≤ j + kn ≤ N, by (5.3.1) and (5.3.2). So, the relation (15.1.6) holds.

■

By (15.1.6), we have the following generalized result. Theorem 15.3. Let J = (j1 , . . . , jn ) be an arbitrary n-tuple in {1, . . . , N } (without considering repetition on entries), for n ∈ N, and n Q xjsll ,kl ∈ XN , induced by generating operators xsjll ,kl ∈ GN , let WJ = l=1

for some sl ∈ N, and kl ∈ {0, ±1, . . . , ±N }, for all l = 1, . . . , n. Then ! n Y sl τN (WJ ) = φN xjl ̸= 0, (15.1.7) l=1

if and only if 1 ≤ jl +

n X

ki ≤ N, ∀l, p = 1, . . . , n,

i=p

 where φN

n Q l=1

xsjll

 ∈ C is characterized by (3.22), or (3.25). Equiva-

lently, τN (WJ ) = 0, if and only if the condition in (15.1.7) does not hold.

Free Probability on XN = (XN ⊗C ΛN , τN ) ■ 189

Proof. Let xjsll ,kl ∈ GN be generating operators of XN , for a fixed ntuple J = (j1 , . . . , jn ) in {1, . . . , N }. Then ! ! n n n Y Y Y kl WJ = xsjll ,kl = xsjll ⊗ βN , l=1

in XN , i.e.,

l=1

l=1

WJ = X[J] ⊗ T in XN ,

where

n Y X[J] = xsjll ∈ XN \ {0N } l=1

is in the sense of (3.7), and n Y kl T = βN in ΛN . l=1

Thus, we have τN (WJ ) = φN (T (X[J])) = φN (X[J]) ̸= 0, if and only if 1 ≤ jl +

n X

ki ≤ N, ∀l, p = 1, . . . , n,

i=p

by Corollary 5.4 and (15.1.6). So, the relation (15.1.7) holds true.

■

The above theorem characterizes the free-distributional data of free reduced words of the λN -semicircular ∗-probability space XN in the generator set GN , by (15.1.4). In the following section, we study freeprobabilistic information on XN more in detail.

15.2

FREE-DISTRIBUTIONAL DATA ON XN

As in Section 15.1, we fix N ∈ N∞ >1 , and the corresponding λN semicircular ∗-probability space XN = (XN , τN ) of (15.1.3). Let T =

N X l=−N

l tl βN ∈ ΛN

190 ■ Banach-Space Operators On C ∗ -Probability Spaces

be a shift operator of ΛN , and let def

xs,T = xsj ⊗ T in XN , j

(15.2.1)

for xj ∈ XN and s ∈ N. Recall that, by (15.2.1), the tensor factor T of xs,T ∈ XN is not only well-defined in ΛN , but also induced fully by j the restricted-shift family λN in the sense that: it is generated by all elements of λN . Lemma 15.4. If xs,T ∈ XN is a free random variable (15.2.1), then j  

τN xs,T j



N −j X




= ωs c 2s 

 tl  .

(15.2.2)

l=1−j

Proof. Let xs,T ∈ XN be in the sense of (15.2.1). Then j  

τN xs,T = φN T xsj j = φN

N X



! l tl βN


xsj

= φN 

N −j X

 tl xsj+l 

l=1−j

l=−N

by (5.3.4)

=

N −j X

tl φN xsj+l = φN


  N −j X

xsj  tl 

l=1−j

l=1−j

by (5.3.5) 
= ωs c 2s 

N −j X

 tl  ,

l=1−j

by the semicircularity of xj ∈ XN in XN . Therefore, the freedistributional data (15.2.2) is obtained. ■ By (15.2.2), one can get the following generalized result.

Free Probability on XN = (XN ⊗C ΛN , τN ) ■ 191

 Theorem 15.5. Let U =

n Q

l=1 XN ,

xsjll



⊗ T ∈ XN be a free random

variable, where xjl ∈ XN ⊂ for l = 1, . . . , n, for n ∈ N, and N P i T = ti βN ∈ ΛN is a shift operator (15.2.1). Then there exists i=−N

  1 ≤ jl + k i ≤ N   K = i ∈ {0, ±1, . . . , ±N } for all ,   l = 1, . . . , n such that τN (U ) =

φN

n Y

!!

where φN

n Q l=1

xsjll

ti

,

i∈K

l=1



! X

xsjll

(15.2.3)

 is characterized by (3.22), or (3.25). Clearly,

τN (U ) = 0, if and only if K = ¨ı¿œ, the empty set. Proof. Let U ∈ XN be given as above. Then ! n !! N X Y s i τN (U ) = τN ti βN xjll i=−N

=

N X

ti

φN

l=1 i βN

i=−N N X

=

!!!

n Y

xsjll

l=1

ti χi φN

n Y

! xsjll+i

i=−N

l=1

  1

if 1 ≤ jl + ki ≤ N for all l ∈ {1, . . . , n} otherwise,

where χi =

  0 and hence, it goes to =

N X i=−N

ti χi φN

n Y l=1

! xsjll

=

X i∈K

ti φN

n Y

! xsjll

,

l=1

by (5.3.6), where K is the subset (15.2.3) of {0, ±1, . . . , ±N }.

■

192 ■ Banach-Space Operators On C ∗ -Probability Spaces

More cases can/may be considered precisely with help of the main results of Section 5. However, the above theorem provides building blocks for computing the free-distributional data on XN , by the structure theorem (15.1.4) of XN .

15.3

FREE-HOMOMORPHIC RELATIONS ON {XN }N ∈N∞

>1

Let XN = (XN ⊗C ΛN , τN ) be the λN -semicircular ∗-probability space for N ∈ N∞ >1 . Here, we consider free-homomorphic relations among {XN }N ∈N∞ . >1

Theorem 15.6. Let N1 ≤ N2 ≤ · · · ≤ Nn in N∞ >1 , for n ∈ N. Then XN1

free−homo

−→

XN2

free−homo

−→

···

free−homo

−→

XNn .

(15.3.1)

Proof. For n ∈ N, and for N1 ≤ N2 ≤ · · · ≤ Nn ∈ N∞ >1 , we have XN1

free−homo

−→

XN2

free−homo

−→

···

free−homo

−→

XN n ,

by (5.4.1). Thus, by the structure theorem (15.1.4), the freehomomorphic relation (15.3.1) holds. More precisely, for any N1 ≤ N2 in N∞ >1 , with N2 − N1

denote

=

n0 ∈ N0 ,

one can define an embedding map, Ψ : XN1 → XN2 by a linear morphism, satisfying
[k+n ] k Ψ W ⊗ βN = W ⊗ βN2 0 in XN2 , 1 where

  0 [k + n0 ] = k + n0   k − n0

(15.3.2)

if k = 0 if k > 0 if k < 0,

in {0, ±1, . . . , ±N2 }, for all free reduced words W ∈ XN1 in the free semicircular family XN1 with their (reduced-)length-|W |, inducing

Free Probability on XN = (XN ⊗C ΛN , τN ) ■ 193 k W ⊗ βN ∈ XN1 . i.e., under Ψ, the above embedding assigns the tensor 1 factors,  k N1 βN1 k=−N of GN1 1

to the tensor factors, n o −N2 −N2 −n0 n0 +1 N2 0 βN , . . . , β , β , β , . . . , β of GN2 , N2 N2 N2 N2 2 while it assigns the other tensor factors, 

xs1 , . . . xsN1

s∈N

of GN1

to the other tensor factors, 

xs1 , . . . , xsN1

s∈N

of GN2 ,

where GN are the generator sets (15.1.5) of XN , for all N ∈ N∞ >1 . So, this embedding map Ψ is indeed a ∗-homomorphism, since Ψ



k1 xsj11 ⊗ βN 1

    k2 s1 s2 k1 k2 xsj22 ⊗ βN = Ψ x x ⊗ β β j1 j2 N1 N 1 [k +n ] [k +n ]

= xsj11 xsj22 ⊗ βN21 0 βN22 0    [k +n ] [k +n ] = xsj11 ⊗ βN21 0 xsj22 ⊗ βN22 0 by (15.3.2)     k1 s2 k2 = Ψ xsj11 ⊗ βN Ψ x ⊗ β j2 N1 , 1

(15.3.3)

and Ψ



k xsj ⊗ βN


∗ 


[−k+n0 ] −k = Ψ xsj ⊗ βN = xsj ⊗ βN ∗ 

∗ [k−n ] k = Ψ xsj ⊗ βN , = xsj ⊗ βN 0

by (15.3.2). Therefore, by (15.3.3), this embedding map Ψ is a ∗-homomorphism from XN1 into XN2 .

194 ■ Banach-Space Operators On C ∗ -Probability Spaces

Observe now that k τN2 Ψ xsj ⊗ βN 1





  [k+n ] = τN2 xsj ⊗ βN2 0   
 [k+n ] = φN2 βN2 0 xsj = φN2 xsj+[k+n0 ] ( ωs c 2s if 1 ≤ j + [k + n0 ] ≤ N 2 = 0 otherwise ( ωs c 2s if 1 ≤ j + k ≤ N1 = 0 otherwise

by (15.3.2)



k k , xsj = τN1 xsj ⊗ βN = φN1 xsj+k = φN1 βN 1 1 for all j ∈ {1, . . . , N1 }, k ∈ {0, ±1, . . . , ±N1 }, and s ∈ N. Therefore, this embedding ∗-homomorphism Ψ is a freehomomorphism. i.e., if N1 ≤ N2 in N∞ >1 , then XN1

free−homo

−→

XN2 .

Inductively, the chain relation (15.3.1) holds.

■

The above theorem also illustrates that the λ∞ -semicircular ∗probability space X∞ is a kind of the enveloping free-probabilistic structure of {XN }N ∈N∞ . >1

Corollary 15.7. For any N ∈ N∞ >1 , the λN -semicircular ∗-probability space XN is free-homomorphic to the λ∞ -semicircular ∗-probability space X∞ . □

CHAPTER

16

Operator-Theoretic Properties on XN

I

N THIS SECTION, WE FIX ARBITRARY N ∈ N∞ >1 , and the corresponding λN -semicircular ∗-probability space, XN = (XN ⊗C ΛN , τN ) .

And, we study operator-theoretic properties on XN as in Section 14.

16.1

OPERATOR-THEORETIC PROPERTIES ON ΛN

Let ΛN be the (restricted-)shift-operator algebra generated by the restricted-shift family λN . By definition, every shift operator T is expressed by (5.2.11). Instead of considering such general shift operators of ΛN , we restrict our interests to the shift operators T ∈ ΛN formed by N X l T = tl βN ∈ ΛN , with tl ∈ C. (16.1.1) l=−N

Let T ∈ ΛN be a shift operator (16.1.1). Then, by (5.2.12), its adjoint T ∗ is N X −k T∗ = tl βN in ΛN . (16.1.2) l=−N

DOI: 10.1201/9781003263487-16

195

196 ■ Banach-Space Operators On C ∗ -Probability Spaces

By (16.1.1), as in Section 14, one can decompose T by 0 T = T− + t0 βN + T+ in ΛN ,

(16.1.3)

with T+ =

N X

l tl βN , and T− =

l=1

N X

−l , t−l βN

l=1

in ΛN . So, by (16.1.2) and (16.1.3), the self-adjointness is characterized. 0 Proposition 16.1. Let T = T− + t0 βN + T+ ∈ ΛN be a shift operator (16.1.1) with its decomposition (16.1.3). Then T is self-adjoint in ΛN , if and only if t0 ∈ R, and T− = T+∗ ∈ ΛN , (16.1.4)

if and only if t0 ∈ R and t−l = tl in C, ∀l = 1, . . . , N. Proof. By (16.1.2), a shift operator T is self-adjoint in ΛN , if and only if −0 0 + T+∗ = T− + t0 βN + T+ , T−∗ + t0 βN if and only if t0 = t0 in C, and T− = T+∗ in ΛN , if and only if t0 ∈ R, and t−l = tl in C, ∀l = 1, . . . , N, by (16.1.3). So, the characterization (16.1.4) holds.

■

By (16.1.4), one obtains the following corollary. Corollary 16.2. A shift operator T ∈ ΛN of (16.1.1) is self-adjoint, if and only if N N X X −l 0 l tl βN + t0 βN + T = tl βN in ΛN , with t0 ∈ R. l=1

l=1

(16.1.5)

Operator-Theoretic Properties on XN ■ 197

Proof. Let T+ =

N P

l tl βN ∈ ΛN be in the sense of (16.1.3). Then the

l=1

operator T is of the form (16.1.5) in ΛN , if and only if 0 T = T+∗ + t0 βN + T+ , with t0 ∈ R,

if and only if T is self-adjoint in ΛN , by (16.1.4).

■

Observe that if a shift operator T ∈ ΛN is of the form (16.1.5), then 0 T 2 = T+∗2 + t0 T+∗ + T+∗ T+ + t0 T+∗ + t20 βN + t0 T+ + T+ T+∗ + t0 T+ + T+2 ,

and hence, this self-adjoint shift operator T is a projection in ΛN , if and only if t20 = t0 in R, (16.1.6) and T+∗2 + t0 T+∗ + T+∗ T+ + t0 T+∗ + t0 T+ + T+ T+∗ + t0 T+ + T+2 = T+∗ + T+ , in ΛN . By (16.1.6), one obtains the following projection-property on ΛN . Proposition 16.3. Let N < ∞ in N∞ >1 . Then the only nonzero 0 projection T ∈ ΛN of (16.1.1) is the identity operator βN in ΛN . ∞ Meanwhile, if N = ∞ in N>1 , then a shift operator T of (16.1.1) is a projection in ΛN , if and only if the condition (16.1.6) holds. Proof. By definition, a shift operator T of (16.1.1) is a projection in ΛN , if and only if it is both self-adjoint and idempotent. So, it is sufficient to consider the projection-property under self-adjointness (16.1.4). A self-adjoint shift operator T of (16.1.5) is a projection in ΛN , if and only if the condition (16.1.6) holds. So, if N = ∞, then T is a projection in ΛN , if and only if (16.1.6) holds. Suppose N < ∞. Then the condition (16.1.6) holds, if and only if the “finite-sum” shift operator T of (16.1.1) satisfy t0 = 0, or t0 = 1, in R, and T+ = 0ΛN = T+∗ in ΛN , 0 implying that T = 0ΛN , or T = βN in ΛN , where 0ΛN is the zero operator of ΛN . So, the only nonzero projection (16.1.1) of ΛN is the 0 identity operator βN . ■

198 ■ Banach-Space Operators On C ∗ -Probability Spaces

Consider that if T ∈ ΛN is in the sense of (16.1.1), then X −l1 l2 T ∗T = tl1 tl2 βN βN ,

(16.1.7)

(l1 ,l2 )∈{0,±1,...,±N }2 , −N ≤l1 +l2 ≤N

and X

TT∗ =

i1 −i2 βN , ti1 ti2 βN

(i1 ,i2 )∈{0,±1,...,±N }2 , −N ≤i+i2 ≤N

by (16.1.2). Proposition 16.4. Let T ∈ ΛN be a shift operator (16.1.1). k (16.1.8) If T = tβN with t ∈ C× = C \ {0}, then it is normal in ΛN , if and only if k = 0. (16.1.9) If T contains more than one summands, then it is normal in ΛN , if and only if it is of the form (16.1.5) in ΛN . n1 n2 n2 n1 Proof. Recall that βN βN ̸= βN βN in ΛN , whenever n1 ̸= n2 in {0, ±1, . . . , ±N }. k First, suppose a shift operator T = tβN has only one summand in × ΛN , for some t ∈ C . Then 2

2

−k k k −k T ∗ T = |t| βN βN , and T T ∗ = |t| βN βN ,

in ΛN . Note that

−k k k −k βN βN ̸= βN βN in ΛN ,

whenever k ̸= 0 in {0, ±1, . . . , ±N }. So, the equality can happen only k when k = 0. It shows that a shift operator T = tβN is normal in ΛN , if and only if k = 0. It proves the statement (16.1.8). Now, assume that T ∈ ΛN is a shift operator (16.1.1), containing more than one nonzero summands. If such a shift operator T is selfadjoint in ΛN , then it is normal, since T ∗ T = T 2 = T T ∗ in ΛN . Conversely, if T is not self-adjoint in ΛN (with multi nonzero summands), then T ∗ T ̸= T T ∗ in ΛN , by (16.1.7). i.e., if a shift operator T of (16.1.1), with multi nonzero summands, is not self-adjoint, then it is not normal in ΛN . i.e., if T has multi summands, then T is normal in ΛN , if and only if T is self-adjoint in ΛN , if and only if it is of the form (16.1.5), by the self-adjointness characterization (16.1.4). Therefore, the statement (16.1.9) holds. ■

Operator-Theoretic Properties on XN ■ 199

By the normality (16.1.8) and (16.1.9), one can obtain the following result. Proposition 16.5. Let T ∈ ΛN be a shift operator (16.1.1). k (16.1.10) If T = tβN ∈ ΛN , with t ∈ C× , then it is a unitary in ΛN , if and only if k = 0 in {0, ±1, . . . , ±N } , and t ∈ T, where T is the unit circle of C. (16.1.11) If T contains more than one nonzero summands, then it is a unitary in ΛN , if and only if it is self-adjoint in ΛN , satisfying X tl1 tl2 = 1, (l1 ,l2 )∈{0,1,...,N }2 , −l1 +l2 =0

and X

tl1 tl2 = 0.

(l1 ,l2 )∈{0,1,...,N }2 , −l1 +l2 ∈{±1,...,±N } k Proof. By the definition of unitaries, a shift operator T = tβN is a unitary in ΛN , if and only if 2

2

−k k k −k T ∗ T = |t| βN βN = I = |t| βN βN = T T ∗ ,

if and only if 2

k = 0, and |t| = 1, if and only if k = 0, and t ∈ T, in {0, ±1, . . . , ±N } , respectively, in C. It proves the statement (16.1.10). Now, suppose a shift operator T of (16.1.1) has multi nonzero summands. By definition, if such an operator T is a unitary in ΛN , then it is normal, and hence, by (16.1.9), it is self-adjoint in ΛN , i.e., it has its expression, T =

T+∗

+

0 t0 βN

N X l + T+ , with t0 ∈ R, T+ = tl βN , l=1

in ΛN , by (16.1.4) and (16.1.5).

200 ■ Banach-Space Operators On C ∗ -Probability Spaces

Thus, by (16.1.5) and (16.1.7), if T is as above, then X −l1 l2 tl1 tl2 βN βN , T ∗T = (l1 ,l2 )∈{0,1,...,N }2 , −N ≤−l1 +l2 ≤N

and TT∗ =

X

i1 −i2 ti1 ti2 βN βN ,

(i1 ,i2 )∈{0,1,...,N }2 , −N ≤i1 −i2 ≤N

Therefore, the above shift operator T is a unitary in ΛN , if and only if X tl1 tl2 = 1, (l1 ,l2 )∈{0,1,...,N }2 , −l1 +l2 =0

and X

tl1 tl2 = 0,

(l1 ,l2 )∈{0,1,...,N }2 , −l1 +l2 ∈{±1,...,±N }

in C. So, the statement (16.1.11) holds true.

■

The above proposition characterizes the unitarity of the shift operators (16.1.1) in the shift-operator algebra ΛN . Proposition 16.6. A shift operator T of (16.1.1) is an isometry in ΛN , if and only if X tl1 tl2 = 1, (16.1.12) (l1 ,l2 )∈{0,±1,...,±N }2 , l1 +l2 =0

and X

tl1 tl2 = 0,

(l1 ,l2 )∈{0,±1,...,±N }2 , l1 +l2 ∈{±1,...,±N }

in C. Proof. A shift operator T is an isometry in ΛN , if and only if T ∗ T = 1ΛN in ΛN , if and only if the condition (16.1.12) holds, by (16.1.7).

■

So, the above condition (16.1.12) characterizes the isometryproperty of the shift operators (16.1.1) of ΛN .

Operator-Theoretic Properties on XN ■ 201

16.2

OPERATOR-THEORETIC PROPERTIES OF XJS ⊗ T IN XN

Let XN be the λN -semicircular ∗-probability space for a fixed N ∈ N∞ >1 . If U = S ⊗ T ∈ XN is an operator with S ∈ XN , and T ∈ ΛN , then it is easily verified by definition that U satisfies (P), ⇐⇒ both S and T satisfy (P),

(16.2.1)

where (P) is one of our operator-theoretic properties: self-adjointness, projection-property, normality, isometry-property and unitarity because (S1 ⊗ T1 ) (S2 ⊗ T2 ) = S1 S2 ⊗ T1 T2 and ∗

(S ⊗ T ) = S ∗ ⊗ T ∗ , in XN . For example, since ∗

(S ⊗ T ) (S ⊗ T ) = S ∗ S ⊗ T ∗ T, and ∗

(S ⊗ T ) (S ⊗ T ) = SS ∗ ⊗ T T ∗ , in XN , this operator S ⊗ T is normal in XN , if and only if S is normal in XN , and T is normal in ΛN . Theorem 16.7. Let U = S ⊗ T ∈ XN , for S ∈ XN , and T ∈ ΛN . Then U satisfies an operator-theoretic property (P) in XN , if and only if S satisfies (P) in XN and T satisfies (P) in ΛN , where (P) is selfadjointness, or projection-property, or normality, or isometry-property, or unitarity. Proof. The proof is clear by the tensor product on Banach ∗-algebras. ■ Now, let def

xs,T = xsj ⊗ T ∈ XN , j

(16.2.2)

202 ■ Banach-Space Operators On C ∗ -Probability Spaces

where xj ∈ XN is a generating semicircular element of XN , for N P l j ∈ {1, . . . , N }, and s ∈ N0 = N ∪ {0}, and T = tl βN ∈ ΛN is l=−N

a shift operator (16.1.1). Remark that the tensor factors T ∈ ΛN of the operators xs,T ∈ XN j of (16.2.2) are assumed to be the shift operators formed by (16.1.1) in ΛN , not generally as (5.2.11). Corollary 16.8. An operator xs,T of (16.2.2) is self-adjoint in XN , j if and only if T is self-adjoint in ΛN , if and only if T is an operator (16.1.5) of ΛN . Proof. Every generating semicircular element xj ∈ XN is self-adjoint in XN , and hence, the operators xsj are self-adjoint in XN , for all s ∈ N. Clearly, x0j = 1N , the identity operator, is self-adjoint in XN . So, by (16.2.1), an operator xs,T is self-adjoint in XN , if and only if the shift j operator T of (16.1.1) is self-adjoint in ΛN . By the self-adjointness (16.1.4), the tensor factor T is expressed by (16.1.5) in ΛN . ■ By the above self-adjointness of xs,T in XN , one obtains the j following result. Corollary 16.9. An operator xs,T of (16.2.2) is a projection in XN , if j and only if s = 0, and T ∈ ΛN satisfies the condition (16.1.6). As special cases, if N < ∞, then the operator xs,T is a “non-zero” projection in j 0 XN , if and only if s = 0, and T = βN = 1ΛN in ΛN , if and only if xs,T = IN , the identity operator of XN . j Proof. By (16.2.1), an operator xs,T of (16.2.2) is a projection in XN , j if and only if xsj is a projection in XN , and a shift operator T is a projection in ΛN , if and only if s = 0 in N0 , and T satisfies the condition (16.1.6), if and only if xs,T = x0j ⊗ T = 1N ⊗ T, j in XN , where T ∈ ΛN satisfies (16.1.6).

Operator-Theoretic Properties on XN ■ 203

Suppose now that N < ∞. Then, by Proposition 16.3, a shift 0 operator T of (16.1.1) is a projection in ΛN , if and only if T = βN . So, in such a case, one has 0 xs,T = 1N ⊗ βN = 1N ⊗ 1ΛN = IN , j

in XN . So, if N < ∞, then the only nonzero finite-sum projection (16.2.2) is the identity operator IN of XN . ■ The above corollary characterizes the projection-property of the operators xs,T ∈ XN of (16.2.2). j Recall that, by (16.1.8) and (16.1.9), a nonzero shift operator 0 T ∈ ΛN of (16.1.1) is normal, if and only if either (i) T = tβN , for × t ∈ C , in ΛN , or (ii) T is self-adjoint in ΛN , whenever T contains multi nonzero summands. Corollary 16.10. A nonzero operator xs,T of (16.2.2) is normal in j XN , if and only if either 0 T = tβN ∈ ΛN , for t ∈ C× ,

(16.2.3)

or T =

T+∗

+

0 t0 βN

N X l + T+ with T+ = tl βN , l=1

in ΛN . Proof. Recall that xsj is self-adjoint, and hence, normal in XN , and a shift operator T is normal in ΛN , if and only if it satisfies either (16.1.8), or (16.1.9) in ΛN . Therefore, an operator xs,T of (16.2.2) is j normal in XN , if and only if T is normal in ΛN , if and only if T satisfies either (16.1.8), or (16.1.9). The first relation in (16.2.3) is obtained by (16.1.8), and the second relation in (16.2.3) is obtained by (16.1.9). ■ The above corollary characterizes the normality of xs,T ∈ XN of j (16.2.2). By Proposition 16.5, a shift operator T ∈ ΛN of (16.1.1) is a unitary, 0 if and only if either (i) T = tβN ∈ ΛN , with t ∈ T in C, by (16.1.10), or (ii) T satisfies the condition (16.1.11) in ΛN .

204 ■ Banach-Space Operators On C ∗ -Probability Spaces

Corollary 16.11. An operator xs,T of (16.2.2) is a unitary in XN , if j and only if it satisfies either 0 s = 0, and T = tβN ∈ ΛN with t ∈ T ⊂ C,

(16.2.4)

or X

s = 0, and (l1 ,l2

)∈{0,1,...,N }2 ,

tl1 tl2 = 1, −l1 +l2 =0

and X (l1 ,l2

)∈{0,1,...,N }2 ,

tl1 tl2 = 0.

−l1 +l2 ∈{±1,...,±N }

xs,T j

of (16.2.2) is a unitary in XN , if and only if

∗ 2s ∗ x2s in XN , j ⊗ T T = IN = xj ⊗ T T

Proof. An operator

if and only if x2s j = 1XN , for s ∈ N0 , and T ∈ ΛN is a unitary, by (16.2.1), if and only if s = 0 in N0 , and a shift operator T of (16.1.1) satisfies either (16.1.10), or (16.1.11). Therefore an operator xs,T of (16.2.2) is a unitary in XN , if and j only if the condition (16.2.4) holds. ■ The above corollary characterizes the unitarity of the operators xs,T j of (16.2.2) in XN . Recall that a shift operator T ∈ ΛN of (16.1.1) is an isometry, if and only if it satisfies the condition (16.1.12), by Proposition 16.6. Corollary 16.11. An operator xs,T of (16.2.2) is an isometry, if and j only if s = 0 in N0 , and the tensor factor T satisfies the condition (16.1.12) in ΛN . Proof. It is not hard to check that xsj is an isometry in XN , if and only if s = 0 in N0 . So, an operator xs,T of (16.2.2) is an isometry in XN , j if and only if xsj is an isometry in XN , and T is an isometry in ΛN , by (16.2.1), if and only if s = 0 in N0 , and a shift operator T satisfies the condition (16.1.12) in ΛN . ■ The above corollary characterizes the isometry-property of the operators xs,T of (16.2.2) in the λN -semicircular ∗-probability space XN . j

Operator-Theoretic Properties on XN ■ 205

16.3

OPERATOR-THEORETIC PROPERTIES OF W ⊗ T IN XN

In this section, we concentrate on studying operator-theoretic properties of operators U of the λN -semicircular ∗-probability space XN , where U = W ⊗ T ∈ XN , (16.3.1) with W =

n Y

xsjll ∈ XN , and T =

i=1

N X

l tl βN ∈ ΛN .

l=−N

In particular, for convenience, we assume W ∈ XN is a free reduced N word with its length-n in the free semicircular family XN = {xj }j=1 , for n ∈ N, in (16.3.1). Consider that if W ∈ XN is a free reduced word with its length-n in XN , then n Y sjn−l+1 W∗ = xjn−l+1 ∈ XN , (16.3.2) l=1

as a free reduced word with its length-n in XN . Definitely, one can check that W ∗ ̸= W in XN , in general, and hence, the operator U of (16.3.1) is not self-adjoint in XN , in general. Theorem 16.12. An operator U = W ⊗ T of (16.3.1) is self-adjoint in XN , if and only if j1 = jn , j2 = jn−1 , . . . , j[ n2 ]−1 = j[ n2 ]+1 , and s1 = sn−1 , s2 = sn−1 , . . . , s[ n2 ]−1 = s[ n2 ]+1 ,

(16.3.3)

and N N X X −l 0 l T = tl βN + t0 βN + tl βN ∈ ΛN with t0 ∈ R, l=1

where

n 2

l=1

is the maximal integer less than, or equal to

n 2,

for n ∈ N.

206 ■ Banach-Space Operators On C ∗ -Probability Spaces

Proof. The tensor factor W ∈ XN of U ∈ XN is self-adjoint, if and only if    j1 = jn , j2 = jn−1 , . . . , j[ n2 ]−1 = j[ n2 ]+1  and   s1 = sn , s2 = sn−1 , . . . , s[ n2 ]−1 = s[ n2 ]+1 . Indeed, if the above condition holds in {1, . . . , N }, respectively, in N, then W ∗ = W in XN , moreover, if the above condition does not hold, then W ∗ ̸= W in XN , by (16.3.2). The self-adjointness of the other tensor factor T ∈ ΛN is characterized by (16.1.4), implying the operator-equality (16.1.5). Therefore, the operator U of (16.3.1) is self-adjoint in XN , if and only if the condition (16.3.3) holds, by (16.2.1). ■ For example, if W = x41 x22 x41 ∈ X5 is a free reduced word with its length-3, satisfying the conditions in (16.3.3), then, W ∗ = x41 x22 x41 = W in XN . The above theorem characterizes the self-adjointness of an operator U ∈ XN of (16.3.1) by (16.3.3). Theorem 16.13. An operator U of (16.3.1) is not a projection in XN . Proof. If an operator U ∈ XN were a projection, then it is self-adjoint in XN , in the sense of (16.3.3). So, let such an operator U is selfadjoint in XN . To satisfy the projection-property, it should satisfy the idempotence: U 2 = W 2 ⊗ T 2 = W ⊗ T = U, in XN , where a shift operator T ∈ ΛN satisfies the condition (16.1.6), as a tensor factor of U . However, one can verify that W 2 ̸= W in XN , for all free reduced words W ∈ XN in the free semicircular family XN . In other words, the tensor factor W ∈ XN cannot be a projection. It implies that an operator U of (16.3.1) is not a projection in XN . ■

Operator-Theoretic Properties on XN ■ 207



n Q

Let U = W ⊗ T =

i=1

xsjii



!

N P

⊗

l tl βN

be an operator

l=−N

(16.3.1) in XN , with its adjoint, ∗

∗

∗

U =W ⊗T =

n Y

! sn−i+1 xjn−i+1

⊗

i=1

N X

! −l tl βN

,

l=−N

in XN , where W ∈ XN is a free reduced word with its length-n, and T ∈ ΛN is a shift operator (10.1.1). Then U ∗ U = W ∗ W ⊗ T ∗ T,

(16.3.4)

in XN , with W ∗W =

n Y

! s

n−i+1 xjn−i+1

i=1

n Y

! xsjii

,

i=1

a free (reduced, or non-reduced) word of XN in XN , where T ∗ T ∈ ΛN is in the sense of (16.1.7). Similarly, U U ∗ = W W ∗ ⊗ T T ∗,

(16.3.5)

in XN , with ∗

WW =

n Y

! xsjii

i=1

n Y

! sn−i+1 xjn−i+1

,

i=1

a free (reduced, or non-reduced) word of XN in XN , where T T ∗ ∈ ΛN is in the sense of (16.1.7). If W is a free reduced word W =

n Y

xsjii ∈ XN , with n > 1,

i=1

of XN with its length-n in XN , then W ∗ W = xsjnn . . . xsj22 xsj11 xsj11 xsj22 . . . xsjnn =

xsjnn

1 s2 . . . xsj22 x2s j1 xj2

. . . xsjnn

(as a free non-reduced word) (as a free reduced word) (16.3.6)

208 ■ Banach-Space Operators On C ∗ -Probability Spaces

and W W ∗ = xsj11 xsj22 . . . xsjnn xsjnn . . . xsj22 xsj11 =

xsj11

sn−1 2sn sn−1 . . . xjn−1 xjn xjn−1

. . . xsj11

(as a free non-reduced word) (as a free reduced word),

in XN , by (16.3.4) and (16.3.5). So, for n ∈ N, it is verified that W ∗ W = W W ∗ in XN ,

(16.3.7)

if and only if    j1 = jn , j2 = jn−1 , . . . , j[ n2 ]−1 = j[ n2 ]+1  and   s1 = sn , s2 = sn−1 , . . . , s[ n2 ]−1 = s[ n2 ]+1 , in {1, . . . , N }, respectively, in N, by (16.3.6). Notice that the above normality (16.3.7) of W ∈ XN is identified with the self-adjointness of W in XN , contained in the condition (10.3.3). Theorem 16.16. An operator U = W ⊗ T of (16.3.1) is normal in XN , if and only if W is self-adjoint in XN , and T is normal in ΛN , if and only if    j1 = jn , j2 = jn−1 , . . . , j[ n2 ]−1 = j[ n2 ]+1  and ,   s1 = sn , s2 = sn−1 , . . . , s[ n2 ]−1 = s[ n2 ]+1 ,

(16.3.8)

and T ∈ ΛN satisfies either (16.1.8), or (16.1.9). Proof. By (16.2.1), a given operator U is normal in XN , if and only if W is normal in XN , and T is normal in ΛN . However, by (16.3.7), a free reduced word W is normal in XN , if and only if it is self-adjoint in XN . Thus, the operator U is normal in XN , if and only if W is self-adjoint in XN , and T is normal in ΛN . Therefore, the condition (16.3.8) characterizes the normality of U in XN . ■

Operator-Theoretic Properties on XN ■ 209

The above theorem characterizes the normality of the operators U of (16.3.1) in XN , by (16.3.8). Observe now that if U = W ⊗ T ∈ XN is in the sense of (16.3.1), then U ∗ U = IN in XN , (16.3.9) if and only if W ∗ W = 1N , and T ∗ T = 1ΛN , in XN , respectively, in ΛN , by (16.2.1). By Proposition 16.6, the isometry-property is characterized by (16.1.12). So, the above isometry-property (16.3.9) for U ∈ XN can be re-stated by that: U is an isometry in XN , if and only if W ∗ W = 1N , and T satisfies (16.1.12), by (16.2.1). However, it is not hard to check that if W ∈ XN is a free reduced word in XN , then, by (16.3.6), W ∗ W ̸= 1N in XN .

(16.3.10)

Theorem 16.17. An operator U of (16.3.1) is not an isometry in XN . Proof. By (16.3.9), an operator U were an isometry in XN , if and only if W is an isometry in XN , and T ∈ ΛN satisfies (16.1.12). However, by (16.3.10), all free reduced words of XN in XN cannot be isometries in XN . It implies that there does not exist an isometry formed by U ∈ XN of (16.3.1), satisfying (16.3.9). ■ By the above theorem, one immediately obtains the following unitarity. Theorem 16.18. An operator U of (16.3.1) is not a unitary in XN . Proof. By definition, every unitary is an isometry. Since an operator U of (16.3.1) is not an isometry by Theorem 10.17, it is automatically not a unitary in XN . ■

CHAPTER

17

Summary

W

E FINISH THIS MONOGRAPH WITH THIS ADDITIONAL SECTION, summarizing our main results. Whenever a C ∗ -probability space (A, φ) contains a free semicircular family X of mutually free, |Z|-many semicircular elements {xj }j∈Z , one can construct the corresponding C ∗ -probability space Xφ = (X, φ) generated by X, as a free-probabilistic sub-structure of (A, φ). The free-distributional data on Xφ are characterized by the combinatorial computations (3.22) and (3.25), up to Catalan numbers. By defining a trivial  k shifting processes on Z, one can define the integer shifts λ = β k∈Z , which are free-isomorphisms on Xφ by (4.19), and the set λ of all such integer shifts forms a subgroup of the automorphism group Aut (Xφ ), which is group-isomorphic to the infinite cyclic abelian group (Z, +) by (4.21). So, with a canonical action α of the group λ acting on Xφ , one obtains the discrete group dynamical system, Γ = (Xφ , λ, α) , and it induces the crossed product Banach ∗-probability space, X = (Xφ ⋊α λ, τ ) , whose free-distributional data are dictated not only by those on Xφ , but also by those on the integer-shift-operator algebra Λ, the group Banach ∗-algebra of λ, which are characterized by (12.3.5). Meanwhile, if there is a C ∗ -probability space (B, ψ) generated by a N free semicircular family {yj }j=1 , for N ∈ N∞ >1 , then it is free-isomorphic to the free-probabilistic sub-structure, XN = (XN , φN ) of Xφ generated N by the free semicircular family XN = {xj }j=1 , which is a free subfamily of the free semicircular family X, generating Xφ . i.e., such a DOI: 10.1201/9781003263487-17

211

212 ■ Banach-Space Operators On C ∗ -Probability Spaces

C ∗ -probability space (B, ψ) is free-isomorphic to XN , whenever the unity 1B of B and the unity 1Xφ of
Xφ have the same free distributions in the sense that: ψ (1B ) = φ 1Xφ by (5.4). By construction, canonically, one can restrict the action α of the integer-shift group λ (on Xφ ) to the action αN of the subset  k N λN = βN of the homomorphism semigroup Hom (XN ), acting k=−N k on XN , where βN = β k |XN , for N ∈ N∞ >1 . Unfortunately, this set λN does not form a suitable algebraic (sub-)structure (of Hom (XN )), but it generates a well-defined Banach ∗-algebra ΛN , the restricted-shiftoperator algebra, in the operator space B (XN ) acting on a Banach space XN equipped with its C ∗ -norm. The (restricted-)shift operators of ΛN “partially” preserve the free probability on XN , by (13.1.7). Since λN does not form an algebraic structure, we cannot have a “nice” dynamical system induced by it (in the natural settings), but, under the usual tensor product of Banach ∗-algebras, the Banach ∗-probability space, XN = (XN ⊗C ΛN , τN )

is well-defined whose free probability is characterized both by that on XN , and the action of ΛN , for N ∈ N∞ >1 . i.e., even though we cannot have nice dynamical systems for {λN }N ∈N∞ , the tensor>1 product Banach ∗-probability spaces {XN }N ∈N∞ have ΛN -depending >1 (and hence, λN -depending) free probabilities, explained by (13.2.3). Independent from free probability, the operator-theoretic properties of free random variables of X, and those of {XN }N ∈N∞ were considered >1 as adjointable Banach-space operators acting on Xφ , respectively, on {XN }N ∈N∞ . In particular, the characterizations of self-adjointness, >1 projection-property, normality, isometry-property, and unitarity of generating free random variables are provided. Acknowledgment. The author specially thanks Prof. P. Jorgensen, Prof. D. Alpay, Prof. W.Y. Lee, Prof. H. Dutta, and Prof. T. Gillespie, for valuable discussions and suggestions. □

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Index action 7 adjoint 6 adjointable 58 automorphism group 7

group-dynamical system 7 group-inverse 8

Banach *-algebra 5, 7 Banach-space operator 57 Banach *-probability space 5 block 9

integer-shift 32 integer-shift group 32 integer-shift operator 33 integer-shift operator algebra 33 isometry 158

Catalan number 2 circular element 105, 110 the circular law 105 conditional-nilpotent 88 C*-algebra 5 C*-probability space 5 crossed product 7

homomorphism semigroup 41

joint joint joint joint

free cumulant 6 free distribution 9 free moment 6 partition 15

dynamical system 7

Kronecker delta 7

free 9 free cumulant 6 free cumulant sequence 6 free distribution 7 free-homomorphism 25 free-isomorphism 25 free moment 6 free moment sequence 6 freeness 5 free Poisson distribution 120 free Poisson element 120 free probability 5 free random variable 5

lattice 9 linear functional 6

group 7 group-action 7

measure space 5 measure theory 5 M¨ obius inversion 6, 11 noncrossing partition 9 normal 158 operator algebra 5 operator sum 171 p-adic number field 2 partial isometry 70 projection 58, 158 217

218 ■ Index

restricted action 41 restricted-shift 43 family 43 operator 47 operator algebra 47

topological *-algebra 5, 6 topological *-probability space 5, 6 unitary 158 von Nuemann algebra 5

self-adjoint 6, 158 semicircular element 6 the semicircular law 7 *-isomorphism 7

w*-probability space 5 weighted semicircular element 113 Zorn’s lemma 86