C*-Algebras and Their Automorphism Groups

By Academic Press

This elegantly edited landmark edition of Gert Kjærgård Pedersen’s C*-Algebras and their Automorphism Groups (1979) carefully and sensitively extends the classic work to reflect the wealth of relevant novel results revealed over the past forty years. Revered from publication for its writing clarity and extremely elegant presentation of a vast space within operator algebras, Pedersen’s monograph is notable for reviewing partially ordered vector spaces and group automorphisms in unusual detail, and by strict intention releasing the C*-algebras from the yoke of representations as Hilbert space operators. Under the editorship of Søren Eilers and Dorte Olesen, the second edition modernizes Pedersen’s work for a new generation of C*-algebraists, with voluminous new commentary, all-new indexes, annotation and terminology annexes, and a surfeit of new discussion of applications and of the author’s later work.

• Covers basic C*-algebras theory in a short and appealingly elegant way, with a few additions and corrections given to the editors by the original author
• Expands coverage to select contemporary accomplishments in C*-algebras of direct relevance to the scope of the first edition, including aspects of K-theory and set theory
• Identifies key modern literature in an updated bibliography with over 100 new entries, and greatly enhances indexing throughout
• Modernizes coverage of algebraic problems in relation to the theory of unitary representations of locally compact groups
• Reviews mathematical accomplishments of Gert K. Pedersen in comments and a biography

• Language: English
• Publisher: Academic Press
• Release date: Aug 8, 2018
• ISBN: 9780128141236

Book preview

art.

Author's Preface

Gert Kjærgård Pedersen     

-algebras is the study of operators on a Hilbert space with algebraic methods. The motivating example is the spectral theorem for a normal operator (which, in effect, is nothing but the Gelfand transformation applied to the algebra generated by an operator). The applications of the theory range from group representations to model quantum field theory and quantum statistical mechanics.

-algebrae.

At the end of each section a few remarks are inserted with references to the bibliography. The intention is to give the reader a rough idea of the development of the subject. Such personal comments are bound to contain errors, and the author humbly asks forgiveness from the mathematicians who have undeservedly not been mentioned.

Many people were important for the completion of this book: Richard Kadison, whose work has been a constant source of inspiration for me; Daniel Kastler, who provided shelter and a two-month raincurtain when the work was begun in 1974; colleagues, who shouldered my teaching load while I was writing; and students at the University of Copenhagen, who were exposed to the first wildly incorrect drafts. It is a pleasure to record my thanks to all of them.

Copenhagen

August 1978

Editors' Preface

Søren Eilers; Dorte Olesen     

-algebraist, along with its fellow contemporary classics -algebraists.

code from a scan of the original), we find the book simultaneously eternally youthful and showing its age. Gert Pedersen's elegant style and careful choice of notation holds up, and we have only found it necessary to change the name of the Pedersen ideal as it conflicts with modern use from K– which nowadays are not omnipresent in the literature – easier to parse. Of course, the many exciting developments since 1978 in the area covered by the book make it desirable to update and complement the original.

To try to maximize the value added to the book for a modern user in the limited space available, we have prioritized as follows:

(i)  Answers to open problems explicitly mentioned in the first edition;

(ii)  Reports of and references to new developments of direct importance for the material in the first edition;

(iii)  Insights into Gert Pedersen's later work.

-algebras since 1978, and in particular we have not been able to include any material on K-theory nor on any of the great strides taken in the von Neumann setting. We also follow the original in affording issues concerning nuclearity and exactness the absolute minima of attention. We recommend all of the modern textbooks [62,27,26,88,342,258] for an introduction to these developments. Also see [283] for further details of the life and works of the author.

The original material is presented as in the first edition, although we have corrected errata known to the author and to the many colleagues who have assisted us in the preparation of this book. In the few instances when the corrections necessary are mathematically significant, we have recorded this at the end of the relevant sections. Desiring to preserve the numbering of all the original results to avoid confusion between the two editions, all added material is placed at the end of the original chapters and/or sections. Such boundary conditions have forced us to deviate from the strict linear order of the original and sometimes employ forward references. There is of course no circularity arising from this unfortunate fact.

It is our great pleasure to record our gratitude to Chuck Akemann, Joel Anderson, Tristan Bice, Nate Brown, Toke Carlsen, Erik Christensen, Marius Dadarlat, George Elliott, Ilijas Farah, Takeshi Katsura, Akitaka Kishimoto, Bartosz Kwaśniewski, Nadia Larsen, Ryszard Nest, Costel Peligrad, Mikael Rørdam, Yasuhiko Sato, Aidan Sims, Masamichi Takesaki, Stuart White, and John Maitland Wright for their invaluable help in this process.

Copenhagen

June 2018

References

[26] B. Blackadar, K-Theory for Operator Algebras. second edition Mathematical Sciences Research Institute Publications. Cambridge: Cambridge University Press; 1998;vol. 5.

[27] B. Blackadar, Operator AlgebrasTheory of -Algebras and von Neumann Algebras. Operator Algebras and Non-commutative Geometry, III Encyclopaedia of Mathematical Sciences. Berlin: Springer-Verlag; 2006;vol. 122.

[49] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. Vol. 1 - and -Algebras, Algebras, Symmetry Groups, Decomposition of States. Texts and Monographs in Physics. New York–Heidelberg: Springer-Verlag; 1979.

[50] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum-Statistical Mechanics. IIEquilibrium States. Models in Quantum-Statistical Mechanics. Texts and Monographs in Physics. New York–Berlin: Springer-Verlag; 1981.

-Algebras and Finite-Dimensional Approximations. Graduate Studies in Mathematics. Providence, RI: American Mathematical Society; 2008;vol. 88.

-Algebras by Example. Fields Institute Monographs. Providence, RI: American Mathematical Society; 1996;vol. 6.

[199] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I: Elementary Theory. Graduate Studies in Mathematics. Providence, RI: American Mathematical Society; 1997;vol. 15 reprint of the 1983 original.

[200] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II: Advanced Theory. Graduate Studies in Mathematics. Providence, RI: American Mathematical Society; 1997;vol. 16 corrected reprint of the 1986 original.

-Algebras and Operator Theory. Boston, MA: Academic Press, Inc.; 1990.

[283] D. Olesen, E. Størmer, The life and works of Gert Kjærgaard Pedersen, J. Operator Theory 2005;20:1–5.

[342] M. Rørdam, F. Larsen, N. Laustsen, An Introduction to K-Theory for -Algebras. London Mathematical Society Student Texts. Cambridge: Cambridge University Press; 2000;vol. 49.

-Algebras and -Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. New York–Heidelberg: Springer-Verlag; 1971;vol. 60.

[388] M. Takesaki, Theory of Operator Algebras. I. New York–Heidelberg: Springer-Verlag; 1979.

Chapter 1

Abstract -Algebras

Abstract

-algebras are presented.

Keywords

Spectral theory; Positivity; Approximate units; Factorization; Hereditary subalgebras; Ideals; Quotients

1.1 Spectral Theory

1.1.1

-algebra is a complex Banach algebra A for all x in Afor each x in A, so that the involution is isometric. An element x in A is normal if it commutes with its adjoint , and it is self-adjoint . The self-adjoint part of a subset B of A . For each x in A(the real and imaginary parts of xis a closed real subspace of A and that each element x in A with y and z .

1.1.2

-algebra A .

, then we say that an element u in A is unitary . Note that each unitary is normal and has norm 1.

1.1.3 Proposition

For each nonunital -algebra A, there is a smallest -algebra with unit containing A as a closed ideal so that .

Proof

Let π denote the left regular representation of A for all x and y in A. It is clear that π . Since

we see that π is an isometry. Let 1 denote the identity operator on Abe the algebra of operators on A with x in A and α , there is a y in A such that

□

1.1.4

For each x -algebra A, we define the spectrum of x in A ) as the set of complex numbers λ if and only if there is a y in A ).

is the spectral radius of xwe obtain

If x is just normal, then from the preceding we have

.

1.1.5 Lemma

If , then . If and u is unitary, then is contained in the unit circle.

Proof

, which proves the second assertion in the lemma.

Take now x ) since

, as desired. □

1.1.6

Let A be a commutative Banach algebra. The spectrum of A is the set of nonzero homomorphisms of A of Aconsisting of functionals t in Ais a locally compact Hausdorff space in the weak⁎ topology. The Gelfand transform on A of A for all x in A and t .

1.1.7 Theorem

If A is a commutative -algebra, then the Gelfand transform is a ⁎-preserving isometry of A onto .

Proof

, then ker t is a maximal ideal of Afor some t by for each x in Ais the spectral radius of xby 1.1.4 as each x in A is a ⁎-preserving isometry of A and does not vanish at any point, we conclude from the Stone–Weierstrass theorem that the image of A . □

1.1.8 Proposition

Let x be a normal element of a -algebra A, and let B denote the smallest -subalgebra of A containing x. Then and .

Proof

Since B by , there is an element b in B , and the proposition follows. □

1.1.9

If x is a normal element of A the element of A corresponding to f into A given by 1.1.8.

If f is a sequence of normal elements converging to x.

1.1.10

Let x be a self-adjoint element of A. By are the positive and negative parts of x is the absolute value of xfor any x in Ato be the absolute value of x.

1.1.11

, u shows that each element in A can be written as a linear combination of (four) unitary elements.

An elementary calculation shows that if x and y are invertible in A is unitary. This is used in the proof of 1.1.12.

1.1.12 Proposition

If A is a -algebra with unit, then the unit ball in A is the closed convex hull of the unitary elements in A.

Proof

is strictly positive, so that the element

exists in A and is invertible for each λ , whence

This expression is unchanged when exchanging x and λ , and we conclude that

It follows that, for each λ is unitary (cf. 1.1.11).

The function

. Moreover,

It follows from Cauchy's integral formula (A.4, Appendix) that

are unitary in A, the open unit ball of A is contained in the closed convex hull of the unitary elements in A, from which the proposition follows. □

1.1.13

If A -algebra with unit, then 1 is an extreme point in the unit ball of A with x and y , then x commutes with y. Thus x and y from spectral theory.

Since multiplication by a unitary element is a linear isometry of A, it follows from the above that every unitary in A is an extreme point in the unit ball of A. From -algebra with unit is the closed convex hull of its extreme points. This is remarkable since the unit ball is not in general compact either in the norm topology or in any other vector space topology on A.

1.1.14 Author's notes and remarks

-algebra were formulated in 1943 by Gelfand and Naimark is positive for every x-algebra was coined by Segal in [366], where the foundations for representation theory were laid. Presumably, the C , whereas the ⁎ recalls the importance of the involution.

The result in 1.1.11 is an early discovery, and that in 1.1.12 is more recent [288].

1.1.15 Editors' notes and remarks

Kadison and the author -algebra as in 1.1.12. Indeed, for any a in A , the unitary rank is the smallest n such that a convex combination

can be found, or ∞ in case a is not in the convex hull of the unitary group. The maximal unitary rank -algebra A . Rørdam , where the finite cases occur precisely when the invertible elements of A are dense in A. This condition, introduced and studied under the name stable rank one by Rieffel [338], has proved to be of great importance as the base case of a noncommutative dimension theory.

1.2 Examples

1.2.1

As mentioned in of bounded linear operators on a (complex) Hilbert space Hmatrices.

1.2.2

-algebras A and B-algebra. We content ourselves here with the case where one of the factors is commutative, so that this unpleasantness does not occur.

Let T be a locally compact Hausdorff space, and let A we understand the set of bounded continuous functions x from T to A the subset of functions x for each x forms a dense subset.

1.2.3

as defined in 1.1.3), or the set of sequences x .

1.2.4

-algebras. The set of functions x from I for each i in I and call it the direct product (with I as a discrete space), we obtain the direct sum . When I .

for all i in I, then

1.2.5

consisting of the compact operators on His simpleis separable if H is separable.

1.2.6 Author's notes and remarks

-algebras can be found in Sakai's book [361]. We return in Chapter 6 to tensor products of matrix algebras as a means to generate new algebras by an inductive limit procedure (infinite tensor products). See also 8.15.15.

1.2.7 Editors' notes and remarks

-algebras and/or mathematical structures originating in other subjects. Although the first edition of this book does describe some of these constructions, most of them are more recent, and it is not possible to even provide an overview in the limited space available here. We refer to other sources such as [88,62,27]. We cannot, however, avoid the concept of universal -algebras, which has proved to be an important source of examples, and introduce this concept in 2.9.

In the first edition of this book, the discussion of the concepts below were postponed to 7.7.10 and 7.9.8. To allow for a discussion of several modern concepts in the next six chapters, we present them here in the second edition.

1.2.8

-algebra A and a natural number n-algebra of Acan be computed in many ways; the simplest is probably to represent A as operators on some Hilbert space H . If K is an infinite-dimensional Hilbert space, then for every nof K, there is a natural embedding ι .

1.2.9

Following Brown -algebras A and B are stably isomorphic (H a separable Hilbert space).

We say A is stable is stable even though A is not [339].

1.3 Positive Elements and Order

1.3.1 Lemma

The following four conditions on an element x in A are equivalent:

(i)  x is normal, and ;

with y in ;

and for any ;

and for some .

Proof

(i) ⇒ (ii). Using .

(ii) ⇒ (i). Embedding x and y .

(i) ⇒ (iii). From for each normal element z , we have

(iii) ⇒ (iv) is immediate.

, whence

. □

1.3.2

The elements x -algebra A satisfying the conditions in 1.3.1 are called positive ), and the positive part of a subset B of A .

1.3.3 Theorem

The set is a closed real cone in , and if and only if for some y in A.

Proof

From is a cone, take x and y . By 1.3.1(iii) we have

by .

by 1.1.8. Moreover,

with a and b . Then

was a cone. But, zero apart, the spectrum of a product does not depend on the order of the factors (A.1, Appendix), whence

. □

1.3.4

. When A is not a vector lattice.

1.3.5 Proposition

If , then for each a in A. Further, .

Proof

, from .

Adjoining a unit to A. □

1.3.6 Proposition

If and x and y are invertible elements in with , then .

Proof

From . □

1.3.7

We say that a continuous real function f is operator monotone whenever the spectra of x and y belong to the interval of definition for f.

by

Since the process of taking inverses is operator monotone decreasing by .

1.3.8 Proposition

If , then the function is operator monotone on .

Proof

as in 1.3.7. Now

. For all t , there are therefore a large n such that

, and since ε . □

1.3.9 Proposition

If implies for some and all in a -algebra A, then A is commutative.

Proof

By iteration we see that if the exponent β for every n . Using .

, whence

, and thus

( )

with a and b . Since (⁎) is valid for any product of positive elements and

( )

.

The set E for all x and y is therefore nonempty. The set is also closed, so if it was bounded, it would have a largest element, say λ, and therefore by (⁎)

( )

From (⁎⁎) we now have

that is,

by ) into the right-hand side, we get

that is,

By for all a and b in contradiction with our choice of λ as the largest element. It follows that E , and A is commutative. □

1.3.10

We say that a continuous real function f is operator convex with spectrum in this interval and any λ ,

We say that f is operator concave if −f is operator convex.

1.3.11 Proposition

The functions , , the functions , , and the functions , , are all operator concave on .

Proof

If a is a positive invertible operator, then from spectral theory we have, for each λ ,

If x and y from both sides, we get by 1.3.5

. It follows immediately from the formula in .

Since operator concavity like operator monotonicity is preserved under limits (uniformly on compact subsets) and under convex combinations, we see exactly as in the proof of put,

is integrable. An elementary calculation yields

are operator concave. Incidentally, the argument also shows that the functions are operator monotone. □

1.3.12 Author's notes and remarks

The result in -algebra (see 1.1.14) was redundant, as Gelfand and Naimark also suspected.

Operator monotone functions were characterized by Löwner . It follows by a slight variation of Herglotz's formula that each operator monotone function f for some positive measure μ . The result in 1.3.9 can be found in [271]. Operator convexity and monotonicity is treated in [22].

1.3.13 Editors' notes and remarks

We use the notation

for commutators -algebras. For later use, we record below a result by the author (from .

After nonquantitative versions of such commutator inequalities had been discovered by Arveson [21], the author produced several exact versions and speculated on the best constants attainable. The version provided has the shortest proof, employing 1.3.14 which is due to Haagerup. The constant 5/4 is known to not be optimal, but it remains an open question whether or not the author was correct in his prediction that, in fact, the constant 1 would suffice. In fact, the conjecture has been backed up by extensive computer experiments [250].

1.3.14 Lemma

For any u and b in a unital -algebra such that u is unitary and , we have

for all .

Proof

The operator monotonicity of root functions 1.3.8 shows that

where the last inequality uses the subadditivity of the root functions in spectral theory, since the two operators commute. It follows that

( )

which is an equivalent formulation of the desired inequality. □

1.3.15 Theorem

If a and b are elements in a -algebra A with , then whenever , we have the commutator estimate

Proof

We may assume, without loss of generality, that A ; for if the inequality holds in the self-adjoint case, then for a general operator a, we define

. Since

the norm estimate for a by simple computations.

, then we define the unitaries

for real λ, we have

from which we obtain the formula

( )

Combining ,

. Consequently,

. □

1.4 Approximate Units and Factorization Theorems

1.4.1

Let A for all λ is called an approximate unit for A for each x in Aas well.

1.4.2 Theorem

Each -algebra contains an approximate unit.

Proof

Consider the set Λ of elements u . To see this, take u and v by . By 1.3.7 we have

.

, which is decreasing by 1.3.5, converges to zero for each x , since

and A .

as in and that

. □

1.4.3

-algebra (or just a Banach algebra) A , nor does it have to be increasing. The more restrictive definition given here facilitates some of the computations later. It is clear that there is nothing unique about an approximate unit; in fact, every subnet of the one constructed in 1.4.2 will work equally well. However, the approximate unit constructed in 1.4.2 contains all other approximate units (if they are scaled down a little so as not to touch the unit sphere of A), and we will refer to it as the canonical approximate unit for A.

If A is separable, then it may be convenient to be able to work with a countable approximate unit for Ain A in the canonical approximate unit for A is an approximate unit for A.

1.4.4 Lemma

Let , and a be elements of a -algebra A such that and , with . Then the sequence with elements is norm convergent to an element u in A with .

Proof

. Then

is norm convergent to an element u in A. Reasoning as above, we have

. □

1.4.5 Proposition

Let x and a be elements in a -algebra A such that and . If , then there is an element u in A with such that .

Proof

. From is convergent to an element u in A with

Furthermore,

. □

1.4.6

(see 1.1.8) we have by 1.4.5, for each x in A-algebra but can be performed in algebras for which spectral theory admits the use of Borel functions (see 2.2.9).

1.4.7 Proposition

The extreme points in the unit ball of a -algebra A are precisely those elements x in A such that . In particular, and are idempotents (projections), and A has a unit .

Proof

. If x (see . If y , then

is nontrivial, so that x is not extreme.

. Then, in particular,

with y and z in the unit ball of A, whence

, and thus by the assumption on p and q we have

so that x is an extreme point.

is an approximate unit for Ain A. □

1.4.8 Proposition

Let and be idempotents in . The extreme points in the unit ball of the subspace are precisely those elements x in such that .

Proof

Replace in the proof of if it stands to the left of x if it stands to the right. The arguments carry over verbatim. □

1.4.9

Recall that a partially ordered Banach space E over the reals satisfies the Riesz decomposition property , and c in E in E . If E is a vector lattice, it has the Riesz decomposition, and if E is a vector lattice. It will later become quite apparent that if A satisfies the Riesz decomposition property if and only if A is commutative.

-algebras replace the ordinary one.

1.4.10 Proposition

Let x, y, and z be elements in a -algebra A. If , then there are elements u and v in A with and such that .

Proof

. By 1.4.4 the sequences with elements

are norm convergent in A with limits u and v, respectively. We have

. Finally,

. □

1.4.11 Author's notes and remarks

-algebras was shown by Segal [366]. The canonical approximate unit was found by Dixmier around 1968. The extreme points were characterized by Kadison [185]; the generalized Riesz decomposition appears in [296].

1.4.12 Editors' notes and remarks

The author characterized in -algebra A as the extremal points whose distance to the set of invertible elements is strictly less than 1. Also, Akemann and the author described all faces of the unit ball in a way we will outline further (3.11.12) as soon as we have introduced the necessary