Classical Clifford Algebras: Operator-Algebraic and Free-Probabilistic Approaches [1 ed.] 9781032637112, 9781032637143, 9781032637211

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Table of Contents
Chapter 1 Motivation: On the Second Clifford Algebra C[sub(2)]=H
1 Introduction of Chapter 1
2 On the Quaternions H
2.1 The Quaternions H
2.2 The Canonical Representation (C[sup(2)],π) of H
3 Spectral Analysis on H Under (C[sup(2)],π)
3.1 Quaternion-Spectral Forms
3.2 Similarity on q-Spectral Forms in H[sub(2)]
3.3 Quaternion-Spectral Equivalence
3.4 Quaternion-Spectral Mapping Theorem
3.5 The Quaternion-Spectralization σ
4 On Noncommutative Field H Under (C[sup(2)],π)
4.1 On the Noncommutative Field H
4.2 On the Unital Subgroup H[sup(×)] [sub(1)] of H[sup(×)]
4.3 Operator-Theoretic Spectral Properties of H[sup(×)]
5 Free-Probabilistic Data Induced by H
5.1 Free Probability
5.2 The C[sup(*)]-Algebra A[sub(2)] Generated by H[sup(×)] under (C[sup(2),π)
5.3 On the C[sup(*)]-Probability Spaces (A[sub(2)],tr) and (A[sub(2)],τ)
References
Chapter 2 On Classical Clifford Algebras
6 Introduction of Chapter 2
7 Classical Clifford Algebras
7.1 The Classical Clifford Algebras {C[sub(k)]}[sub(k∈N)]
7.2 The Clifford Algebra C
7.3 The Group-Hilbert Space H of the Clifford Group G
8 Free Probability on the Clifford-Group C[sup(*)]-Probability Space (M[sub(G)],τ)
9 Free Probability on Certain Sub-structures of (M[sub(G)],τ)
References
Chapter 3 The Clifford Group G and the Semicircular Law
10 Introduction of Chapter 3
11 On the Tensor-Product C[sup(*)]-Probability-Spaces (M[sub(g)] ⊗ A, τ ⊗ φ))
12 Deformed Semicircular Laws on (C ⊗ A, τ ⊗ φ))
13 Applications
References
Chapter 4 Representations of the Clifford Algebra C
14 Introduction of Chapter 4
15 The Clifford Algebra C Embedded in the CG C[sup(*)]-Algebra M[sub(G)]
16 On the R-Banach *-Algebra C
17 R-Adjointable-Operator-Theoretic Properties on C
References
Index

Citation preview

Classical Clifford Algebras Classical Clifford Algebras: Operator-Algebraic and Free-Probabilistic Approaches offers novel insights through operator-algebraic and freeprobabilistic models. By employing these innovative methods, the author sheds new light on the intrinsic connections between Clifford algebras and various mathematical domains. This monograph should be an essential addition to the library of any researchers interested in Clifford algebras or algebraic geometry more widely. Features • Includes multiple examples and applications • Is suitable for postgraduates and researchers working in algebraic geometry • Takes an innovative approach to a well-established topic Ilwoo Cho is currently a professor at St. Ambrose University, Iowa. He earned his PhD in Mathematics from the University of Iowa in 2005 and his master’s degree from Sungkyunkwan University in 1999. His research interests include free probability, operator algebra and theory, combinatorics, and groupoid dynamical systems.

Monographs and Research Notes in Mathematics Series Editors, John A. Burns, Thomas J. Tucker, Miklos Bona, and Michael Ruzhansky 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 appealing not only to the mathematical community, but also to 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. Inverse Scattering Problems and Their Applications to Nonlinear Integrable Equations, Second Edition Pham Loi Vu Generalized Notions of Continued Fractions: Ergodicity and Number Theoretic Applications Juan Fern´andez S´anchez, Jer´onimo L´opez-Salazar Codes, Juan B. Seoane Sep´ ulveda, Wolfgang Trutschnig Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals Ronald B. Guenther and John W. Lee Direct and Projective Limits of Geometric Banach Structures Patrick Cabau and Fernand Pelletier Classical Clifford Algebras: Operator-Algebraic and Free-Probabilistic Approaches Ilwoo Cho

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Classical Clifford Algebras Operator-Algebraic and Free-Probabilistic Approaches

Ilwoo Cho

Designed cover image: ©ShutterStock Images First edition published 2024 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 CRC Press is an imprint of Taylor & Francis Group, LLC © 2024 Ilwoo Cho 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: Classical Clifford algebras : operator-algebraic and free-probabilistic approaches / Ilwoo Cho. Description: First edition. | Boca Raton : C&H, CRC Press, 2023. | Series: Chapman and Hall/CRC monographs and research notes in mathematics | Includes bibliographical references and index. | Identifiers: LCCN 2023046152 (print) | LCCN 2023046153 (ebook) | ISBN 9781032637112 (hbk) | ISBN 9781032637143 (pbk) | ISBN 9781032637211 (ebk) Subjects: LCSH: Clifford algebras. | Geometry, Algebraic. Classification: LCC QA199 .C46 2023 (print) | LCC QA199 (ebook) | DDC 512/.57–dc23/eng/20231031 LC record available at https://lccn.loc.gov/2023046152 LC ebook record available at https://lccn.loc.gov/2023046153 ISBN: 9781032637112 (hbk) ISBN: 9781032637143 (pbk) ISBN: 9781032637211 (ebk) DOI: 10.1201/9781032637211 Typeset in Minion by codeMantra

To Okhwa, Jeanelle, and Alika

Contents Chapter 1 ■ Motivation: On the Second Clifford Algebra C2 =H

1

1

INTRODUCTION OF CHAPTER 1

2

2

ON THE QUATERNIONS H

3

2.1 2.2 3

4

5

The Quaternions H

3 2

The Canonical Representation (C , π) of H

5

SPECTRAL ANALYSIS ON H UNDER (C2 , π)

7

3.1

Quaternion-Spectral Forms

7

3.2

Similarity on q-Spectral Forms in H2

9

3.3

Quaternion-Spectral Equivalence

12

3.4

Quaternion-Spectral Mapping Theorem

13

3.5

The Quaternion-Spectralization σ

16 2

ON NONCOMMUTATIVE FIELD H UNDER (C , π)

17

4.1

17

On the Noncommutative Field H H× 1

×

4.2

On the Unital Subgroup

of H

21

4.3

Operator-Theoretic Spectral Properties of H×

23

FREE-PROBABILISTIC DATA INDUCED BY H 5.1 5.2 5.3

26

Free Probability ∗

The C -Algebra A2 Generated by H ∗

27 ×

2

under (C , π) 28

On the C -Probability Spaces (A2 , tr) and (A2 , τ )

REFERENCES

29 35

vii

viii ■ Contents

Chapter 2 ■ On Classical Clifford Algebras

37

6

INTRODUCTION OF CHAPTER 2

37

7

CLASSICAL CLIFFORD ALGEBRAS

38

8 9

7.1

The Classical Clifford Algebras {Ck }k∈N

39

7.2

The Clifford Algebra C

45

7.3

The Group-Hilbert Space H of the Clifford Group G

50

FREE PROBABILITY ON THE CLIFFORD-GROUP C ∗ -PROBABILITY SPACE (MG , τ )

58

FREE PROBABILITY ON CERTAIN SUB-STRUCTURES OF (MG , τ )

73

REFERENCES

77

Chapter 3 ■ The Clifford Group G and the Semicircular Law 79 10 11

INTRODUCTION OF CHAPTER 3

80

∗

ON THE TENSOR-PRODUCT C -PROBABILITY -SPACES (MG ⊗ A, τ ⊗ φ)

81

12

DEFORMED SEMICIRCULAR LAWS ON (C ⊗ A, τ ⊗ φ)

87

13

APPLICATIONS

95

REFERENCES

Chapter 4 ■ Representations of the Clifford Algebra C

102

105

14

INTRODUCTION OF CHAPTER 4

105

15

THE CLIFFORD ALGEBRA C EMBEDDED IN THE CG C ∗ -ALGEBRA MG

107

16

ON THE R-BANACH ∗-ALGEBRA C

113

17

R-ADJOINTABLE-OPERATOR-THEORETIC PROPERTIES ON C

117

REFERENCES

Index

124

125

CHAPTER

1

Motivation: On the Second Clifford Algebra C2 =H

n this first chapter, we study the fundamental algebraic, analytic, operator-theoretic, and free-probabilistic properties on the quaternions H, under a suitable canonical Hilbert-space representation of H. The key idea for
the study is the construction of the Hilbert-space representation C2 , π of the quaternions H, allowing us to understand each quaternion q ∈ H as a (2 × 2)-matrix over the complex field C. Those properties provide motivations and technical approaches for studying algebra, analysis, operator theory, and free probability on classical Clifford algebras {Ck+1 }k∈N in the following text, because the quaternions H is isomorphic to the second Clifford algebra C2 . From below, we will understand each n-th Clifford algebra as a C ∗ algebra over the real field R (or, in short, R-C ∗ -algebra), for n ∈ N. For instance, the matricial algebra Mk (R) of all (k × k)-matrices whose entries are R-quantities forms a R-C ∗ -algebra, for k ∈ N. Indeed, it is a vector space over R (or a R-vector space), since

I

r1 A1 + r2 A2 ∈ Mk (R), ∀r1 , r2 ∈ R and A1 , A2 ∈ Mk (R), having a well-defined matricial multiplication,

DOI: 10.1201/9781032637211-1

1

2 ■ Classical Clifford Algebras



" k #   X [aij ]k×k [bij ]k×k = ail blj l=1

, k×k

and its R-adjoint (∗) on Mk (R), which is the transpose of matrices, ∗

[aij ]k×k = [aji ]k×k , in Mk (R), satisfying ∗

A∗∗ = A, (rA) = rA∗ , for all r ∈ R, and A ∈ Mk (R), and ∗

∗

(A1 + A2 ) = A∗1 + A∗2 , (A1 A2 ) = A∗2 A∗1 , for all A1 , A2 ∈ Mk (R). Clearly,
this ∗-algebra Mk (R) is ∗-isomorphic to the operator ∗-algebra B Rk of all linear transformations on the k-dimensional R-Hilbert space Rk , equipped with the operator norm,
∥T ∥ = sup {∥T v∥k : ∥v∥k = 1} , ∀T ∈ B Rk , where ∥v∥k =

√

v • v, ∀v ∈ Rk ,

where (•) is the usual dot product. (Note that, in fact, all norms are equivalent from each other, e.g., [7] and [8].) The continuity or the
k boundedness (under linearity) of all elements of B R is guaranteed by the finite dimensionality
of Rk . Furthermore, the above operator
norm is complete on B Rk , by the finite dimensionality of B Rk = Mk (R). So, indeed, the matricial algebra Mk (R) is a well-defined RC ∗ -algebra, for k ∈ N. From the construction of our Clifford algebras {Ck }k∈N , they form complete ∗-subalgebras of {M2k (R)}k∈R , and hence, they are R-C ∗ algebras.

1

INTRODUCTION OF CHAPTER 1

In this chapter, we study algebra, analysis, operator theory, and free probability on the quaternions H, under a canonical representation
C2 , π of H, where C2 is the two-dimensional Hilbert space and π is the algebra action of H acting on C2 , making us understand every quaternion q ∈ H as a (2 × 2)-matrix of M2 (C), the (2 × 2)-matricial

Motivation: On the Second Clifford Algebra C2 =H ■ 3

algebra over C, which is ∗-isomorphic to the operator C ∗ -algebra
B C2 consisting of all (bounded linear) operators on C2 . Since the quaternions H is ∗-isomorphic to the classical second Clifford algebra C2 (see Chapter 2), the main results of this chapter not only provide the motivations for studying algebra, analysis, representation theory, operator theory, and free probability on the classical Clifford algebras {Ck+1 }k∈N but also give algebraic and analytic technics for such extended theories of Chapters 2–4 later. The quaternions H is an interesting object not only in pure mathematics (e.g., [9], [10], and [16]) but also in applied mathematics (e.g., [3], [18], and [19]). Algebra on H is considered in Ref. [21], analysis on H is studied in Refs. [11] and [17], and physics on H is investigated in Ref. [6]. Also, the matrices over the quaternions H have been studied (e.g., [4], [5], [13], and [17]), and the eigenvalue problems on such matrices form a branch of linear or multilinear algebra (e.g., [12], [14], and [15]). Note that our spectral analysis of Ref. [2] is done “over C,” which is different from the quaternion-eigenvalue problems of quaternion matrices mentioned here. In Refs. [1] and [2], we studied the maximal noncommutative multiplicative group H× = H \ {0}, its subgroup H× 1 of all unital quaternions, its normal subgroup R× of all nonzero real numbers, and the corresponding operators acting on separable Hilbert
spaces under the extension of the canonical representation C2 , π under “direct product.” Not only operator-theoretic spectral properties (normality, self-adjointness, unitarity, and projection property) but also certain natural free-distributional data of such operators are characterized by the quaternion moduli and the spectral values of the quaternions.

2

ON THE QUATERNIONS H


In this section, we consider a representation C2 , π of the quaternions H, introduced in Ref. [21] (also see Refs. [1], [2], [15], and [21]). 2.1

The Quaternions H

Let a and b be complex numbers, a = x + yi and b = u + vi in C,

4 ■ Classical Clifford Algebras

√ where x, y, u, v ∈ R, and i = −1 in C. From such a, b ∈ C, the corresponding quaternion q ∈ H is canonically constructed by q = a + bj = (x + yi) + (u + vi)j = x + yi + uj + vij = x + yi + uj + vk in H, satisfying

(2.1.1) i2 = j 2 = k 2 = ijk = −1.

The set H has a well-defined addition (+) and multiplication (·); for any ql = al + bl j ∈ H, with al , bl ∈ C, (in the sense of Eq. 2.1.1) for l = 1, 2, one has q1 + q2 = (a1 + a2 ) + (b1 + b2 )j and

(2.1.2) q1 q2 = (a1 a2 − b1 b2 ) + (a1 b2 + a2 b1 )j

in H, where z is the conjugate of z ∈ C. By Eq. 2.1.2, H is noncommutative under its multiplication, i.e., q1 q2 ̸= q2 q1 in H, in general.

(2.1.3)

Under the operations of Eq. 2.1.2, the quaternions H forms a “noncommutative field” (according to Ref. [21]) as a ring. A noncommutative field (F, +, ·) is an algebraic structure satisfying that (i) the algebraic pair (F, +) is an abelian (or a commutative group); (ii) the pair (F × , ·) forms a “noncommutative” group, where F × = F \ {0F }, where 0F is the (+)-identity of (F, +); and (iii) (+) and (·) are left distributive and right distributive. If q ∈ H is a quaternion (Eq. 2.1.1), then one can define the quaternion conjugate q ∈ H by q = x − yi − ui − vi.

(2.1.4)

So, one has that 2

2

qq = qq = |a| + |b| = x2 + y 2 + u2 + v 2

(2.1.5)

Motivation: On the Second Clifford Algebra C2 =H ■ 5

by Eq. 2.1.4, satisfying that qq = qq ≥ 0 in R ⊂ H, ∀q ∈ H,

(2.1.6)

by Eq. 2.1.5. This non-negativity (Eq. 2.1.6) allows us to define the quaternion modulus ∥.∥ on H by √ ∥q∥ = qq, for all q ∈ H. (2.1.7) This quaternion modulus ∥.∥ of Eq. 2.1.7 is a well-defined norm on H. If q ̸= 0 in H, then the quaternion reciprocal (or the inverse) q −1 of q,     a q −1 = |a|2 +|b| + |a|2−b j, (2.1.8) 2 2 +|b| satisfying q −1 q = 1 = 1 + 0i + 0j + 0k = qq −1 , is well defined in H, by Eqs. 2.1.5 and 2.1.7. 2.2

The Canonical Representation (C2 , π) of H

As in Eq. 2.1.1, let’s understand each quaternion q ∈ H as q = a + bj in H with a, b ∈ C, where a = x + yi and b = u + vi in C. Define an injective representation, π : H → M2 (C), by

(2.2.1)  π(q) = π (a + bj) =

a −b b a

 ,

where M2 (C) is the matricial algebra of all (2
× 2)-matrices over C (understood to be the operator algebra B C2 of all bounded linear transformations on C2 ). Indeed, this morphism π of Eq. 2.2.1 is a representation of the noncommutative field H, because π (q1 + q2 ) = π(q1 ) + π(q2 )

6 ■ Classical Clifford Algebras

and

(2.2.2) π (q1 q2 ) = π(q1 )π(q2 ),

for all q1 , q2 ∈ H, by Eqs. 2.1.2 and 2.2.1. Then, the quaternion conjugate q of q ∈ H satisfies that   a b π (q) = π (a − bj) = −b a (2.2.3)  ∗ a −b = = π(q)∗ b a in M2 (C) by Eq. 2.1.4, where A∗ is the
conjugate transpose of A ∈ M2 (C), which is the adjoint on B C2 = M2 (C). Furthermore,   a −b 2 2 det (π(q)) = det = |a| + |b| , b a implying that ∥q∥ =

p

det (π(q)), for all q ∈ H,

(2.2.4)

by Eqs. 2.1.6 and 2.1.7. Proposition 1. Let π be
in the sense of Eq. 2.2.1. Then C2 , π is a representation of H.

(2.2.5)

Proof. The proof is done by Eqs. 2.2.2 and 2.2.3. If we understand C2 as an inner-product space (over C) equipped with the inner product, ⟨(t1 , t2 ), (s1 , s2 )⟩ = t1 s1 + t2 s2 ,
then the pair C2 , π is a well-determined Hilbert-space representation of H by Eq. 2.2.5, because ∥π (q)∥ = ∥q∥ , ∀q ∈ H, by Eq. 2.2.3, where ∥.∥ in the left-hand side is the operator norm (or the matrix norm) on M2 (C), and ∥.∥ in the right-hand side is the quaternion modulus, which is a norm on H. Notation From below, we denote π(q) by [q], for all q ∈ H. □

Motivation: On the Second Clifford Algebra C2 =H ■ 7

Let’s define a subset H2 of M2 (C) by the set of all realizations of H, i.e., def

H2 = π (H) = {[q] ∈ M2 (C) : q ∈ H}.

(2.2.6)

Proposition 2. (see Ref. [2]) The quaternions H and the set H2 of Eq. 2.2.6 are isomorphic noncommutative fields, i.e., NF

H = H2 ,

(2.2.7)

NF

where “ = ” means “being noncommutative-field-isomorphic to.” 2

3

SPECTRAL ANALYSIS ON H UNDER (C2 , π)

Let H2 be the noncommutative field (Eq. 2.2.6). In this section, we regard each quaternion q ∈ H as a (2 × 2)-matrix [q] ∈ H2 in M2 (C) by Eq. 2.2.7 and review the spectral analysis on H2 (and hence, that on H), introduced in Ref. [2]. 3.1

Quaternion-Spectral Forms

Suppose q = a + bj ∈ H is a quaternion with a = x + yi, b = u + vi ∈ C, and let  [q] =

a −b b a



 =

x + yi −u − vi u − vi x − yi

 ∈ H2

be the realization of q. For a complex variable z ∈ C,     a −b z 0 det ([q] − zI2 ) = det − 0 z b a
= z 2 − 2xz + x2 + y 2 + u2 + v 2 .

(3.1.1)

By Eq. 3.1.1, one has det ([q] − zI2 ) = 0, ⇐⇒

(3.1.2)
z 2 − 2xz + x2 + y 2 + u2 + v 2 = 0,

8 ■ Classical Clifford Algebras

implying that z =x±i

p y 2 + u2 + v 2 in C

(3.1.3)

(see Ref. [2] for details). Proposition 3. Let q = a + bj ∈ H be a quaternion, realized to be [q] ∈ H2 . Then, the spectrum spec ([q]) of [q] is the subset,  spec ([q]) = λ, λ of C, (3.1.4)

where p λ = x + i y 2 + u2 + v 2 in C.

Proof. The spectrum spec ([q]) of Eq. 3.1.4 is obtained by Eq. 3.1.3. Motivated by Eq. 3.1.4, we define the following concept. Definition 4. Let q = x + yi + uj + vk ∈ H be a quaternion, realized to be [q] ∈ H2 . If u = 0 = v in R, equivalently, if q ∈ C in H, then the matrix     x + yi 0 q 0 denote q = = = [q] ∈ H2 0 x − yi 0 q is called the quaternion-spectral form (in short, the q-spectral form) of q. Meanwhile, if either u ̸= 0 or v ̸= 0 in R, equivalently, if q ∈ (H \ C) in H, then the matrix   λ 0 denote q = ∈ H2 , 0 λ with p λ = x + i y 2 + u2 + v 2 ∈ C, is called the quaternion-spectral form (in short, the q-spectral form) of q. For instance, if q1 = 2 − 3i + 0j + 0k ∈ H, then   2 − 3i 0 q1 = 0 2 + 3i

Motivation: On the Second Clifford Algebra C2 =H ■ 9

is the q-spectral form of q1 ; meanwhile, if q2 = 2 − 3i + j + 0k ∈ H, then √   2 + 10i 0√ q2 = 2 − 10i 0 is the q-spectral form of q2 . 3.2

Similarity on q-Spectral Forms in H2

Throughout this section, let a = x + yi, b = u + vi ∈ C, with x, y, u, v ∈ R and

(3.2.1) q = a + bj = x + yi + uj + vk ∈ H.

In Section 3.1, we showed that every quaternion q ∈ H of Eq. 3.2.1, realized to be [q] ∈ H2 , has its q-spectral form   p λ 0 q= with λ = x + i y 2 + u2 + v 2 , (3.2.2) 0 λ if either u ̸= 0 or v ̸= 0 in R, and   x + yi 0 q = [q] = 0 x − yi

(3.2.3)

in H2 , if u = 0 = v in R by Definition 4. Suppose b ∈ C× in Eq. 3.2.1. For t ∈ C× , define a (2 × 2)-matrix Qt (q) by 
 t − t a−λ b  Qt (q) =  (3.2.4)
a−λ t b t in M2 (C), where q ∈ H is in the sense of Eq. 3.2.1 and λ is in the sense of Eq. 3.2.2. Then, by the straightforward computation, we have [q]Qt (q) = Qt (q)q, whenever t, b ∈ C× , in M2 (C) (see Ref. [2] for details).

(3.2.5)

10 ■ Classical Clifford Algebras

Theorem 5. Let q = a + bj ∈ H be a quaternion Eq. 3.2.1 with its realization [q] ∈ H2 , and let q ∈ H2 be the q-spectral form of q. If b ̸= 0 in C, then q = Qt (q)−1 [q]Qt (q) ⇐⇒ [q] = Qt (q)qQt (q)−1 in H2 , where

(3.2.6) 

−

t

Qt (q) =  a−λ b




t

a−λ b


 t  ∈ H2 ,

t

for any t ∈ C× . Meanwhile, if b = 0 in C, then q = [w]−1 q[w] = [w]−1 [q][w] in H2 ,

(3.2.7)

where w = w + 0j + 0k ∈ C× in H.

(3.2.8)

Proof. First, suppose that b = 0 in C, and hence, q = a + 0j in H. Then, by Eq. 3.2.2, the quaternion q has its q-spectral form   a 0 = [q] in H2 , q= 0 a by Eq. 3.2.3. Suppose w ∈ C× , and w = w + 0j + 0k ∈ H, realized to be [w] ∈ H2 . Then,   wa   0 a 0 w
= q = [q] = wa 0 a 0 w  =

w 0 0 w



a 0 0 a



w−1 0 0 w−1



= [w][q][w−1 ] = [w]q[w]−1 in H2 . Therefore, the relation of Eq. 3.2.7 holds true, whenever w ∈ C× ⊂ H are in the sense of Eq. 3.2.8.

Motivation: On the Second Clifford Algebra C2 =H ■ 11

Assume now that b ̸= 0 in C. Then, for any t ∈ C× , the corresponding matrices Qt (q) of Eq. 3.2.3 satisfy Qt (q)q = [q]Qt (q), by Eqs. 3.2.2 and 3.2.5. Thus, Qt (q)−1 (Qt (q)q) = Qt (q)−1 [q]Qt (q) in H2 , if and only if

(3.2.9) q = Qt (q)−1 [q]Qt (q) in H2 .

So, the relation of Eq. 3.2.6 holds true by Eq. 3.2.9, whenever b ̸= 0 in C. Theorem 5 shows that for a quaternion q ∈ H with its q-spectral form q ∈ H2 , there exists at least one nonzero matrix A ∈ H2 , such that q = A−1 [q] A or [q] = A q A−1 , (3.2.10) “in H2 .” Equivalent to Eq. 3.2.10, for any q ∈ H, there exists at least one nonzero q0 ∈ H, such that q = q0 λ q0−1 in H,

(3.2.11)

where spec ([q]) = {λ, λ} in C. Definition 6. Let q ∈ H be a quaternion with its realization [q] ∈ H2 , and let   λ 0 q= = [λ] ∈ H2 0 λ be the q-spectral form. Then, the (1, 1)-entry λ ∈ C of q is called the quaternion-spectral value (in short, the q-spectral value) of q. For example, if q1 = 2 − i − j + k ∈ H, then the q-spectral value is p √ 2 + i (−1)2 + (−1)2 + 12 = 2 + 3i, and meanwhile, if q2 = 2 − i + 0j + 0k ∈ H, then the q-spectral value is 2 − i = q in C, by Definitions 4 and 6.

12 ■ Classical Clifford Algebras

3.3

Quaternion-Spectral Equivalence

In this section, we let an arbitrary fixed quaternion q ∈ H be in the sense of Eq. 3.2.1. Define a relation R on H by def

q1 R q2 ⇐⇒ λ1 = λ2 in C,

(3.3.1)

where λl is the q-spectral value of ql , for l = 1, 2. It is trivial to show that this relation R of Eq. 3.3.1 is an equivalence relation on H. Definition 7. The equivalence relation R of Eq. 3.3.1 is called the quaternion-spectral equivalence relation (in short, the q-spectral relation) on H. In addition, two q-spectral equivalent quaternions q1 and q2 are said to be q-spectral related, if q1 Rq2 . Let ql = al + bl j be q-spectral-related quaternions in H, with bl ̸= 0 in C, and let λ ∈ C be the identical q-spectral value of ql , for l = 1, 2. Then, there exists yl ∈ H such that ql = yl λ yl−1 in H, ∀l = 1, 2,

(3.3.2)

by Eqs. 3.2.10 and 3.2.11. So, we have

q2 = y2 λ y2−1 = y2 y1−1 y1 λ y1−1 y1 y2−1 by Eq. 3.3.2 = y2 y1−1




y1 λy1−1




y1 y2−1




by Eq. 2.2.7

−1 = y2 y1−1 q1 y2 y1−1

(3.3.3)

in H. Recall that two matrices A1 and A2 are similar in Mn (C), for n ∈ N, if there exists an invertible matrix U ∈ Mn (C), such that A2 = U A1 U −1 in Mn (C).

(3.3.4)

It is also well known that if two matrices A1 and A2 are similar in the sense of Eq. 3.3.4, then spec (A1 ) = spec (A2 ) in C and vice versa in Mn (C).

(3.3.5)

Motivation: On the Second Clifford Algebra C2 =H ■ 13

Definition 8. Let ql ∈ H be quaternions realized to be [ql ] ∈ H2 , for l = 1, 2. The realizations [q1 ] and [q2 ] are said to be similar “in H2 ,” if there exists a nonzero matrix U “in H2 ,” such that [q2 ] = U [q1 ] U −1 “in H2 .”

(3.3.6)

By abusing notation, two quaternions q1 and q2 are said to be similar in H, if their realizations [q1 ] and [q2 ] are similar in the sense of Eq. 3.3.6. Since the similarity (Eq. 3.3.6) on H2 (and hence, the similarity on H) is determined by the similarity (Eq. 3.3.4) on M2 (C), it is an equivalence relation, because the similarity (Eq. 3.3.4) on M2 (C) is an equivalence relation (e.g., [7]). Theorem 9. Two quaternions q1 and q2 are q-spectral related, if and only if they are similar in the sense of Eq. 3.3.6 in H, i.e., as equivalence relations, The q-spectral relation on H = The similarity on H. (3.3.7) Proof. The relation (Eq. 3.3.7) is shown by Eq. 3.3.3. For details, see Refs. [1] and [2]. 3.4

Quaternion-Spectral Mapping Theorem

Let C[z] be the (pure-algebraic) polynomial ring over a field C in a variable z , C[z] = {f (z) : f is a polynomial in z over C}, i.e., C[z] is a ring equipped with the polynomial addition and the polynomial multiplication of all polynomial, k P

an z n with an ∈ C, ∀n = 1, ..., k,

(3.4.1)

n=0

in a variable z , for all k ∈ N0 = N ∪ {0 }. It is well known that if A is a matrix in Mn (C) for n ∈ N, and if f ∈ C[z] is a polynomial (Eq. 3.4.1), then spec (f (A)) = {f (t) : t ∈ spec (A)} in C,

(3.4.2)

by the spectral mapping theorem (e.g., [7] and [8]), where a new matrix f (A) ∈ Mn (C) is

14 ■ Classical Clifford Algebras

ak Ak + ak−1 Ak−1 + ... + a2 A2 + a1 A + a0 In , where In is the identity (n × n)-matrix of Mn (C). Lemma 10. Let g(z) ∈ C[z], and let q ∈ H realized to be [q] ∈ H2 “in M2 (C).” Then, 
(3.4.3) spec (g ([q])) = g (λ), g λ in C, where λ is the q-spectral value of q. Proof. The relation (Eq. 3.4.3) is proven by Eq. 3.4.2 by considering [q] as an element of M2 (C). Even though the spectral mapping theorem gives the relation (Eq. 3.4.3) on M2 (C), this relation (Eq. 3.4.3) does not hold “on H2 ,” in general. Now, let’s define the subset Cr [z] of C[z] by ∞

Cr [z] = ∪ {

N P

N =0 n=0

an z n ∈ C[z] : a0 , a1 , ..., aN ∈ R}.

(3.4.4)

Theorem 11. Let q ∈ H be a quaternion (Eq. 3.2.1) with its q-spectral value λ ∈ C. If N X f (z ) = an z n ∈ Cr [z ], n=0

then f (λ) ∈ C is the q-spectral value of f (q) ∈ H, where Cr [z] is the subset (Eq. 3.4.4) of C[z], and f (q) =

(3.4.5) N P

an q n in H.

n=0

Proof. Let q ∈ H be a quaternion (Eq. 3.2.1) with its q-spectral value λ ∈ C, and let h(z) ∈ C[z]. If [q] ∈ H2 is the realization of q, then 
spec (h ([q])) = h (λ), h λ , in C, by Eqs. 3.1.4 and 3.4.3. Note, however, that for h(z) ∈ C[z],
h λ ̸= h(λ) in C, in general. (For instance, if h(z) = iz in C[z], then h(1 + i) = −1 − i ̸= 1 + i =
h 1 + i .)

Motivation: On the Second Clifford Algebra C2 =H ■ 15

However, if f (z) =

N P

an z n ∈ Cr [z] with a0 , a1 , ..., aN ∈ R, then

n=0

f λ




=

N P

  N
n P an λ = an λn

n=0

=

N P n=0

n=0

(an λn ) =

N P

an λn = f (λ)

n=0

in C, implying that o 
n spec (f ([q])) = f (λ), f λ = f (λ), f (λ) . Therefore, the statement (Eq. 3.4.5) holds. Now, let R[x] be the polynomial ring over R in a variable x , i.e., ∞

R[x] = ∪ {

N P

N =0 n=0

an xn : a0 , a1 , ..., aN ∈ R}.

(3.4.6)

Corollary 12. Let f (x) be a polynomial in the polynomial ring R[x] of Eq. 3.4.6. If q ∈ H is a quaternion with its q-spectral value λ ∈ C, realized to be [q] ∈ H2 , then spec (f ([q])) = {f (λ), f (λ)} in C.

(3.4.7)

Proof. The set equality (Eq. 3.4.7) holds by Eqs. 3.4.5 and 3.4.6. We may call the relation (Eq. 3.4.7), the quaternion-spectral mapping theorem. Theorem 13. Let q1 and q2 be q-spectral related in H, with their q-spectral value λ ∈ C. If f (x) ∈ R[x], then f (q1 ) and f (q2 ) are q-spectral related in H, too, with their identical q-spectral value f (λ) ∈ C. Equivalently, if q1 and q2 are similar in H, then f (q1 ) and f (q2 ) are similar in H, for all f (x) ∈ R[x]. Proof. It is proven by Eqs. 3.3.7 and 3.4.7. See Refs. [1] and [2] for details.

16 ■ Classical Clifford Algebras

3.5

The Quaternion-Spectralization σ

Define a function σ : H → H by def

σ(q) = the q-spectral value of q, ∀q ∈ H.

(3.5.1)

For example, p σ (1 + 0i + 2j − 3k) = 1 + i√ 02 + 22 + (−3)2 = 1 + 13i and σ (−2 − i + 0j + 0k) = −2 − i. Definition 14. We call the function σ of Eq. 3.5.1, the quaternionspectralization (in short, the q-spectralization) on H. Let’s consider the range of the q-spectralization σ. Proposition 15. If σ is the q-spectralization (Eq. 3.5.1), then σ(H) = C.

(3.5.2)

Proof. If q ∈ C in H, then σ (q) = q, and if q ∈ H \ C in H, then σ (q) is contained in the upper-half plane C+ of C, by Eq. 3.5.1. Thus, σ (H) = C ∪ C+ ⊆ C. Also, if z ∈ C, then it is understood to be z + 0j + 0k in H, satisfying [z] = [σ (z)] in H2 , and hence, z = σ (z) in H. So, C ⊆ σ (H). Therefore, the set equality (Eq. 3.5.2) is shown. By Eq. 3.5.2, the q-spectralization σ is onto C, and hence, it is not surjective. Also, it is not injective. Indeed, even though −2 − i + j − k ̸= −2 + i − j + k in H, one has p σ (−2 − i + j − k) = −2 + √ i (−1)2 + 12 + (−1)2 = −2 + p 3i = −2 + i 12 + (−1)1 + 12 = σ (−2 + i − j + k).

Motivation: On the Second Clifford Algebra C2 =H ■ 17

Corollary 16. Let σ be the q-spectralization (Eq. 3.5.1). Then, σ (q1 ) = σ (q2 ) in C, if and only if q1 and q2 are similar in H, and if and only if they are q-spectral related in H. Proof. The q-spectralization σ satisfies σ (q1 ) = σ (q2 ) in C for q1 , q2 ∈ H. Then, by definition, the quaternions q1 and q2 are qspectral related. Conversely, if q1 and q2 are q-spectral related, then σ (q1 ) = σ (q2 ). So, σ (q1 ) = σ (q2 ), if and only if q1 and q2 are q-spectral related, if and only if they are similar in H by (3.3.7). The above corollary confirms that all quaternions are classified by the q-spectralization σ. For a fixed quaternion q ∈ H, define the subset, q o = {h ∈ H : σ (h) = σ (q)}

(3.5.3)

in H. Then, by Corollary 16, it forms the equivalence class of q ∈ H, under the similarity (or, the q-spectral relation). And one has o

q o = (σ(q)) in H, set-theoretically. Thus, σ (q) ∈ C ⊂ H becomes a representative of all quaternions of q o in H, by Eqs. 3.5.2 and 3.5.3. So, H = ⊔ zo, z∈C

o

where z is in the sense of Eq. 3.5.3. Therefore, C = H/R, where R is the q-spectral relation on H.

4

ON NONCOMMUTATIVE FIELD H UNDER (C2 , π)


In this section, we consider the canonical representation C2 , π of a noncommutative field, the quaternions H, and the corresponding matrices of H2 , as elements of M2 (C). 4.1

On the Noncommutative Field H

As we considered in Eq. 2.1.8, if q = a + bj ∈ H× = H \ {0} with a, b ∈ C, then the quaternion reciprocal, ! ! a −b q −1 = + j, 2 2 2 2 |a| + |b| |a| + |b|

18 ■ Classical Clifford Algebras

is well determined uniquely in H× , and equivalently, the realization [q] = π (q) ∈ H2 always has its inverse matrix,   1 1 [q] a b −1 [q] = 2 [a − bj] = , = 2 −b a ∥q∥ ∥q∥ |a| + |b| in H2× . Therefore, the algebraic pair H× =
(H× , ·) is a well-defined group, which is isomorphic to H2× = H2× , · , where H2× = H2 \ {[0]}.
× × A subgroup H× 1 = H1 , · of H is well determined where  × H× 1 = q ∈ H : ∥q∥ = 1 (see Ref. [1]). Note that if q ∈ H× 1 in H, then [q −1 ] = [q]−1 = [q] = [q]∗ , by the above formula. Motivated by the above discussions, we concentrated on studying the maximal multiplicative group H× and the corresponding operators under the “direct product” representations of our canonical represen
tation C2 , π in Ref. [1]. As one can see above, for any arbitrarily fixed q ∈ H× , there exists a unique qo ∈ H× 1 , such that q = ∥q∥ qo in H× , with qo =

q ∈ H× 1 ∥q∥

and

(4.1.1) q −1 =

qo qo−1 = in H× . ∥q∥ ∥q∥

In particular, the first formula of Eq. 4.1.1 is a quaternion version of the polar decomposition on C, and the second formula of Eq. 4.1.1 illustrates the relation between the quaternion reciprocals (or the inverses) and the quaternion conjugates on H (e.g., also see Ref. [1]). As we considered in Ref. [1], the group H× is understood as a matrix subgroup
 H2× = [q] ∈ H2 : q ∈ H× , · of the general linear group GL2 (C), the matrix subgroup of M2 (C) consisting of all invertible (2 × 2)-matrices.

Motivation: On the Second Clifford Algebra C2 =H ■ 19

Define the usual trace on Mn (C), tr : Mn (C) → C, by

(4.1.2) tr ([aij ]n×n ) =

n X

akk , ∀[aij ]n×n ∈ Mn (C),

k=1

for n ∈ N. Recall that the trace tr of Eq. 4.1.2 satisfies tr (AB) = tr (BA), ∀A, B ∈ Mn (C) and

(4.1.3)
tr U AU −1 = tr (A), ∀A ∈ Mn (C), U ∈ GLn (C),

on Mn (C), where GLn (C) is the general linear group. Since the quaternions H = H2 is a subring of the matricial ring M2 (C), one can restrict the trace tr of Eq. 4.1.3 on H, i.e., tr

denote

=

tr |H : H → C,

defined by

(4.1.4) tr (q) = tr ([q]), ∀q ∈ H.

Observe that if q = a + bj ∈ H with a, b ∈ C, then   a −b = a + a = 2Re (a), tr (q) = tr ([q]) = tr b a where Re (z) means the real chapter of z ∈ C. Theorem 17. Let q = ∥q∥ qo ∈ H with qo ∈ H× 1 . If σ is the q-spectralization on H, then tr (q) = 2 ∥q∥ Re (σ (qo )).

(4.1.5)

Proof. By Eqs. 3.2.10 and 3.2.11, for any quaternion q ∈ H, there exists h ∈ H× , such that q = hσ (q) h−1 in H.

20 ■ Classical Clifford Algebras

Thus, by Eqs. 4.1.3 and 4.1.4, tr (q) = tr

 

hσ (q) h−1 = tr [h][σ (q)][h]−1 = tr (σ (q))

and hence,  tr (q) = tr (σ (q)) = tr

σ (q) 0 0 σ (q)

 ,

implying that tr (q) = σ (q) + σ (q) = 2Re (σ (q)), ∀q ∈ H. If q = ∥q∥ qo ∈ H, with qo ∈ H× 1 , then tr (q) = 2Re (σ (∥q∥ qo )) = 2Re (∥q∥ σ (qo )), by Eqs. 3.4.5 and 3.4.7; therefore, Eq. 4.1.5 holds because tr (q) = 2Re (∥q∥ σ (qo )) = 2 ∥q∥ Re (σ (qo )). The above theorem characterizes the relation between the usual trace tr and our q-spectralization σ on H by Eq. 4.1.5. For instance, one has  tr (1 − 4i + 0j + 0k) = tr

√

 17

1 4 √ −√ i 17 17



√ 2 17 = √ =2 17

by Eq. 4.1.5. Now, we consider the normalized trace τ : H → C, defined by def 1 2 tr (q)

τ (q) =

= 12 tr ([q]), ∀q ∈ H,

(4.1.6)

where tr is the usual trace (Eq. 4.1.4) on H. Corollary 18. Suppose τ is the normalized trace (Eq. 4.1.6) on H. If q = ∥q∥ qo ∈ H with qo ∈ H× 1 , then τ (q) = τ (σ (q)) = ∥q∥ Re (σ (qo )). Proof. Equation 4.1.7 is shown by Eqs. 4.1.5 and 4.1.6.

(4.1.7)

Motivation: On the Second Clifford Algebra C2 =H ■ 21

One may satisfy the formula tr (q) = 2Re (σ (q)) = 2τ (q), provided in the proof of Theorem 17. However, Eq. 4.1.5 is more useful than the above equation. Indeed, if q ∈ H× with its inverse q −1 , then



tr q −1 = 2Re σ q −1 = 2τ q −1 and equivalently,

2 Re (σ (qo )) = 2τ q −1 , tr q −1 = ∥q∥ by Eqs. 4.1.5 and 4.1.7. As one
can check, computing σ (qo ) is relatively easier than computing σ q −1 . Definition 19. The maximal multiplicative group H× is called the quaternion group. The subgroup H× 1 is called the unital (quaternion-) × subgroup of H . 4.2

× On the Unital Subgroup H× 1 of H

Let H× 1 =




q ∈ H× : ∥q∥ = 1 , ·

be the unital subgroup of the quaternion group H× . Observe that if q ∈ H× 1 with ∥q∥ = 1, and if τ is the normalized trace (Eq. 4.1.6) on H, then τ (q) = τ (σ (q)) = Re (σ (q)), by Eq. 4.1.7, and hence,

(4.2.1)

∥τ (q)∥ = |τ (σ(q))| ≤ 1, in general. Indeed, for any q ∈ H× 1 satisfying q = q0 σ (q) q0−1 for some q0 ∈ H, one has
2 1 = ∥q∥ = det ([q]) = det [q0 ][σ (q)][q0 ]−1 = det ([σ (q)]),

22 ■ Classical Clifford Algebras

because det (AB) = det (A) det (B), ∀A, B ∈ Mn (C), for all n ∈ N, and hence,

(4.2.2)

2

2

1 = ∥q∥ = det ([σ (q)]) = σ (q) σ (q) = |σ (q)| . By Eq. 4.2.2, |τ (q)| = |Re (σ (q))| ≤ |σ (q)| = 1, implying the inequality (Eq. 4.2.1). Proposition 20. For any group element q ∈ H× , one has |τ (q)| ≤ |σ (q)| = ∥q∥. As application, if q ∈

H× 1,

(4.2.3)

then |τ (q)| ≤ 1.

Proof. For any q ∈ H× , one can have that 2

2

2

∥q∥ = det ([q]) = det ([σ (q)]) = ∥σ (q)∥ = |σ (q)| , implying that 1 = ∥q∥ = ∥σ (q)∥ = |σ (q)|, by the q-spectral relation of q and σ (q). Therefore, |τ (q)| = ∥q∥ |Re (σ (qo ))| ≤ ∥q∥ |σ (qo )| = ∥q∥, implying Eq. 4.2.3, having its special case (Eq. 4.2.1). The equality in Eq. 4.2.3 is interesting, because if T ∈ B (H) is an operator on a Hilbert space H, then in general, |r (T )| ≤ ∥T ∥ , for T ∈ B (H), where r (T ) = sup {|t| : t ∈ spec (T )} is the spectral radius of T (e.g., [8]). However, if q ∈ H× , then ∥q∥ = ∥[q]∥ = |σ (q)| = r ([q]) in H2 . Corollary 21. For every element q of the quaternion group H× , is an element of the unital subgroup H× 1. Proof. Since |σ (q)| = ∥q∥ for all q ∈ H× by Eq. 4.2.3, one has q q = ∈ H× 1. ■ |σ (q)| ∥q∥

q |σ(q)|

Motivation: On the Second Clifford Algebra C2 =H ■ 23

4.3

Operator-Theoretic Spectral Properties of H×

In this section, we study operator-theoretic spectral properties of our group elements, which are the nonzero quaternions of H× , by representing them as matrices of H2× in M2 (C). Recall that, for any matrix A ∈ Mn (C), there exists a unique adjoint (or the conjugate transpose) A∗ ∈ Mn (C), for all n ∈ N. Consider now that if q = a + bj ∈ H× with a, b ∈ C, then

(4.3.1) [q]∗ =



a −b b a

∗

 =

a b −b a

 = [a − bj] = [q]

is well determined, where q is the quaternion conjugate of q. Proposition 22. The adjoint (∗) restricted from M2 (C) is closed on H2× as a unary operation. Proof. Since [q] = [q]∗ in H2× by Eq. 4.3.1, the adjoint (∗) is closed on H2× . Since the quaternion group H× and the matrix group H2× are isomorphic, there is a well-defined adjoint (∗) on H× : def

q ∗ = q in H× ,

(4.3.2)

by Proposition 22. Notation. From below, if there are no confusions, then we write q by q ∗ as in Eq. 4.3.2, for all q ∈ H. 2 Recall that an operator A ∈ B (H) is said to be normal on a Hilbert space H, if A∗ A = AA∗ on H.

(4.3.3)

Since the adjoint is a closed unary operation on H× , in the sense of Eq. 4.3.2, one can obtain the following normality condition (Eq. 4.3.3) on H× .

24 ■ Classical Clifford Algebras

Theorem 23. Every group element q ∈ H× is normal in the sense that [q ∗ ][q] = [q]∗ [q] = [q][q]∗ = [q][q ∗ ] in H2× and moreover,

(4.3.4) 2

[q]∗ [q] = ∥q∥ [1] = [q][q]∗ in H2× . 2

Proof. Since q ∗ q = ∥q∥ = qq ∗ by Eqs. 2.1.6 and 2.1.7, and since [q ∗ ] = [q]∗ in H2 , for all q ∈ H, the normality in Eq. 4.3.4 holds. In particular, one can get that h i 2 2 [q]∗ [q] = [q ∗ q] = ∥q∥ = ∥q∥ [1] = [qq ∗ ] = [q][q]∗ , and hence, the second formula of Eq. 4.3.4 holds true, too. Recall that an operator U ∈ B (H) is said to be unitary on a Hilbert space H, if U ∗ U = I = U U ∗ in B (H), where I is the identity operator of B (H); equivalently, U ∗ = U −1 in B (H), where U −1 is the inverse (operator) of U . × Theorem 24. Let H× 1 be the unital subgroup of H . Then every × element q of H1 is unitary in the sense that

where

× H2:1

× [q ∗ ][q] = [1] = I2 = [q][q ∗ ] in H2:1 ,
× = π H1 in H2 .

(4.3.5)

Proof. For any q ∈ H× 1 , one has [q ∗ ][q] = [q]∗ [q] = [q ∗ q] = [1] = [qq ∗ ] = [q][q]∗ = [q][q ∗ ], by the normality (Eq. 4.3.4) in H2× , because ∥q∥ = 1. We showed so far that all quaternions of H are normal, and in particular, the unital quaternions of H× 1 are unitary. Recall now that an operator A ∈ B (H) is self-adjoint, if A∗ = A in B (H).

Motivation: On the Second Clifford Algebra C2 =H ■ 25

Theorem 25. A quaternion q ∈ H is contained in R, if and only if q is self-adjoint in H in the sense that [q]∗ = [q] in H2 . Proof. Let q = a + bj ∈ H be a quaternion with a, b ∈ C, having its realization,   a −b [q] = in H2 . b a Since ∗

∗



[q] = [q ] = [a − bj] =

a b −b a

one has

 , (4.4.6)

[q ∗ ] = [q]∗ = [q], if and only if a = a, and b = −b in C, if and only if a ∈ R and b = 0 in C, if and only if [q]∗ = a[1] with a ∈ R. Therefore, the self-adjointness (Eq. 4.4.6) holds, if and only if q ∈ R in H. Recall that, in Ref. [1], we showed that R× = (R \ {0} , ·) is a normal subgroup of the quaternion group H× , consisting of all nonzero self-adjoint quaternions of H. Also, recall that an operator A ∈ B (H) is a projection on a Hilbert space H, if it is both self-adjoint and idempotent, i.e., A∗ = A = A2 , in B (H). By the above self-adjointness characterization, we have only one “nonzero” projection 1 in the quaternion group H× . Theorem 26. The elements 0 and 1 are the only projections of H in the sense that [0] and [1] are the only projections in H2 .

26 ■ Classical Clifford Algebras

Proof. Let q ∈ H with its realization [q] ∈ H2 and assume that [q] is a projection in H2 . Then, [q]∗ = [q ∗ ] = [q] = [q]2 in H2 , and hence, it must be self-adjoint in H2 . So, we restrict our interests to self-adjoint quaternions q ∈ H× . By Theorem 25, such a self-adjoint quaternion q is a real number in H, and hence, such self-adjoint q = q + 0i + 0j + 0k (with q ∈ R) is a projection, if and only if [q]2 = [q] in H2 , if and only if 

q 0 0 q

2

 =

q2 0 0 q2



 =

q 0 0 q

 ,

if and only if q 2 = q in R, implying that q = 0 or q = 1 in R ⊂ H.

5

FREE-PROBABILISTIC DATA INDUCED BY H

In Section 4, we studied spectral
properties of quaternions under the 2 canonical representation C , π of H. Also, we defined natural linear functionals, the usual trace tr, and the normalized trace τ , on H2 = π (H), by restricting them from those on M2 (C). Recall that tr (q) = 2 ∥q∥ Re (σ (qo )) = 2τ (q) and

(5.1) ∥q∥ = |σ (q)| = ∥q∥ |σ (qo )|,

implying ∥qo ∥ = 1 = |σ (qo )|, for all q = ∥q∥ qo ∈ H with qo ∈ H× 1 , where σ is the q-spectralization on H. In this section, we consider free-distributional data induced by
H under the canonical representation C2 , π .

Motivation: On the Second Clifford Algebra C2 =H ■ 27

5.1

Free Probability

For more about fundamental free probability theory, see Refs. [17] and [20]. Roughly speaking, free probability theory is a generalization of classical measure theory on noncommutative operator algebras. Let A be a noncommutative algebra over C, and φ : A → C, a linear functional on A. Then, the pair (A, φ) is called a (noncommutative) free probability space. By definition, free probability spaces are the noncommutative version of classic measure spaces (X, µ) consisting of a set X and a measure µ on the σ-algebra of X. As in measure theory, the (noncommutative) free probability on (A, φ) is dictated by the linear functional φ. Meanwhile, if (A, φ) is unital in the sense that (i) the unity 1A of A exists, and (ii) φ (1A ) = 1, then it is called a unital free probability space. And unital free probability spaces are the noncommutative analog of classical probability spaces (X, µ), where the given measures are the probability measures µ satisfying µ (X) = 1. If A is a topological algebra, and if φ is bounded (or continuous under linearity), then the corresponding free probability space (A, φ) is said to be a topological free probability space. Similarly, if A is a topological ∗-algebra equipped with the adjoint (∗), and φ (a∗ ) = φ (a) for all a ∈ A, then the topological free probability space (A, φ) is said to be a topological (free) ∗-probability space. Especially, if A is a C ∗ algebra, or a von Neumann algebra, or a Banach ∗-algebra, we call (A, φ) a C ∗ -probability space, a W ∗ -probability space, and a Banach ∗-probability space. For our main purposes, we focus on C ∗ -probability spaces from below. If (A, φ) is a C ∗ -probability space, and a ∈ A, then the algebra element a is said to be a free random variable of (A, φ). For any arbitrarily fixed free random variables a1 , ..., as ∈ (A, φ) for s ∈ N, one can get the corresponding free distribution of a1 , ..., as , characterized by the joint free moments, ! n Y
ri φ ail = φ ari11 ari22 ...arinn , l=1 n

n

for all (i1 , ..., in ) ∈ {1, ..., s} and (r1 , ..., rn ) ∈ {1, ∗} , for all n ∈ N, where a∗l are the adjoints of al , for all l = 1, ..., s. For instance, if a ∈ (A, φ) is a free random variable, then the free distribution of a is fully characterized by the joint free moments of {a, a∗ },

28 ■ Classical Clifford Algebras n Y φ ar l

! = φ (ar1 ar2 ...arn ),

l=1 n

for all (r1 , ..., rn ) ∈ {1, ∗} , for all n ∈ N (e.g., [17] and [20]). Thus, if a given free random variable a ∈ (A, φ) is self-adjoint in A in the sense that a∗ = a in A, then the free distribution of a is fully characterized by the free-moment sequence,


∞ (φ (an ))n=1 = φ (a), φ a2 , φ a3 , ... , in (A, φ). 5.2

The C ∗ -Algebra A2 Generated by H× under (C2 , π)

In this section, we consider the group C ∗ -algebra A2 generated by the quaternion group H× . Recall that since the generating group H× is noncommutative, the noncommutative polynomial algebra, 
   def A2 = C π H× = C H2× , over C is a well-defined ∗-algebra; furthermore, it is topological
since A2 ⊂ M2 (C), by considering M2 (C) as a C ∗ -algebra B C2 , the operator algebra on C2 equipped with its operator norm, i.e., A2 is a C ∗ -subalgebra of M2 (C) over C. Definition 27. The polynomial algebra A2 = C [π (H× )] over C is called the quaternion-group (C ∗ -)algebra (acting on C2 ). Each element A ∈ A2 has its expression, P A= tq [q], with tg ∈ C, in A2 ⊂ M2 (C),

(5.2.1)

q∈H×

where [q] = π (q) ∈ H2× are the realizations of q ∈ H× . Note that the summands tq [q] of Eq. 5.2.1 mean that  tq [q] = tq

a −b b a



 =

tq a −tq b tq b tq a

 “in A2 ⊆ M2 (C), ”

whenever q = a + bj ∈ H× , with a, b ∈ C,

Motivation: On the Second Clifford Algebra C2 =H ■ 29

i.e., z[q] ̸= [z][q] in M2 (C), in general, for z ∈ C and q ∈ H× . But, notice that if r ∈ R, then r[q] = [r][q] = [rq], ∀q ∈ H. It shows that H2 ⫋ A2 in M2 (C). Remark and Discussion. At this moment, we need to emphasize that, by the very definition of A2 , this C ∗ -algebra A2 is different from our second Clifford algebra C2 , which is identical (or isomorphic) to the quaternions H. It is the group C ∗ -algebra generated by the group H× . For instance, for any q = a + bj ∈ H× , with a, b ∈ C, realized to be [q] ∈ H2× ,     a −b za −zb z [q] = z = ∈ A2 , b a zb za but it is not contained in H2 , in general (especially, if z ∈ C \ R in C). So, A2 ̸= H = C2 ! 2 5.3

On the C ∗ -Probability Spaces (A2 , tr) and (A2 , τ )

Let A2 be the quaternion-group algebra of H× in the sense of Definition 27, which is
a C ∗ -subalgebra of M2 (C), under the canonical representation C2 , π . Since the usual trace tr on M2 (C) is a welldefined linear functional satisfying the additional properties (Eq. 4.1.4), the restricted trace tr = tr |A2 becomes a well-defined linear functional on A2 , satisfying tr (T1 T2 ) = tr (T2 T1 ) and

(5.3.1)
−1

tr T1 T2 T1

= tr (T2 ) (if T1 is invertible),

for T1 , T2 ∈ A2 . As a special case of Eq. 5.3.1, one has tr (q1 q2 ) = tr (q2 q1 ), respectively,

(5.3.2)
tr q1 q2 q1−1 = tr (q2 ),

30 ■ Classical Clifford Algebras

for all generating group elements q1 , q2 ∈ H× , where tr (q) = tr ([q]) is in the sense of Eq. 4.1.4, for all q ∈ H× . So, as in Section 5.1, one can construct a C ∗ -probability space, def

Atr 2 = (A2 , tr).

(5.3.3)

So, similar to Eq. 5.3.3, one has a C ∗ -probability space, def

Aτ2 = (A2 , τ ),

(5.3.4)

where τ = 21 tr is the normalized trace on A2 . Recall that tr (q) = 2Re (σ (q)) = 2 ∥q∥ Re (σ (qo )) = 2τ (q),

(5.3.5)

H× 1,

×

for all q = ∥q∥ qo ∈ H , with qo ∈ in A2 , where σ is the q-spectralization on H. Let q = ∥q∥ qo ∈ H× be an arbitrarily fixed generating element of A2 , for qo ∈ H× 1 . Then, for any n

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

n Q

r

q rl = q r1 q r2 ...q rn = (∥q∥ qo ) 1 ... (∥q∥ qo )

rn

l=1


= (∥q∥ ∥q∥ ... ∥q∥) qoi1 qoi1 ...qoin , ∗

since (zA) = zA∗ , for all z ∈ C and A ∈ M2 (C), where  1 qo = qo if rl = 1 qoi1 = qo−1 = qo∗ if rl = ∗, × by the unitarity of the subgroup H× 1 of H , for all l = 1, ..., n  P  n il

n = (∥q∥ ) qol=1 .

(5.3.6)

Lemma 28. Let q = ∥q∥ qo ∈ H× , with qo ∈ H× 1 , be a generating element of the C ∗ -probability space Atr of Eq. 5.3.3. Then 2   P  n n  il Q rl n , tr q = 2 ∥q∥ Re σ ql=1 (5.3.7) o l=1 n

where (i1 , ..., in ) ∈ {±1} satisfies the condition (Eq. 5.3.6), for all n (r1 , ..., rn ) ∈ {1, ∗} , for all n ∈ N.

Motivation: On the Second Clifford Algebra C2 =H ■ 31 n

Proof. Under hypothesis, if (r1 , ..., rn ) ∈ {1, ∗} , for n ∈ N, then ! n n Y X n rl tr q = tr (∥q∥ qoη ) with η = il ∈ Z, l=1

l=1 n

where (i1 , ..., in ) ∈ {±1} satisfies the condition (Eq. 5.3.6). So, ! n Y n tr q rl = 2 ∥q∥ Re (σ (qη )), l=1

by Eq. 5.1. Therefore, the free-distributional data (Eq. 5.3.7) holds. The above lemma characterizes the free distributions of generating elements of Atr 2 by the joint free-moment equation (Eq. 5.3.7). Lemma 29. Let q = ∥q∥ qo ∈ H× be a generating element of the C ∗ -probability space Aτ2 of Eq. 5.3.4. Then    P n n  il Q rl n , (5.3.8) τ q = ∥q∥ Re σ ql=1 o l=1 n

where (i1 , ..., in ) ∈ {±1} satisfies the condition (Eq. 5.3.6), for all n (r1 , ..., rn ) ∈ {1, ∗} , for all n ∈ N. Proof. The free-distributional data (Eq. 5.3.8) is obtained by Eqs. 5.1 and 5.3.7. The above lemma characterizes the free distributions of generating elements of the C ∗ -probability space Aτ2 by the joint free moments (Eq. 5.3.8). Now, let q ∈ H× 1 with ∥q∥ = 1, as an element of A2 . Then, by Eqs. 5.3.7 and 5.3.8, we have !! n n  n  P il Q rl Q rl l=1 tr q = 2Re σ q = 2τ q , (5.3.9) l=1

l=1 n

for all (r1 , ..., rn ) ∈ {1, ∗} , where (i1 , ..., in ) ∈ {±1} Eq. 5.3.6, for all n ∈ N.

n

satisfies

32 ■ Classical Clifford Algebras

By Eq. 5.3.9, computing the free-distributional data of q ∈ H× 1 is η to consider the q-spectral value σ (q ), for η ∈ Z. As we discussed in detail in Ref. [2], the q-spectralization σ is not multiplicative, i.e., σ (q1 q2 ) ̸= σ (q1 ) σ (q2 ) in general, for q1 , q2 ∈ H× . Lemma 30. The q-spectralization σ satisfies that if q, h ∈ H× , then
σ hqh−1 = σ (q) and

(5.3.10) n

n

σ (σ (q) ) = σ (q) , ∀n ∈ Z. Proof. For any h, q ∈ H× , the elements q and hqh−1 are similar in H, and hence, they are q-spectral related from each other. Since σ (q) is the q-spectral value of q,
σ (q) = σ hqh−1 . n

Also, since σ (q) ∈ C× = C\{0} in H× , the powers σ (q) are contained in C× ⊂ H× , and hence, n

σ (σ(q)n ) = σ (q) , for all n ∈ Z, because σ (a) = a, for a = a+0j +0k ∈ H, for all a ∈ C. By Eq. 5.3.10, one obtains the following relation. Theorem 31. If q, h ∈ H× , then
n
n σ hqh−1 = σ (q) , for all n ∈ Z. Proof. Under hypothesis, observe that   hq n h−1
n −1 hqh =  −1 n h q h

if n ≥ 0 if n < 0,

for all n ∈ Z. So, σ

(5.3.11)

(5.3.12) 

hqh−1


n 


= σ h0 q n h−1 = σ (q n ), o

Motivation: On the Second Clifford Algebra C2 =H ■ 33

by the first formula of Eq. 5.3.10, for all n ∈ Z, where h0 = h if n ≥ 0, and h0 = h−1 if n < 0. Note that q and σ (q) are q-spectral related in H× , and hence, there exists h ∈ H× , such that q = hσ (q) h−1 in H× and hence, n

× q n = h0 σ (q) h−1 0 ∈ H , ∀n ∈ Z,

where h0 = h if n ≥ 0, and h0 = h−1 if n < 0, by (5.3.12). Therefore,
n σ (q n ) = σ h0 σ(q)n h−1 = σ (σ(q)n ) = σ (q) , 0 for all n ∈ Z. In particular, the last equality holds by the second formula of Eq. 5.3.10. Therefore, we have that, for any q, h ∈ H× , 
n  n σ hqh−1 = σ (q n ) = σ (σ(q)n ) = σ (q) , for all n ∈ Z, proving the relation (Eq. 5.3.11). The relation (Eq. 5.3.11) is interesting because, even though σ (q1 q2 ) ̸= σ (q1 ) σ (q2 ), in general, for q1 , q2 ∈ H× , we have n

σ (q n ) = σ (q) , for all n ∈ Z. Suppose q ∈ H× 1 is a generating element of the quaternion-group algebra A2 . Then, one has ! !! ! n n P P n Y il il tr arl = 2Re σ ql=1 = 2Re σ(q)l=1 l=1

and hence,

(5.3.13) τ

n Y arl

!

n P

il

= Re σ (q)l=1

! ,

l=1 n

by Eqs. 5.3.7, 5.3.8, and 5.3.11, for all (r1 , ..., rn ) ∈ {1, ∗} , where n (i1 , ..., in ) ∈ {±1} satisfies the condition (Eq. 5.3.6), for all n ∈ N.

34 ■ Classical Clifford Algebras

Theorem 32. Let q = ∥q∥ qo ∈ H× be a generating element of the quaternion-group algebra A2 with qo ∈ H× 1 . Then, ! n n  n  P il Q rl Q rl n tr q = 2 ∥q∥ Re σ (qo )l=1 = 2τ q , (5.3.14) l=1

l=1 n

for all (r1 , ..., rn ) ∈ {1, ∗} , where (i1 , ..., in ) ∈ {±1} Eq. 5.3.6, for all n ∈ N.

n

satisfies

Proof. The free-distributional data (Eq. 5.3.14) is obtained by Eqs. 5.3.11 and 5.3.13. The refined equation (Eq. 5.3.14) not only re-characterizes the free distributions (Eq. 5.3.7) of generating elements of C ∗ -probability τ spaces Atr 2 , and those (Eq. 5.3.8) of A2 , but also provides an easier way to compute the joint free moments of {q, q ∗ } in both Atr and Aτ2 ,
2 k k because computing σ (q) is easier than computing σ q , for q ∈ H× 1 and k ∈ Z. For example, let q = 1 − i + 2j − k ∈ H× , as a generating element of the C ∗ -probability space Aτ2 . Then,   √ 1 i 2j k q = ∥q∥ qo = 7 √ − √ + √ − √ , 7 7 7 7 in H× , and 1 σ (qo ) = √ + i 7

r

√ √ 1 4 1 7 42 + + = + i. 7 7 7 7 7

Therefore, 2 ∗

2∗ 3

σ q q qq q




√ 2+1+1+2+3   2−1+1−2+3 = 7 Re σ (qo ) ,

identifying with σ q 2 q ∗ qq 2∗ q


3

  √ √ !3 √ 9 7 42 = 7 Re  + i . 7 7

Motivation: On the Second Clifford Algebra C2 =H ■ 35

REFERENCES [1] I. Cho, Certain Nonlinear Functions Action on the Vector Space Hn Over the Quaternions H, J. Nonlinear Sci. Appl., 15, (2022) 14–40. [2] I. Cho, and P. E. T. Jorgensen, Multi-Variable Quaternionic Spectral Analysis, Opusc. Math., 41, no. 3, (2021) 335–379. [3] C. J. L. Doran, Geometric Algebra for Physicists, ISBN: 978-0-521-480222, (2003) Published by Cambridge University Press. [4] F. O. Farid, Q. Wang, and F. Zhang, On the Eigenvalues of Quaternion Matrices, Lin. & Multilin. Alg., 59, no. 4, (2011) 451–473. [5] C. Flaut, Eigenvalues and Eigenvectors for the Quaternion Matrices of Degree Two, Analele Stiint. Univ. Ovidius Constanta, 10, no. 2, (2002) 39–44. [6] P. R. Girard, Einstein’s Equations and Clifford Algebra, Adv. Appl. Clifford Alg., 9, no. 2, (1999) 225–230. [7] P. R. Halmos, Linear Algebra Problem Book, ISBN: 978-0-88385-322-1, (1995) Published by Mathematical Association of America. [8] P. R. Halmos, Hilbert Space Problem Book, ISBN: 978-038-790685-0, (1982) Published by Springer-Verlag. [9] W. R. Hamilton, Lectures on Quaternions, Available on http://books. google.com, (1853) Published by Cambridge University Press. [10] I. L. Kantor, and A. S. Solodnikov, Hypercomplex Numbers, an Elementary Introuction to Algebras, ISBN: 0-387-96980-2, (1989) Published by Springer-Verlag. [11] V. Kravchenko, Applied Quaternionic Analysis, ISBN: 3-88538-228-8, (2003) Published by Heldemann Verlag. [12] N. Mackey, Hamilton and Jacobi Meet Again: Quaternions and the Eigenvalue Problem, SIAM J. Matrix Anal. Appl., 16, no. 2, (1995) 421–435. [13] S. Qaisar, and L. Zou, Distribution for the Standard Eigenvalues of Quaternion Matrices, Int. Math. Forum, 7, no. 17, (2012) 831–838. [14] S. D. Leo, G. Scolarici, and L. Solombrino, Quaternionic Eigenvalue Problem, J. Math. Phys., 43, (2002) 5815–5829.

36 ■ Classical Clifford Algebras [15] L. Rodman, Topics in Quaternion Linear Algebra, ISBN: 978-0-69116185-3, (2014) Published by Prinston University Press. [16] B. A. Rozenfeld, The History of Non-Eucledean Geometry: Evolution of the Concept of a Geometric Spaces, ISBN: 978-038-796458-4, (1988) Published by Springer. [17] A. Sudbery, Quaternionic Analysis, Math. Proc. Camb. Phil. Soc., 85, (1979) 199–225. [18] L. Taosheng, Eigenvalues and Eigenvectors of Quaternion Matrices, J. Central China Normal Univ., 29, no. 4, (1995) 407–411. [19] J. A. Vince, Geometric Algebra for Computer Graphics, ISBN: 978-184628-996-5, (2008) Published by Springer. [20] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, ISBN: 978-0-8218-1140-5, (1992) Published by American Mathematical Society. [21] J. Voight, Quaternion Algebra, Available on http://math.dartmouth. edu/˜jvoight/quat-book.pdf, (2019) Department of Mathematics, Dartmouth University.

CHAPTER

2

On Classical Clifford Algebras

n this chapter, motivated by the main results of Chapter 1, we study algebra, analysis, operator theory, and free probability induced by the classical Clifford algebras {Ck }k∈N and generalize them to those induced by the limit C ∗ -algebra C = − lim → Ck , simply called the Clifford algebra. To do that, from Clifford algebras {Ck }k∈N , we construct the embedded multiplicative groups {Gk }k∈N . The construction of such groups allows us to have a group G from the Clifford algebra C . Since this group G is identified with the set ±E = {±w : w ∈ E}, where E is the R-basis of C , the mathematical properties on C are inherited to those on G . From this group G , the corresponding group C ∗ -algebra C = C ∗ (G ) is constructed, and the operator theory and free probability on C are studied.

I

6

INTRODUCTION OF CHAPTER 2

In this chapter, we review well-known classical Clifford algebras {Ck }k∈N over the real field R = C0 (see Section 7 or Ref. [2]), study the fundamental properties on them, and establish suitable free-probabilistic models on {Ck }k∈N to study free-distributional data induced by them. Then, we extend such results to those of the limit

DOI: 10.1201/9781032637211-2

37

38 ■ Classical Clifford Algebras

C ∗ -algebra C chain,

denote

=

lim − → Ck , called the Clifford algebra, of the increasing C = C1 ⊂ C2 ⊂ C3 ⊂ · · · ,

of algebras and study certain analytic data on C . The quaternions H is an interesting object not only in mathematics (e.g., [1], [2], [8], [9], and [14]) but also in scientific fields (e.g., [3] and [17]). The quaternions H have been studied algebraically, analytically and physically in Refs. [6], [10], and [15]. Also, the matrices over the quaternions H and the corresponding quaternionic-eigenvalue problems have been considered in linear or multilinear analysis (e.g., [4], [5], [10], [11], [12], [13], and [16]). Independently, as we have seen in Chapter 1, or Ref.
[1], the spectral analysis on H up to the canonical representation C2 , π of Ref. [17] allows us to study the quaternions H structurally by considering all quaternions q as (2 × 2)-matrices over the complex field
C. In particular, the representation C2 , π of H not only makes the study of H relatively easier but also let each complex number z ∈ C as a certain representative of all quaternions whose eigenvalues are {z, z}, where z is the conjugate of z (e.g., see Chapter 1 or Ref. [1]). Meanwhile, the construction of classical Clifford algebras {Ck }k∈N0 , with C0 = R, the real numbers (as the 1-dimensional R-algebra over itself), let us consider the Clifford algebras Ck+1 as finite-dimensional algebras over R = C0 with manageable bases Ek+1 , see Ref. [2]. The recursive construction of the increasing chain of Clifford algebras, R = C0 ⊂ C1 ⊂ C2 ⊂ C3 ⊂ ..., gives motivations to understand {Ck }k∈N as finite-dimensional substructures embedded in certain C ∗ -algebras over the complex field C. As R-algebras, C1 is isomorphic to the 2-dimensional R-algebra C, the set of all complex numbers; C2 is isomorphic to the quaternions H, as a 4-dimensional R-algebra; C3 is isomorphic to an 8-dimensional Ralgebra M2 (C); and C4 is isomorphic to a 16-dimensional R-algebra M4 (R) (e.g., [2]).

7

CLASSICAL CLIFFORD ALGEBRAS

In this section, we review the construction of classical Clifford algebras {Ck }k∈N0 (over the real field R = C0 ), where N0 = N ∪ {0}. Since

On Classical Clifford Algebras ■ 39

each Ck is regarded as a ring, the corresponding matricial ring M2 (Ck ) over the ring Ck is well defined, for all k ∈ N. We consider certain morphisms, {trk : M2 (Ck ) → Ck }k∈N , to establish free-probabilistic structures. 7.1

The Classical Clifford Algebras {Ck }k∈N

In this section, we construct the classical Clifford algebras {Ck }k∈N as algebras over R as in Ref. [2]. The construction of them is done recursively, starting from C0 = R. So, the initial algebra C0 is the 1-dimensional R-algebra, isomorphic to R (over R). We let def

C1 = C0 + C0 e1 , with e21 = −1,

where C0 + C0 e1 means the algebra generated by {C0 , C0 e1 }, with its addition (+) defined by (x1 + x2 e1 ) + (y1 + y2 e1 ) = (x1 + y1 ) + (y1 + y2 ) e1 , and its multiplication (·) defined by (x1 + x2 e1 ) (y1 + y2 e1 ) = (x1 y1 − x2 y2 ) + (x1 y2 + x2 y1 ) e1 , for all xl , yl ∈ C0 = R, for all l = 1, 2. Then, by considering all real numbers x ∈ R = C0 , as x + 0e1 , the R-scalar product, (x, z) ∈ R × C1 7−→ xz ∈ C1 is well defined on C1 . Thus, C1 forms a 2-dimensional algebra over R, having its R-basis, E1 = {1, e1 } (by considering the algebra C1 as a vector space over R). Similarly, define an algebra, C2 = C1 + C1 e2 ,

40 ■ Classical Clifford Algebras

with e22 = −1 and e1 e2 = −e2 e1 , having its addition, (z1 + z2 e2 ) + (w1 + w2 e2 ) = (z1 + w1 ) + (z2 + w2 ) e2 , and its multiplication, (z1 + z2 e2 ) (w1 + w2 e2 ) = (z1 w1 − z2 w2 ) + (z1 w2 + w1 z2 ) e2 , for all zl , wl ∈ C1 , for all l = 1, 2. Then, by regarding x ∈ R as x + 0e1 + 0e2 + 0e1 e2 ∈ C2 (because R = C0 ⊂ C1 ), one has a well-defined R-scalar product, (x, a) ∈ R × C2 7−→ xa ∈ C2 , and hence, C2 is a well-defined 4-dimensional algebra over R, having its R-basis, E2 = {1, e1 , e2 , e1 e2 } (by regarding the algebra C2 as a vector space over R). From C2 , define an algebra, C3 = C2 + C2 e3 ,

with

(7.1.1) e23 = −1, and ei ej = −ej ei , ∀i ̸= j ∈ {1, 2, 3},

having its R-basis, E3 = {1, e1 , e2 , e3 , e1 e2 , e1 e3 , e2 e3 , e1 e2 e3 }, under Eq. 7.1.1. So, every element of C3 is expressed by X tw w ∈ C3 , with tw ∈ R. w∈E3

Then, it has its addition, ! X tw w + w∈E3

! X w∈E3

sw w

=

X w∈E3

(tw + sw ) w

On Classical Clifford Algebras ■ 41

and

!

! X

X

tw w

identified with ! X tw w

sw w

tw1 sw2 (w1 w2 ),

(w1 ,w2 )∈E32



! X

X

=

w∈E3

w∈E3

w∈E3

sw w

=

w∈E3

 X

X  w∈E3

tw1 sw2  w,

(w1 ,w2 )∈E3 , w=w1 w2

in C3 . So, it becomes an 8-dimensional algebra over R. Inductively, for k ∈ N, define C0 = R, and def

Ck = Ck−1 + Ck−1 ek ,

with e2i = −1, for all i = 1, ..., k and

(7.1.2) ei ej = −ej ei , ∀i ̸= j ∈ {1, ..., k},

for all k ∈ N, having its R-basis (by considering the algebra Ck as a vector space over R), Ek = {1} ∪ Ik , with

(7.1.3) k

Ik = ∪

 l Y

l=1  j=1

ekj

  : k1 < k2 < ... < kl , 

under Eq. 7.1.2, whose cardinality is 2k , i.e., |Ek | = 2k , for all k ∈ N. It has its addition, ! ! X X X tw w + sw w = (tw + sw ) w w∈Ek

w∈Ek

w∈E3

and ! X w∈Ek

tw w



! X w∈Ek

sw w

=

X

 X

 w∈Ek

(w1 ,w2 )∈Ek , w=w1 w2

tw1 sw2  w,

42 ■ Classical Clifford Algebras

where tw , sw ∈ R, under the relation (Eq. 7.1.2) on the R-basis Ek of Eq. 7.1.3. Let w = ek1 ...ekn ∈ Ik , for 1 ≤ k1 < ... < kn ≤ k. Then, the length l (w) of w is defined to be n. We axiomatize the length l (1) of 1 ∈ Ek to be 0. Definition 33. The algebra Ck of (7.1.1) is called the k-th (classical) Clifford algebra (over R), for all k ∈ N0 . By the definition (Eq. 7.1.1), one has the following isomorphic relations: iso

iso

iso

iso

C1 = C, C2 = H, C3 = M2 (C), C4 = M4 (R), iso

as R-algebras (e.g., see [2]), where “ =” means “being Ralgebraisomorphic to,” and H is the quaternions of Chapter 1. By definition, one has an increading chain, R = C0 ⊂ C1 ⊂ C2 ⊂ C3 ⊂ · · · , of the Clifford algebras {Ck }k∈N , where “⊆” is the “subalgebra inclusion” over R. Lemma 34. The k-th Clifford algebra Ck is a 2k -dimensional Banach algebra over R (or, R-Banach algebra) with its unity (or the multiplication identity), X 1=1+ 0 · w ∈ Ck . w∈Ik

Proof. For any k ∈ N, the corresponding k-th Clifford algebra Ck is a 2k -dimensional algebra over R, with its R-basis,   k Ek = {1} ∪ ∪ {ek1 ek2 ...ekl : k1 < k2 < ... < kl } , l=1

by Eqs. 7.1.2 and 7.1.3. By the finite dimensionality of Ck , all norms on Ck are equivalent and complete. So, this R-algebra Ck forms a RBanach algebra. Moreover, this R-Banach algebra Ck contains 1 ∈ Ek , satisfying 1 · η = η = η · 1, ∀η ∈ Ck , i.e., 1 ∈ Ek is the unity of Ck .

On Classical Clifford Algebras ■ 43

Now, observe that if w = ek1 ...ekn ∈ Ik , with its length-n, with 1 ≤ k1 < ... < kn ≤ k. If n = 1, then w2 = e2k1 = −1, by Eq. 7.1.2, and meanwhile, if n > 1, then 2

w2 = (ek1 ...ekn ) = (ek1 ...ekn ) (ek1 ...ekn ) = ek1 ...ekn−1 (ekn ek1 ) ek2 ...ekn = (−1) ek1 ...ekn−1 (ek1 ekn ) ek2 ...ekn

= (−1) ek1 ... ekn−1 ek1 ekn ek2 ...ekn

2 = (−1) ek1 ... ek1 ekn−1 ekn ek2 ...ekn
n−1 = ... = (−1) e2k1 ek2 ... (ekn ek2 ) ...ekn
n−1 n−2 e2k1 e2k2 ... (ekn ek3 ) ...ekn = ... = (−1) (−1)  
n−1 n−2 1 = ... = (−1) (−1) ... (−1) e2k1 e2k2 ...e2kn   = (−1)

(n−1)((n−1)+1) 2

(−1) (−1) ... (−1) | {z } n-times

= (−1) = (−1)

n(n−1) 2 n2 +n 2

n

(−1) = (−1)

= (−1)

n(n+1) 2

n(n−1) +n 2

= (−1)

n2 −n+2n 2

.

(7.1.4)

Note that Eq. 7.1.4 covers the case where w = 1 in Ek , with its length l (1) = 0, indeed, 0

12 = 1 = (−1) = (−1)

0(0+1) 2

.

Thus, one can obtain the following result. Lemma 35. Let w ∈ E be a R-basis element of Ck , with its length l (w) ∈ N0 = N ∪ {0}. Then w2 = (−1)

l(w)(l(w)+1) 2

, in Ck .

(7.1.5) 0(0+1)

Proof. Suppose w = 1 ∈ Ek with l (1) = 0. Then 12 = 1 = (−1) 2 . Therefore, the relation (Eq. 7.1.5) holds. Meanwhile, if w ∈ Ik = Ek \ {1} in Ck , then w2 = (−1)

l(w)(l(w)+1) 2

∈ {±1}, in Ck ,

by Eq. 7.1.5, implying the relation (Eq. 7.1.5).




44 ■ Classical Clifford Algebras

Now, as in complex and quaternion analyses, we define a bijective function (†) on Ck , for an arbitrarily fixed k ∈ N, by (†) : Ck → Ck , satisfying

(7.1.6) † (η)

denote

=

def

η † = t1 −

X

tw w,

w∈Ik

for all η=

X

tw w = t1 +

w∈Ek

X

tw w ∈ Ck , with tw ∈ R,

w∈Ik

where Ek = {1} ∪ Ik is in the sense of Eq. 7.1.3. By the definition (Eq. 7.1.6), one has that: 1 ∈ Ek \Ik satisfies 1† = 1; meanwhile, w ∈ Ik , if and only if w† = −w in Ck . We call this function (†), the conjugate on Ck . In particular, if t = 1, 2, then this function (†) forms the adjoint on C1 and C2 , since it is the C-conjugate on C1 , respectively, the Hconjugate on C2 (over the complex field C), satisfying
η †† = η, zη † = zη † and †

†

(η 1 + η 2 ) = η †1 + η †2 , and (η 1 η 2 ) = η †2 η †1 , for all η 1 , η 2 ∈ C1 ∪ C2 = C2 , and z ∈ C (e.g., see Chapter 1). However, if k ≥ 3 in N, then (†) of Eq. 7.1.6 is simply a bijection, which is “not” an adjoint, on Ck . For instance, if w1 = e1 e3 , and w2 = e2 e4 ∈ C5 , then w1 w2 = (e1 e3 ) (e2 e4 ) = e1 (e3 e2 ) e4 = −e1 e2 e3 e4 , by Eq. 7.1.2, and hence, †

(w1 w2 ) = − (−e1 e2 e3 e4 ) = e1 e2 e3 e4 , by Eq. 7.1.6; meanwhile, w2† = −e2 e4 , and w1† = −e1 e3 ,

On Classical Clifford Algebras ■ 45

by Eq. 7.1.6, and hence, w2† w1† = (−e2 e4 ) (−e1 e3 ) = e2 e4 e1 e3 = e1 e2 e4 e3 = −e1 e2 e3 e4 , by Eq. 7.1.2, showing that †

(w1 w2 ) = e1 e2 e3 e4 ̸= −e1 e2 e3 e4 = w2† w1† , in C5 . It demonstrates that the conjugate (†) of Eq. 7.1.6 is not an adjoint on C5 . Readers have to keep in mind that the conjugate (†) is simply a bijection defined by Eq. 7.1.6, which is not an adjoint, in general. 7.2

The Clifford Algebra C

By definition, the Clifford algebras {Ck }k∈N satisfy the subalgebra chain, R = C0 ⊂ C1 ⊂ C2 ⊂ C3 ⊂ ...,

(7.2.1)

as R-Banach algebras. So, one can construct the “enveloping” or the “limit” Banach algebra C over R of this chain, i.e., C

denote

=

lim − → Ck = the limit Banach algebra over R,

i.e.,

(7.2.2) def

C = R



 ∪ Cn ,

n∈N

where R [X] means the R-polynomial algebra in a set X, and Y means the completion of a topological space Y . Theorem 36. Let Ca be a vector space over R generated by E = ∪ Ek = {1} ∪ I, with I = ∪ Ik , k∈N

k∈N

as a vector space over R, where Ek = {1} ∪ Ik are the R-basis of the k-th Clifford algebras Ck , for all k ∈ N, equipped with its addition, X X X ηk + ξk = (η k + ξ k ), k∈N

k∈N

k∈N

46 ■ Classical Clifford Algebras

and its multiplication, !

! X

X

ηk

ξk

k∈N

k∈N

X

=

η k1 ξ k2 ,

(k1 ,k2 )∈N2

with the norm,

) ( k

X

X

def



η k = lim η l : k ∈ N ,

k→∞ k∈N

for all

P

ηk ,

k∈N

P

l=1

k

ξ k ∈ C , for η k , ξ k ∈ Ck , where ∥.∥k are the Banach-

k∈N

space norms on Ck , for all k ∈ N. Then, the ∥.∥-completion Ca Ca is the limit R-Banach algebra of the chain (Eq. 7.2.1), i.e., C =− lim → Ck = Ca

∥.∥

.

∥.∥

of

(7.2.3)

Proof. By the definition (Eq. 7.2.2) of the enveloping or R the limit  Banach algebra C , this algebra C = Ca , where Ca = R ∪ Cn is the n∈N

polynomial algebra in ∪ Cn . So, every element η ∈ Ca is a finite sum, n∈N

X

η=

η n with η n ∈ Cn ,

n∈N

 generated by the generator set ∪ En = {1} ∪



∪ In . Since each Ck

n∈N

n∈N

is 2k -dimensional R-Banach algebra, it has a well-defined norm ∥.∥k on it, which is equivalent to all norms on Ck by the finite dimensionality. So, the norm ∥.∥ on Ca , defined by

n

X

X



η n = lim η l ,

n→∞

n∈N

l=1

n

is well determined on Ca , i.e., (Ca , ∥.∥) forms a normed space. By the completeness of {∥.∥n }n∈N , one can take the completion Ca (Ca , ∥.∥), which is a R-Banach space. Therefore, Ca

∥.∥

=C =− lim → Ck .

∥.∥

of

On Classical Clifford Algebras ■ 47

Indeed, all R-Banach algebras {Ck }k∈N are subalgebras of C , by the injective homomorphism, equivalently, the embedding, ϵk : Ck → C , defined by ! ϵk

X

tw w

w∈Ek

! def

=

X

tw w

w∈Ek

 +

 X

0 · w,

w∈E\Ek

for all k ∈ N. So, the R-Banach algebra C is the boundary (or, the limit) of the chain (Eq. 7.2.1). The above theorem shows that there exists the R-Banach algebra ∥.∥ as the limit R-Banach algebra of the chain (Eq. 7.2.1) of C = Ca the Clifford algebras {Ck }k∈N . Observation. All mathematical properties on {Ck }k∈N are preserved to those on C . For instance, all results of Section 7.1 hold on C . 2 Definition 37. The limit R-Banach algebra C of the chain (Eq. 7.2.1) of Clifford algebras {Ck }k∈N , in the sense of Eq. 7.2.3, is called the Clifford (R-Banach-) algebra. Let C be the Clifford algebra, and E = {1}∪I, the R-basis of C . Up to the multiplication on C , let’s construct an embedded multiplicative group G in C by the presented group, G = ⟨X |R ⟩ ,

with the generator set, X = {ek }k∈N ⊂ I, the length-1 basis elements, and the relation,

(7.2.4)

R : e2k = −1, ∀k ∈ N; ek1 ek2 = −ek2 ek1 , ∀k1 , k2 ∈ N. One can easily verify that this presented group G is the maximal multiplicative group induced by the length-1 basis elements embedded in the Clifford algebra C .

48 ■ Classical Clifford Algebras

Theorem 38. Set-theoretically, the group G of Eq. 7.2.4 is identical to ±E, i.e., set

G = {±w : w ∈ E}

denote

=

±E.

(7.2.5)

Proof. This group G of Eq. 7.2.4 contains its group identity 1 ∈ G ,
2 2 since e4k = e2k = (−1) = 1 in G , satisfying 1 · g = g = g · 1, ∀g ∈ G , because G ⊂ C and 1 is the unity of C . If w = 1 in E, and ±1 ∈ G , then 2 −1 (±1) = 1 =⇒ 1−1 = 1 and (−1) = −1, in G , where g −1 means the inverse of g ∈ G . Since −1 = e2k ∈ G , for all k ∈ N, ±E = {±w : w ∈ E} ⊆ G , since G is generated by X = {ek }k∈N , inducing E under the relation R (in C ). For any w ∈ I, observe that w† w = (−w) w = −w2 = − (−1)

l(w)(l(w)+1) 2

= ww† ,

(7.2.6)

by Eqs. 7.1.5 and 7.1.6, where l (w) is the length of w, identical to w† w = (−1)

l(w)(l(w)+1) +1 2

= ww† ,

showing that

†

†

w w = ww =

   −1

if

l(w)(l(w)+1) 2

∈ 2N

  1

if

l(w)(l(w)+1) 2

∈ / 2N,

and equivalently, w† w = ww† =

  −1 

It shows that

1

if l (w) (l (w) + 1) ∈ 4N otherwise.

On Classical Clifford Algebras ■ 49

w

−1

=

  −w† = − (−w) = w 

if l (w) (l (w) + 1) ∈ 4N (7.2.7)

w† = −w

otherwise,

for all w ∈ I. By Eq. 7.2.7, one has G ⊆ ±E.

Therefore, G = ±E, set-theoretically. By the set equality (Eq. 7.2.5) of the group G of Eq. 7.2.4, we alternatively understand G as ±E, from below. From the proof of the above theorem, one obtains the following inverse property on G . Corollary 39. Let G be the group (Eq. 7.2.4) induced by the Clifford algebra C , and g ∈ G . If g = sg w ∈ G , with sg ∈ {±1} and w ∈ E, then −1 w = 1 =⇒ g −1 = (sg · 1) = sg = g and

(7.2.8) w ∈ I =⇒ g

−1

=

  g 

if l (w) (l (w) + 1) ∈ 4N

−g

otherwise.

Proof. Clearly, if g = ±1, then g −1 = ±1 = g in G . Now, let g = sg w ∈ G , for w ∈ I, where sg ∈ {±1}, by Eq. 7.2.5. By considering g = sg w ∈ G as an element of the Clifford algebra C , we have that †

g † g = (sg w) (sg w) = −s2g w2 = −w2 = w† w and †

gg † = (sg w) (sg w) = −s2g w2 = −w2 = ww† , since s2g = 1, by Eq. 7.2.6. Therefore, we have g † g = w† w = (−1)

l(w)(l(w)+1) +1 2

= ww† = gg † ,

implying that g

−1

= (sg w)

−1

= sg w

−1

=

 

sg w = g

if l (w) (l (w) + 1) ∈ 4N



sg (−w) = −g

otherwise,

50 ■ Classical Clifford Algebras

by Eq. 7.2.7. Indeed, if l (w) (l (w) + 1) ∈ 4N, then gg = s2g w2 = (−1)

l(w)(l(w)+1) 2

= 1;

meanwhile, if l (w) (l (w) + 1) ∈ / 4N, then g (−g) = −s2g w2 = − (−1)

l(w)(l(w)+1) 2

= − (−1) = 1 = (−g) g,

in G . Therefore, the inverse characterization (Eq. 7.2.8) holds on G . The above equation (Eq. 7.2.8) on the group G fully characterizes the group inverses in G by the length of R-basis elements of E. One can see that there are many self-invertible elements in G . Definition 40. The group G of Eq. 7.2.4 induced by the Clifford algebra C is called the Clifford (multiplicative) group. 7.3

The Group-Hilbert Space H of the Clifford Group G

In this section, we construct the group-Hilbert space H of the Clifford group G of Eq. 7.2.4 and then understand each group element of G as a Hilbert-space operator acting on H . Let G be a discrete group with its group identity e. Then, the groupHilbert space HG = l2 (G) is defined by a Hilbert space equipped with its orthonormal basis, BG = {hg : g ∈ G}, satisfying ⟨hg1 , hg2 ⟩2 = δ g1 ,g2 , the Kronecker delta and ∥hg ∥2 =

q

⟨hg , hg ⟩2 = 1,

for all g1 , g2 , g ∈ G, where ⟨, ⟩2 is the l2 -inner product on HG , *

+ X

g∈G

tg hg ,

X g∈G

def

sg hg

=

2

X g∈G

tg sg ,

On Classical Clifford Algebras ■ 51

and ∥.∥2 is the l2 -Hilbert-norm induced by ⟨, ⟩2 ,

v* +

X

u X sX X

u 2

t t h = t h , t h = |tg | , g g g g g g

g∈G g∈G g∈G g∈G 2

2

for all

P g∈G

tg hg ,

P

sg hg ∈ HG , with tg , sg ∈ C, where sg is the

g∈G

conjugate of sg , and |tg | is the modulus of tg in C, i.e.,   tg ∈ C, hg ∈ B, ∀g ∈ G   X  and HG = tg hg , P 2   |tg | < ∞ g∈G  g∈G

set-theoretically. Remark that such a Hilbert space HG has the identity denote vector 1 = he induced by the group identity e of G, satisfying 1h = h = h1, ∀h ∈ HG . Indeed, if g1 , g2 ∈ G, then hg1 hg2 = hg1 g2 , in B ∪ {1 = he }, in HG . If we consider the operator algebra B (HG ) of all (bounded linear) operators on the Hilbert space HG , equipped by the operator norm ∥.∥, defined by ∥T ∥ = sup {∥T h∥2 : ∥h∥2 = 1}, then it forms a well-defined C ∗ -algebra over the complex field C (e.g., [7]). By the very construction, the given group G acts on the Hilbert space HG in the sense that there is a group action, m : g ∈ G 7−→ mg ∈ B (HG ), where X

mg (h) = hg h, ∀h =

tw hw ∈ HG ,

w∈G

i.e.,

(7.3.1) mg (h) =

X w∈G

tw hg hw =

X w∈G

tw hgw ∈ HG .

52 ■ Classical Clifford Algebras

In other words, the group action m assigns each group element g to the multiplication operator mg ∈ B (HG ) acting on HG , satisfying ∥mg ∥ = ∥hg ∥2 = 1, ∀g ∈ G. If we construct a subset, M = {mg ∈ B (HG ) : g ∈ G}, then one can construct the C ∗ -subalgebra MG of B (HG ), by MG = C ∗ (M ) = C [M ], in B (HG ),

(7.3.2)

where C ∗ (Z) means the C ∗ -subalgebra of B (HG ) generated by a subset Z of B (HG ), and C[Y ] means the polynomial algebra of a subset Y of B (HG ) over C, and X is the operator-norm completion of the subset X of B (HG ). Such a C ∗ -sibalgebra MG of B (HG ) is called the group C ∗ -algebra of G. By the multiplication rule on the group-Hilbert space HG , elements of MG satisfy that mg1 mg2 = mg1 g2 in MG , ∀g1 , g2 ∈ G and

(7.3.3) m∗g = mg−1 in MG , ∀g ∈ G.

So, from our Clifford group G , one can construct the group-Hilbert space, H = l2 (G ) = l2 (⟨X |R ⟩), with its orthonormal basis B = {hg : g ∈ G },

satisfying ⟨hg1 , hg2 ⟩2 = δ g1 ,g2 , ∀g1 , g2 ∈ G and ∥hg ∥2 =

q

⟨hg , hg ⟩2 = 1, ∀g ∈ G ,

with the multiplication rule, hg1 hg2 = hg1 g2 , ∀g1 , g2 ∈ G ,

On Classical Clifford Algebras ■ 53

in H . On this group-Hilbert space H , one can have the corresponding operator algebra B (H ) of all operators acting on H , which is a C ∗ algebra under its operator norm. And, as a C ∗ -subalgebra of B (H ), the group C ∗ -algebra MG of the Clifford group G is well determined as in Eq. 7.3.2, generated by all multiplication operators, {mg ∈ B (H ) : g ∈ G }, of Eq. 7.3.1, i.e.,

(7.3.4) MG = C ∗ (m (G )) ⊂ B (H ) .

Definition 41. The group C ∗ -algebra MG of Eq. 7.3.4, which is a C ∗ -subalgebra of the operator algebra B (H ) of the group-Hilbert space H = l2 (G ) of the Clifford group G , is called the Clifford-group C ∗ -algebra (in short, the CG algebra) of G . Note that the CG algebra MG and the Clifford algebra C are totally different operator-algebraic structures. As we considered earlier, the Clifford algebra C is a Banach algebra over the real field R, but the newly defined CG algebra MG is a C ∗ -algebra over the complex field C. Indeed, if we take w = ek1 ...ekn ∈ E, with its length-n, in C , and z = a + bi ∈ C = C1 with a, b ∈ R = C 0 , then zw = (a + bi) w = (a + be1 ) (ek1 ...ekn ), identical to aek1 ...ekn + be1 ek1 ...ekn , in the Clifford algebra C ; however, zw = (a + bi) w

denote

=

(a + bi) mw ,

in the CG algebra MG by regarding w ∈ E as a group element of set ±E = G . So, we need to keep in mind that C and MG are structurally different operator-algebraic structures. Note also that, by the general rule (Eq. 7.3.3) on group C ∗ -algebras, we have mg1 mg2 = mg1 g2 ∈ MG , ∀g1 , g2 ∈ G

54 ■ Classical Clifford Algebras

and

(7.3.5) m∗g = mg−1 ∈ MG , ∀g ∈ G ,

where 1−1 = 1, and (−1)

−1

= −1

and g

−1

=

  g 

if l (w) (l (w) + 1) ∈ 4N

−g

otherwise,

for all g ∈ G = ±E. Now, let G be an arbitrary discrete group, and MG , the group C ∗ algebra of G acting on the group-Hilbert space HG . By definition, every element T of MG is expressed by T =

X

tg mg with tg ∈ C.

g∈G

The C-quantities tg from the operator T ∈ MG are called the g-th coefficients of T , for all g ∈ G. Remark that there exists a canonical trace trG , which is a linear functional on MG defined by ! P P def trG tg mg = te , ∀ tg mg ∈ MG , (7.3.6) g∈G

g∈G

satisfying trG (T1 T2 ) = trG (T2 T1 ), ∀T1 , T2 ∈ MG . Thus, the pair (MG , trG ) forms a well-defined C ∗ -probability space (e.g., see Section 5.1). Such a C ∗ -probability space (MG , trG ) is called the (canonical-tracial-)group C ∗ -probability space of G. So, if MG is the CG algebra of the Clifford group G , then one can have the corresponding group C ∗ -probability space (MG , τ ), where τ = trG is the canonical trace on MG . Definition 42. The group C ∗ -probability space (MG , τ ) of the Clifford group G is called the Clifford-group C ∗ -probability space (in short, CG C ∗ -probability space) of G .

On Classical Clifford Algebras ■ 55

Notation and Assumption. From below, for convenience, if T = P tg mg ∈ (MG , τ ), then we denote it by

g∈G

T =

X

tg g in MG ,

g∈G

i.e., we denote the multiplication operators mg generating the CG algebra MG simply by g, if there are no confusions. 2 By definition,  τ

 X

tg g  = t1 , the 1-th coefficient,

g∈G

since 1 = 1 +

P

0 · w ∈ C is the group identity of the Clifford group

w∈E

G . In the rest of this section, we study basic free-distributional data on the CG C ∗ -probability space (MG , τ ). denote

Theorem 43. Let w ∈ E in the Clifford group G , and let w = mw ∈ (MG , τ ). If w = 1 ∈ G , then the free distribution of 1 is the 1-universal distribution, fully characterized by the free-moment sequence, ∞

(τ (1n ))n=1 = (1, 1, 1, 1, 1, 1, ...) . If w ∈ I, then the free distribution of w is fully characterized by the joint free moments,   τ (wr1 wr2 ...wrn ) = τ wn−2#(∗) , with #(∗) = the number of ∗ ’s in (r1 , ..., rn ), and

(7.3.7) ∞

(τ (wn ))n=1


= 0, u, 0, 1, 0, u, 0, 1, 0, u, 0, 1, ... ,

with u = (−1) n

l(w)(l(w)+1) 2

∈ {±1},

for all (r1 , ..., rn ) ∈ {1, ∗} , for all n ∈ N, where the expression “0, u, 0, 1” in Eq. 7.3.7 means the first four entries repeatedly or periodically appeared in the sequence.

56 ■ Classical Clifford Algebras

Proof. Trivially, if w = 1 ∈ E in G = ±E, then it is the group identity denote of G , inducing the unity 1 = m1 of the CG algebra MG , which is self-adjoint in MG because 1∗ = 1−1 = 1 in MG , by Eq. 7.3.5. By the self-adjointness of 1 in MG , the free distribution of 1 is characterized by the free-moment sequence, ∞

(τ (1n ))n=1 = (1, 1, 1, 1, 1, ...), by Eq. 7.3.6. Meanwhile, if w ∈ I be a R-basis element of C with its length-N in G , then w2 = (−1)

N (N +1) 2

∈ ±E= G ,

by Eq. 7.1.5. It shows that w3 = w2 w = (−1)

N (N +1) 2

w,

2  N (N +1) , w4 = w3 w = (−1) 2  2 N (N +1) w5 = w4 w = (−1) 2 w, and

3  N (N +1) , w6 = w5 w = (−1) 2

in G . Inductively,    n2 N (N +1)  2  (−1)   n w =    n−1    (−1) N (N2+1) 2 w

if n ∈ 2N (7.3.8) if n ∈ 2N − 1,

in G , for all n ∈ N. By Eqs. 7.3.6 and 7.3.8, one has    n2 N (N +1)   (−1) 2 if n ∈ 2N τ (wn ) =   0 if n ∈ 2N − 1, for all n ∈ N, i.e.,
∞ (τ (wn ))n=1 = 0, u, 0, 1, 0, u, 0, 1, 0, u, 0, 1, ... ,

On Classical Clifford Algebras ■ 57

with

(7.3.9) u = (−1)

N (N +1) 2

∈ {±1} ,

because u2 = 1 (since u ∈ {±1}), u3 = u2 u = u, and u4 = u3 u = u2 = 1, inductively  u if k ∈ 2N − 1 k u = 1 if k ∈ 2N, for all k ∈ N, where the expression “0, u, 0, 1” in Eq. 7.3.9 means the first four entries repeatedly or periodically appeared in the sequence. n Now, let (r1 , ..., rn ) ∈ {1, ∗} , for n ∈ N, and let W

denote

=

wr1 wr2 ...wrm ∈ ±E = G .

Then, one can verify that W = wn−2#(∗) ∈ G , where

(7.3.10) #(∗) = the number of ∗ ’s in (r1 , ..., rn ),

since w∗ = m∗w = mw−1 = w−1 by Eq. 7.3.5. (For instance, w−1 wwww−1 w = w2 = w6−2(2) .) Therefore,   τ (wr1 wr2 ...wrn ) = τ wn−2#(∗) , by Eq. 7.3.10, where
∞ (τ (wn ))n=1 = 0, u, 0, 1, 0, u, 0, 1, 0, u, 0, 1, ... , with u = (−1)

N (N +1) 2

n

, for all (r1 , ..., rn ) ∈ {1, ∗} , for all n ∈ N.

The following corollary is obtained by the above theorem as a special case. Corollary 44. Let w ∈ I in G = ±E, regarded as the free random variable w = mw ∈ MG in the CG C ∗ -probability space (MG , τ ). If w is self-adjoint in MG , then the free distribution of g is characterized by the free-moment sequence,
∞ (τ (g n ))n=1 = 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ... . (7.3.11)

58 ■ Classical Clifford Algebras

Proof. Under hypothesis, assume that w is self-adjoint in MG . Then, w∗ = w in MG ⇐⇒ w−1 = w in G , by Eq. 7.3.5, where w

−1

=

  w 

if l (w) (l (w) + 1) ∈ 4N

−w

otherwise,

in the Clifford group G , by Eq. 7.2.8. So, w∗ = w in MG ⇐⇒ l (w) (l (w) + 1) ∈ 4N. By the self-adjointness of w in MG , the free distribution of w is fully characterized by the free-moment sequence,
∞ (τ (wn ))n=1 = 0, u, 0, 1, 0, u, 0, 1, ... , with u = (−1)

l(w)(l(w)+1) 2

, by Eq. 7.3.7. However,

l (w) (l (w) + 1) ∈ 4N ⇐⇒

l (w) (l (w) + 1) ∈ 2N, 2

and hence, u = 1 in {±1}. Thus, the free-distributional data (Eq. 7.3.11) holds.

8

FREE PROBABILITY ON THE CLIFFORD-GROUP C ∗ -PROBABILITY SPACE (MG , τ )

In this section, we study more free-distributional data on the CG C ∗ -probability space (MG , τ ) of the Clifford group, + * e2k = −1, ∀k ∈ N, and , G = {ek }k∈N ek ek = −ek ek , ∀k1 ̸= k2 ∈ N 1 2 2 1 embedded in the Clifford algebra C . In Section 7.3, we showed that if w = 1 ∈ E in G = ±E, then the free distribution of 1 is the 1-universal distribution; meanwhile, if w ∈ I in G , then the free distribution of w = mw ∈ MG is characterized by the joint free moments, ! n   Y τ wrl = τ wn−2#(∗) , l=1

On Classical Clifford Algebras ■ 59

where #(∗) = the number of ∗ ’s in (r1 , ..., rn ) and

(8.1)
∞ (τ (wn ))n=1 = 0, u, 0, 1, 0, u, 0, 1, 0, u, 0, 1, ... ,

with u = (−1)

l(w)(l(w)+1) 2

∈ {±1},

n

in (MG , τ ), for all (r1 , ..., rn ) ∈ {1, ∗} , for all n ∈ N, because g∗

denote

=

m∗g = mg−1

denote

=

g −1 , ∀g ∈ G ,

in MG , i.e., the adjoint of all generating group elements of G on MG are identified with the group inverses in G . In particular, the operator w ∈ I is self-adjoint in MG , if and only if the free distribution of w is characterized by
∞ (τ (wn ))n=1 = 0, 1, 0, 1, 0, 1, 0, 1, ... . Now, let’s consider a free random variable, η = ek1 + ek2

denote

=

mek1 + mek2 ∈ (MG , τ ),

where ek1 and ek2 are the length-1 R-basis elements in E ⊂ G , satisfying k1 ̸= k2 in N. Then, η 2 = e2k1 + (ek1 ek2 + ek2 ek1 ) + e2k2 = (−1) + 0 + (−1) = −2, in MG , since e2k1 = −1 = e2k2 , and ek1 ek2 = −ek2 ek1 in MG . So,

η 3 = η 2 η = (−2) η = −2η, 2

η 4 = η 3 η = −2η 2 = (−2) , 2

η 5 = η 4 η = (−2) η, 2

(8.2) 3

η 6 = η 5 η = (−2) η 2 = (−2) , etc., in MG . Thus, by the induction on Eq. 8.2, we have

60 ■ Classical Clifford Algebras

ηl =

 l−1   (−2) 2 η  

(−2)

if l ∈ 2N − 1 (8.3)

l 2

if l ∈ 2N,

in MG , for all l ∈ N. Motivated by Eq. 8.3, we consider more general cases. Let η=

m X

ekj ∈ MG , with 1 ≤ m ≤ k + 1,

j=1

where ek1 , ..., ekm ∈ I in G are the mutually distinct elements with their length-1 in Ck+1 . Then η2 =

m P j=1

=

m P

e2kj +

P j1 ̸=j2 ∈{1,...,m}

ekj1 ekj2 !

P

(−1) +

j=1

kj1 kj2

ekj1 ekj2 !

= (−m) +

P kj1