BOUNDARIES OF REDUCED FREE GROUP C*-ALGEBRAS
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
IN
MATHEMATICS
UNIVERSITY OF REGINA
By
Hongyun Dong
Regina, Saskatchewan
January 2009
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1+1 Canada UNIVERSITY OF REGINA
FACULTY OF GRADUATE STUDIES AND RESEARCH
SUPERVISORY AND EXAMINING COMMITTEE
Hongyun Dong, candidate for the degree of Master of Science in Mathematics, has presented a thesis titled Boundaries of Reduced Free Group C*-Algebras, in an oral examination held on October 28, 2008. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material.
External Examiner: Dr. Chang-Nian Zhang, Department of Computer Science
Supervisor: Dr. Martin Argerami, Supervisor
Committee Member: Dr. Douglas Farenick, Department of Mathematics and Statistics
Chair of Defense: Dr. Philip Fong, Department of Computer Science Abstract
The thesis shows the inclusion C*(T) c C;(T,dT) C r(C*(T)), where T is the free groups and dT its boundary. The reduced group C*-algebra C*(T) is separable and exact. The reduced crossed product C*(T, dT) is a separable nuclear C*-algebra.
I(C*(T)) is the injective envelope of C*(T). This proof by N. Ozawa was published in 2007. In order to understand his proof, there is need for a good knowledge of
Operator theory, functional analysis, group theory, C*-algebra and general topology.
The first chapter of this thesis provides the introduction of this interesting question and other related results. In Chapter 2, the necessary preliminaries of C*-algebra are presented. Chapter 3 introduces the topological groups, including free groups, amenable groups, hyperbolic groups and amenable group actions. More advanced materials are presented in the remaining chapters. In Chapter 4, we introduce the concept and properties of group C*-algebras. In particular, we focus on reduced and full group C*-algebras for the free groups. Chapter 5 presents the necessary knowledge of the crossed products of C*-algebras. In Chapter 6, we introduce injective envelope
i of C*-algebras which is very important for this thesis. Ozawa's orginal proofs for this question are given in Chapter 7. The final chapter reviews the result and further developments of this question.
n Acknowledgements
I would like to express my deep gratitude to my supervisor, Dr. Martin Argerami, for his guidance and support through the past two years of my study. His detailed explanations and constructive comments have not only assisted me through my course work but proved an important factor in the independent research and study needed to complete this thesis. I am, also, deeply grateful to Dr. Douglas Farenick for his advice on the structural design of my thesis. Dr. Douglas Farenick kindly devoted additional time to mark the homework I handed in on the courses I audited and was always available to answer my questions. I warmly thank, Dr. Remus Floricel, who regularly attended my seminars and provided friendly help on a number of questions.
I also wish to thank my friends, Sadia Mwangangi and Abdullah Al-Ahamari for their friendship and on-going support through our regular discussions on a variety of math problems, proofs and solution directions.
Lastly and most importantly, I wish to thank my family. Without their encour agement and understanding it would have been impossible for me to finish this work.
iii Contents
Abstract i
Acknowledgements iii
Table of Contents iv
1 Introduction 1
2 Preliminaries on C*-algebras 3
2.1 Gelfand Theory 3
2.2 Representations of C*-algebras 12
2.3 Topologies in B{H) 15
2.4 Nuclear and Exact C*-algebras 18
3 Topological Groups 20
3.1 Free Groups 20
3.2 Topological Groups 22
iv 3.3 Amenable Groups 24
3.4 Hyperbolic Groups 31
4 Group C*-algebras 36
5 Crossed Products 52
6 Injective Envelopes 60
7 Main Result 73
8 Conclusion 82
v Chapter 1
Introduction
In mathematics, embedding is an interesting question which has different meanings in different contexts. In [13, 14], Kirchberg proves that any separable exact C*-algebra can be embedded into a separable nuclear C*-algebra as a C*-subalgebra.
N. Ozawa conjectures that, in general, for any separable exact C*-algebra A there exists a nuclear C*-algebra B between A and its injective envelope 1(A) and B is the unique nuclear C*-algebra that contains A rigidly, which is a tighter embedding. In
[21], he proved that this holds for the free groups with rank 2 < r < oo. That is
C;(T) C Cr*(r,ar) C 7(C;(r)), where r is the free group and DT its boundary.
In [10], Germain proved the result for another particular example. He showed that if T is a non-amenable free product of two countable discrete groups F\ and 1^ such that for some measures on each group the set of extremals in the corresponding Martin boundaries is closed, then there exists a nuclear C*-algebra that sits between C*(T)
1 and its injective envelope. The projective special linear group PSL2(Z) satisfies the assumption of this theorem. Chapter 2
Preliminaries on C*-algebras
2.1 Gelfand Theory
A C*-algebra is a particular type of Banach algebra that is connected with the theory of operators on a Hilbert space.
Before defining C*-algebras, we need to define the Gelfand transform for com mutative unital Banach algebras. There is a stronger form of Gelfand transform for
C*-algebras.
Let V be a Banach space and let B(V) be the space of all continuous and bounded operators on V. The spectrum of x € B(V) is the set a(x) C C of all A G C for which x — XI is not invertible. The set of characters (i.e., non-zero complex-valued homomorphisms) of a commutative unital Banach algebra A is called the spectrum
(or maximal ideal space) of A, and is denoted by SpA. The spectral radius of x € A is the quantity spr(x) = max{|A|, A E c(:r)}, the radius of the smallest closed
3 disc D C C with center 0 G D such that a(x) C D.
Theorem 2.1.1. (Gelfand transform) Let A be a commutative unital Banach algebra.
For x G A, define x : SpA —> C by
x(l) = l(x), I e SpA
Then the range of the function x on SpA is OA{X), where CTA{X) = {A € C | x —
XI has no inverses in A}. Furthermore, the map A is a homomorphism from A to
C(SpA)
A:A^ C{SpA)
x ^ x e C(SpA)
and ||x||oo < II^HJ for every x G A. The homomorphism A is called the Gelfand transform.
An involution on the Banach algebra A is a bijection x i—> x* such that the following properties hold for any x, y G A and A G C:
1. (x*)* = x;
2. (\x)* = Xx*;
3. (x + yY=x*y*;
4. (xj/)* = y*x*.
4 Definition 2.1.2. A C*-algebra A is a Banach algebra with an involution such that
\\x*x\\ = ||x||2 for every x € A.
The following are some examples of C*-algebras.
1. If Ti is a Hilbert space, the space B(H) of all continuous bounded operators on
H is a C*-algebra, where the involution is given by the usual adjoint.
2. If X is a compact space, then C(X) is a C*-algebra, where /* is defined by
P(t) = J{t), and ll/H = sup{/(*), t € X}.
3. The space K(7i) of compact operators in a Hilbert space Ti. is a C*-algebra.
K(7i) is unital if and only if 7i is finite-dimensional.
4. If B is any C*-algebra, and if A is a Banach subalgebra of B such that x £ A
implies that x* G A, then A is a C*-algebra.
If A is a C*-algebra and x £ A, then ||x*|| = \\x\\. Indeed, ||x||2 = ||a;*a;|| <
||x*||||a;||, so ||x|| < ||x*||; Since this is true also for x*, \\x*\\ < ||(x*)*|| = ||x||.
Definition 2.1.3. If A is a C* -algebra and x e A, then
1. x is hermitian if x — x*.
2. x is normal if
3. x is unitary if A is unital and xx* — x*x — 1 .
5 4- x is positive if x = x* and a{x), the spectrum of x, is contained in [0,oo).
For any x € A, we can write x as the linear combination x = a+ib where a — ^f- and b = ^f-, also a, b are self-adjoint and unique. These are often referred to as the real and imaginary part of x, respectively.
Now we prove the Gelfand theorem for C*-algebras.
Theorem 2.1.4. (Gelfand) Suppose that A is a commutative unital Banach *-algebra.
The Gelfand transform A : A —>• C(SpA) is an isometric ^-isomorphism if and only if A is a C*-algebra.
Proof. Suppose that A is an isometric *-isomorphism. Since C(SpA) is a C*-algebra,
A must be a C*-algebra.
Conversely, assume that A is a C*-algebra.
Let h — h* in A, then aA(h) C R. By Gelfand transform, ran(/i) = cr^h), so h is real-valued.
Let x be an arbitrary element in A, then one can write x = a + ib where a, b are
6 self-adjoint. Then, for I E SpA,
x*(l) = l(x*) = l(a - ib)
= 1(a) - il{b)
= l(a) + il(b)
= l(a + ib) = W) = W)
Note that we used the fact that 1(h) € R if h = h* in the third line.
Thus, A is a *-homomorphism.
Now we show that A is isometric.
Again, let h = h* in A, then \\hh*\\ = \\h2\\ = \\h\\2, since A is a C*-algebra. Therefore,
||/i||2n = ||/i2n||. Then
l^lloo = spr(h) = lim \\h2n\\1/2" = \\h\\ n—»oo which shows that A is isometric on self-adjoint elements in A.
For an arbitrary x G A we have
II I loo — ||^'*^'||oo — U^ *^ 11 oo -~ ||^ ^|| — ll^ll
Thus A is isometric, hence is injective.
It still remains to show that A is surjective.
7 Since A is complete and A is isometric, ran(A) is closed in C(SpA). Now, A is a
*-homomorphism, so ran(A) is a closed *-subalgebra of C(SpA) which contains the constant functions. Moreover, ran(A) separates points of SpA. Indeed, if l\, /2 in
SpA and l\ ^ /2, then there exists x 6 A such that /i(.x) 7^ h{x), that is x(li) ^ .x(£2).
By the Stone-Weierstrass theorem, ran(A) = C(SpA). D
Definition 2.1.5. Given unital C*-algebras A,B, a function 4> '• A —»• B is a C*-
algebra homomorphism if, for x,y € A and a, (3 € C,
1.
2. 4>{xy) =
3. 4>{x*) =
I 0(U) = 0(1B).
Let A be a C*-algebra without a unity. We can find a C*-algebra A1 with a unity such that there exists an isometric homomorphism 4> '• A —> A1 with 4>{A) an ideal of
A1 of codimension 1, that is, the quotient Banach space A1 /4>(A) has dimension one.
Now we give the construction oi A1:
Let A1 = A x C = {(a, A) : a € A,\ e C}. Define addition and multiplication as follows:
• (ai, Ai) + (a2, A2) = (ax + a2, Aj + A2)
8 • («i, Ai) * (a2, A2) = (040,2 + Aia2 + X2o.i, AiA2)
The identity of A1 is (0,1). Let
Lemma 2.1.6. If J is an ideal of A, then J is a C*-algebra and A/ J is a C*-algebra
with the quotient norm \\[x]\\ = infj6j ||.x + j|| for [x] G AjJ, and with the involution,
[a]'= [
Example 2.1.7. Let Ti, be a separable infinite-dimensional Hilbert space, the quotient
subalgebra B(Ti.)/K(7i) is called Calkin algebra ofH.
The following theorem states several properties of C*-algebra homomorphisms.
Theorem 2.1.8. Assume that A,B are C*-algebras and
1.
\\ < 1;
2. 3. ker0 is an ideal of A; 4- ran (0) is a C*-subalgebra of B. Definition 2.1.9. A state on a C* -algebra A is a positive linear functional of norm 1, that is, u> : A —* C, such that 9 • ||w|| = 1; • a E A, a > 0 implies u(a) > 0; • uj is linear. The set of states of A is denoted by S(A). The following are two basic examples. 1. A state UJ on C(X) has the form ui(f) = Jx f d/i where n is a positive Borel x a probability measure on X. In particular, for each x E X, u>x(f) = f{ ) is state. 2. If H is a Hilbert space and A C B(7i) is a C*-subalgebra of B(7i), then for every unit vector ^ ETC, the map an-* (a£,£) is a state on A, which is called a vector state. Proposition 2.1.10. (Schwarz' Inequality) Let UJ : A —> C be a positive linear func tional on the C*-algebra A. Then uj(a*) = u>(a), for any a £ A. In particular, uj(h) E R for h = h* E A. Moreover, UJ satisfies Schwarz' Inequality \u{y*x:)\2 < oj(x*x)uj(y*y) for any x,y E A. 10 Proposition 2.1.11. Let A C B be unital C*-algebras with the same unity, and let ui be a state on A. Then u has an extension to a state on B. Proof. Since w is a state, we know that ||c<;|| = w(l) = 1. By the Hahn-Banach theorem, there is a continuous linear functional Q on B, such that ||Q|| = ||o>|| and £1\A = u). So 0,(1) = UJ(1) = 1, since A and B have the same unity, and hence ll^ll — IMI = w(l) = ^(1) = 1- Therefore Vt is a state on B. • Theorem 2.1.12. The set of states on a unital C*-algebra A separates the points of A, i.e., for any a,b € A with a ^ b, there is state u> on A with co(a) ^ u(b). Proof. Let a, b € A with a ^ b. Set x — a — b and write x = h + ik with h = h* and k = k*. Then either h ^ 0 or k ^ 0, because otherwise x = 0. If a; is a state, then u)(h) and u(k) are both real, by the previous proposition, and so u(x) ^ 0 if and only if either ui(h) ^ 0 or u(k) ^ 0. Therefore, the theorem is verified if we can show that there is a state UJ with co(h) ^ 0 for any h = h* € A with h ^ 0. To prove this, by Gelfand transformation, C*(/i) ^ C{Sp(C*(h)). Since /i ^ 0, there is /« e Sp(C*(h)) such that /I(K) 7^ 0. Define p on C*(/i) by p{a) = d(«) for a G C*(h). Clearly, p is a state on C*(h), and so, has an extension to A by the previous proposition. Then uj(h) = p(h) = 1I(K) 7^ 0. • 11 2.2 Representations of C*-algebras Next we study the representations of C*-algebras and the important GNS con struction so that we will be able to prove that every C*-algebra can be regarded as a C*-subalgebra of B(7i) for some Hilbert space H. Definition 2.2.1. A representation of a C*-algebra A is a pair (H,TT) where 7i is a Hilbert space and n : A —> B(7t) is a *-homomorphism. If S is a subset of an algebra A, we define its commutant S to be the set of all elements of A that commute with all the elements of S. Note that S is a subalgebra of A Definition 2.2.2. A representation IT : A —> B(H) is • faithful if ir(x) ^ 0 for every nonzero x G A, that is, if IT is one-to-one and hence isometric. • cyclic, if there exists a unit vector £ € H. such that {7r(a)^ : a € A} is dense in H; such £ is called a cyclic vector . • irreducible, ifn(a) = CI, equivalently {0} andTL are only invariant subspaces ofir(A). • nondegenerate, ifn(A)7i is dense inH. Note that if a representation is irreducible then it is cyclic. 12 A C*-algebra is said to be simple if it has no non-trivial closed two-sided ideals. Recall that the kernel of each homomorphism B is a closed two sided ideal of A. Therefore, if A is simple, all homomorphisms are injective and hence isometric, and so all representations are faithful. The key to representing a C*-algebra on a Hilbert space is to build representations from states and this is achieved by the GNS construction due to Gelfand, Naimark and Segal. Theorem 2.2.3. (GNS Construction) Let A be a unital C*-algebra and to a state on A. Then there is a cyclic representation (TC,TT) of A with unit cyclic vector £ G Ji such that u(a) = (7r(a)£,£) for all a G A. The triple (H,7r,£) is unique up to unitary equivalence, i.e., if (H ,7r , £ ) is another such triple, then there is a unitary operator u : TL —> Ti such that u^ = £ and uix (a)u~l — Tt(a) for all a G A. As a consequence of the GNS construction, we have the following theorem. Theorem 2.2.4. Any C*-algebra A is isometrically ^-isomorphic to a C*-algebra of operators on a Hilbert space. Proof. We may assume that A is unital. Let S(A) be the set of states on A. For each u) G S(A), let ('Hu,,7raJ,£w) be the triple from the GNS construction of A. Let an (7i, 7r) be their direct sum, which means that H. = ©U;e.s(/i)'^w d vr = ©wes(/t)7rw, 13 and £ = (B^es^A)^- Suppose that n(a) = 7r(&) for some a, b G A. Then 0 = (n(a) - TT(6))£ = 0w(7rw(o) - TTW(6))£. Hence 7rw(o — 6)£w = 0 for all to G 5(A). In particular, (^(a — b)^,^} = 0, that is, cu(a — b) = 0 for all w G 5(^4). Since S(A) separates points, we conclude that a = b. Thus, 7r is injective (hence isometric) and so A is isometrically *-isomorphic to TT(A). D Remark 2.2.5. The representation n in the above proof is called the universal representation of the C*-algebra A. Note that this representation is faithful. We know that a subset C of a linear space V is convex if for any vi, v2 G C. A point v G C is an extreme point of C if the equation v = tvi + (1 — t)v2 for some vi, v2 G C and 0 < t < 1, implies fi = ^2 = v. Thus, if v G C is an extreme point, then v G <9C. Some convex sets do not have extreme points, such as£> = {AGC:|A| Let A be a C*-algebra, and let S(A) be the set of states on A, then Theorem 2.2.6. Let (j) G S(A), and 1. (j) is pure. 14 2. n Thus, the GNS representation of a unital C*-algebra A induced by a state is pure. The point is that if a representation (7i, ir) is not irreducible, then IT (A) contains a non-trivial projection p. In this case, pH and (1 — p)7i are invariant subspaces. If {7i,ir) is irreducible, then there are no such non-trivial invariant subspaces of Ji under 7t(A). Theorem 2.2.7. Let A be a C* -algebra, and 0 ^ a € A, then there exists an irre ducible representation n of A such that ||7r(a)|| = ||a||. Theorem 2.2.8. If A is a C*-algebra and if 0 ^ a e A, then there is a pure state 4> of A such that 7r^(a) ^ 0, where ir^ is the representation obtained by the GNS construction. Note that every nondegenerate representation of a C*-algebra is the direct sum of cyclic representations. 2.3 Topologies in B{H) We know that B(7t) is a C*-algebra. There are five important topologies defined on B{H). Let A e B(H) and let (Ba) be a net in B(H). 15 The norm (uniform) topology on B(H) is given by the open neighborhood base at A N(A, e) = {BE B{H) : ||,4 - B\\ < e} That is, Ba —> A in norm if ||5a — A\\ —• 0. The strong operator topology on B(7i) is given by the open neighborhood base at A n 2 N(A; Xl,..., xn] e) = {BE B(7i) : £ ||(A - B)xl\\ < e} for A G B(H), e > 0 and vectors Xi,x2, ...,£„ G 7t. That is, Ba —» .A in the strong operator topology if ||(^4 — J3a):r|| —* 0 for each x ETC. The lueafc operator topology on B(7i) is given by the open neighborhood base at A n N(A; (.Ti)?; (y^; e) = {B E B(H) : | £ The ultrastrong topology on B(Ti) is given by the open neighborhood base at A oo 2 N(A; e; {Xi)?) = {5 G fl(W) : £ \\(A - B)xi\\ < t} t=i for /I G B(H), e > 0 and any sequence(.Xj) in H such that X^Si ll-T*ll2 < °°- That is, 2 J3Q —» /I ultrastrongly if ^Si II(-^* ~ A)xi|| ~»• 0 for each sequence (xi) in 7i with 16 E£i INI2 < °°- The ultraweak topology on JB(7"0 is given by the open neighborhood base at A 00 N(A; e; (Xi)?; (yOf) = {^ #(W) : | ]T (^ (^ - B)Xi) \ < c} for A e B{7i), e > 0 and any sequence(xj), (y;) in Ji such that Ei=i ||^i||2 < oo and 2 That is ESi bill < °°- ' Ba-+ A ultraweakly if oo Y/(yi,(A-B)xi)^0 for each sequence (XJ), (y^) in 7i with Ei=i IWI2 ^ °° and Ei=i ll^ll2 < °°- If W is finite-dimensional, these topologies coincide. When H is infinite-dimensional, the norm topology is stronger than the other four. The ultrastrong topology is stronger than the strong operator topology, the weak operator topology and ultraweak topologies. The strong operator topology and the ultraweak topology are stronger than the weak operator topology. That is, if Ba —• A ultraweakly, then Ba —* A weakly. One can ask whether A*a —• A* or whether AaBa —• AB in the above topologies if Aa —> v4 and Ba —> £? in one of the five topologies introduced above. If H is finite-dimensional the involution and product maps are continuous. However, when H is infinite-dimensional, the situation is different. The involution map is continuous in the norm, ultraweak and weak operator topologies, but not in terms of the strong operator and ultrastrong topologies. The product map is jointly continuous with 17 respect to norm topology but only separately continuous with respect to the other topologies. On bounded sets, the product is jointly continuous in the ultrastrong and the strong operator topology, but not in the weak operator and ultraweak topologies. 2.4 Nuclear and Exact C*-algebras Let A, B be C*-algebras and A *-algebra over a complex field. A C* -norm on A C*-algebra. It is true for any C*-norm that 7(0 ® b) = ||a||||6|| for a G A and b £ B. In [15], Lance constructed the minimal and maximal C*-norms as follows. Let 7i,K, be Hilbert spaces, the algebraic tensor product has a positive definite inner product given by i 3 i,j where &,£• G "H and 7^,7^ G /C. The completion of this inner product space is a Hilbert space denoted by Ji The completion of >1 ® B with respect to this norm is denoted by A ®TOi„ B. The 18 maximal C*-norm is defined as the supremum of all C*-norms on A & B. A C*-algebra A is called nuclear if the maximal and minimal C*-norms coincide on A ® B for any B. For example, finite-dimensional C*-algebras and abelian C*- algebras are nuclear [17]. Moreover, quotients of nuclear C*-algebras are nuclear. A C*-algebra A is said to be exact if taking the minimal tensor product with A preserves exactness of short exact sequences; namely, A is exact if and only if whenever 0 —• / --»• B A. B/I —>Q is a short exact sequence of C*-algebras where / is a closed and two-sided ideal of B and IT is a *-homomorphism, it follows that the resulting sequence 0 —• I ®min A^B ®min A A (B/I) ®min A —• 0 is exact. Every nuclear C*-algebra is exact and every C*-subalgebra of a nuclear C*-algebra is exact. 19 Chapter 3 Topological Groups To understand Ozawa's work [21], it is necessary to introduce some basic defini tions and facts about free groups and C*-algebras most of which can be found in the standard text books [5, 8, 25, 17]. 3.1 Free Groups We start with an arbitrary set S of symbols, say S = {a, b, c,...} which may be finite or infinite, and define a word to be a finite string of symbols from S, in which repetition is allowed. For example, ab, b, and aabaa are words. Two words can be composed by juxtaposition: abbb, ab =£> abbbab In this way the set W of all words has an associative law of composition. Moreover, the identity element is the empty word, denoted by e. 20 Let S' be the set consisting of the symbols in S and also of symbols a 1 for every aeS: S' = {a,a~x, b, 6_1,c, c-1,...} Let W be the set of words made using the symbols in 5". If a word w € W looks like ... xx~l...or... x~lx ... for some x € S, then we can cancel the two symbols x, x~l and reduce the length of the word. The word is called reduced if no cancellation can be made. Next we state two propositions about reduced words. Proposition 3.1.1. There is only one reduced form of a given wordw. We can say two words w,w' in W are equivalent, and write w ~ w', if they have the same reduced form. Clearly, this is an equivalence relation. Proposition 3.1.2. The product of equivalent words is equivalent. Theorem 3.1.3. Let F§ denote the set of equivalence classes of words in W. Then F§ is a group with the law of composition induced from W. Proof. The facts that multiplication is associative and that the class of the empty word is an identity follow from the corresponding facts in W. Now we need to check that all elements of F are invertible. If w € F and w = xy .. . z, then the inverse of w is w"1 — z-1... y-1^"1. • 21 Definition 3.1.4. The group F§ of equivalence classes of words is called the free group on the set S. In particular, the free group on the generating set S = {a} is F = {an} for nsZ, which is an infinite cyclic group. Furthermore, it can be shown that two free groups ¥s and FT, where S, T are generating sets, are isomorphic if and only if S and T have the same cardinality. This cardinality is called the rank of the free group. Thus, for every cardinality number r there is, up to isomorphism, exactly one free group of rank r, that we denote by Fr. In this paper we focus on the free group of rank 2 < r < oo. These are non-abelian and countable. 3.2 Topological Groups Crossed products will be constructed from locally compact group actions on C*- algebras, so we need to have some knowledge about each component. Definition 3.2.1. A topological group is a group (G, •) together with a topology r such that, for s,r 6 G, 1. points are closed in (G,T), and 2. the map (s, r) i—> sr~l is continuous from G x G to G We see that condition (2) is equivalent to 22 a. the map (s,r) i—>• sr is continuous and b. the map s i—> s_1 is continuous. For example, any group G equipped with the discrete topology is a topological group. The additive groups Rn and Cn are topological groups with their usual topolo gies. If G and H are topological groups, then G x H is a topological group in the product topology. A more interesting example is the unitary group U(7i) on a Hilbert space. Let H. be a complex Hilbert space of dimension at least two and let U(H) = {u e B(H) : u*u = uu* = ln} with the strong operator topology, then U(7i) is a topological group. If G is a topological group, then G is Hausdorff and regular [25]. The definition of regular spaces is as follows: suppose that one-point sets are closed in a space X, then X is said to be regular if for each pair consisting of a point x and a closed set B disjoint from x, there exists disjoint open sets containing x and B, respectively. Definition 3.2.2. A topological space X is called locally compact if for each x £ X there is an open set U C X such that x € U and U is compact. Lemma 3.2.3. If X is Hausdorff, then X is locally compact if and only if every point in X has a compact neighborhood. Note that R, K" and Cn are all locally compact. Moreover, any discrete group G is a locally compact group, so the free groups Fr, 2 < r < oo, are locally compact. 23 3.3 Amenable Groups The goal of this section is to understand that free groups of rank 2 < r < oo are not amenable. A measure // on a locally compact space G is called a Borel measure if each open set is measurable. If for each open set V C G, n{V) = sup{/^(C) : C C V and C is compact}, and for each measurable set A, fi(A) = mi{u(V) : A CV and V is open}, then 11 is called a Radon measure [25]. If G is a group, then we say that // is left-invariant if fx(sA) = u(A) for all s e G and a measurable set A. A left-invariant Radon measure on a locally compact group G is called a left Haar measure. Every Haar measure assigns finite measure to each compact set and strictly positive measure to each nonempty open set [25]. Theorem 3.3.1. Every locally compact group G has a left Haar measure which is unique up to a strictly positive scalar, that is, if //j and [i-i are both such measures, then there exists a > 0 such that u\ = a/i2 The proof of this theorem and the following examples can be found in [25]. Example 3.3.2. If G is discrete, then the counting measure is a Haar measure on G, thus ifG = Z, then Jz f(n) du(n) = £„ez /(")• 24 Example 3.3.3. if G is R" or Tn, then the Haar measure is Lebesgue measure. Definition 3.3.4. A group G is called amenable if there is a left translation in variant mean for G. A mean is a state m on L°°(G) and left invariance indicates that m(Ts(g)) =m(g), 1 for all g £ L°°(G) and s.t € G, where Tsg(t) = g(s' t). Example 3.3.5. Compact groups are amenable. The mean is given by integration against Haar measure [8j. To prove that abelian groups are also amenable, we need a fixed point theorem due to Markov and Kakutani [8]. Theorem 3.3.6. (Markov-Kakutani) let T be a commuting family of continuous linear maps of a topological vector space X into itself. Suppose that K is a compact convex subset of X such that TK C K for every T in T. Then there is a point x in K such that Tx = x for all T in T. Corollary 3.3.7. Every abelian locally compact group is amenable. Proof. Consider the state space S of L°°(G) equipped with the weak-* topology, which is a compact convex set. For each s in G, consider the left translation operator Ts(/) = fs on L°°{G). The dual operator T* acts on the dual of L°°{G) and is weak-* 25 continuous. Moreover, it is easy to see that if ^ is a state, then T* Indeed, it is positive because T*M) = Hence ||r>||=TXi) = ^(i) = i. So each operator T* maps S into itself. Since G is abelian, the family {T* : s 6 G'} is abelian. Thus by the Markov-Kakutani Theorem, there is a fixed point m in {T* :seG}. Hence m(fs) = (T»(/) - m(f) for all / e L°°(G). Therefore m is an invariant mean for G. • Theorem 3.3.8. For discrete groups, amenability is preserved under taking sub groups, quotients, direct limits and extensions [8]. Proof. Suppose that G is a discrete amenable group with left invariant mean mc and that H is a subgroup of G. Fix a set of elements {sx : A G A} with one element from each left coset of H in G. Then there is a positive, unital, isometric embedding r\ of L°°(H) into L°°(G) by (r]f)(tsx) = f(t) for all teH,\e A 26 Clearly it is positive, unital and isometric. Moreover, if / G L°°(H), then and (Vf)t(s) = (vm-'s) = fir1) for all t G H. That is, rj(ft) = (/?/)<. Let run — met], then m# is an invariant mean on H, because mHft = mG(r]ft) = 'mG{(vf)t) = mG{r}j) = mHf. Therefore, H is amenable. Now suppose that H is a normal subgroup. Let G/H be the quotient group. Similarly, we define a positive, unital, isometric embedding q of L°°{G/H) into L°°(G) by qf{s) = f(sH) So, if tH G G/H and / G L°°(G/H), then g(/**)(s) = qf{{tH)-ls) = /((tHyhH) = /(r1*//) = (?/),(*). Let mG/H — mGq, then mG/H(ftH) = mGq(ftH) = mG((qf)t) = mG(qf) = mG/H(f). Hence mG/H is a left invariant mean for G/H. Consider the case of direct limits. Suppose that G = lim^ Gn, where each Gn is amenable. By replacing each Gn by its image in G, we may assume that the embed- dings of Gn into G are injective. This image is a quotient of Gn, and thus is amenable. 27 Let mn = mGnrn where rn is the restriction map of L°°(G) onto L°°(Gn), and m,Gn is invariant mean on Gn. Let m be any weak-* cluster point of this net of states. Every s in G lies in Gn for n sufficiently large. Thus it follows that m is translation invariant. For the case of group extensions. Suppose that H is an amenable normal subgroup of G with amenable quotient G/H. Let ra# and mc/H be translation invariant means on these groups. Define a map $ of L°°(G) into L°°(G/H) by *f(sH) = mH(fa-i\H), which is well defined because of the translation invariance of m#. Then set mG(f) = mG/if ($/). This is positive because $ is positive; and is norm one because $1 = 1, hence mG(l) = mG/H{\) = 1. Finally, it is translation invariant because ($/i)(s#) - mHUs-AH) = ($f)MsH). Thus J7io(/t) = rnG/H($ft) = mG/H($f) = mG(f). Therefore, mG is a left invariant mean. • 28 For example, we consider the discrete Heisenberg group [8], denoted by H3 which is a multiplicative group of matrices of the form lac 0 1 b v° ° V where a,b,c € Z. The inverse of the above element is 1 —a —c + ab 0 1 \° ° : / This group is generated by 'no^ ',0 0^ u — 0 1 0 and v — 0 1 1 0 0 1 0 0 1 ! ) and relations w = uvu v ,wu = uw, and wv = vw, where 'lOi^ w = 0 1 0 0 0 1 29 Clearly, w generates the center of the group, Z(H3) which is a normal subgroup of H3 and is isomorphic to Z. Also, M3/Z(M3) is isomorphic to Z x Z. Thus, H3 is an extension of Z by Z x Z. Hence H3 is amenable by the above theorem. The free group is a prototypical example of a non-amenable group [8]. To see this, it is sufficient to show that the free group of rank 2, denoted by F2, is not amenable. Let the generating set of F2 be {u,v}. Consider the set UQ (and U\) of elements of F2 which start with an even (odd) power of u followed by e or a word beginning with a power of v. Similarly, define VQ,V\ and V2 starting with a power of v congruent 2 to 0,1 or 2 modulo 3. Clearly, Ui = uU0, Vx = vV0 and V2 = v V0. Also, F2 is the disjoint union of UQ and Ui, and also of Vo,V\ and V2. Hence an invariant mean must satisfy 2 1 = m(xF2) = ^(Xt/i) = 3m(xvb) where x is the characteristic function. But U\ is a proper subset of Vo, thus m 2 = (Xux) < m(Xv0) = g which is impossible. Therefore F2 is not amenable. There is another proof involving group C*-algebras to show that F2 is not amenable, which will be introduced in a later chapter. 30 3.4 Hyperbolic Groups Hyperbolic groups is one of the main subjects of study in geometric group theory. Started in 1980's, there have been important topics developed, particularly, influential works were done by Mikhail Gromov. To understand why free groups are O-hyperbolic groups, we need to introduce some basic definitions. Definition 3.4.1. A segment in a metric space (X,d) is the image of an isometry a : [a, b] —* X, where [a, b] is an interval. The end points of the segment are a(a) and a(b). Definition 3.4.2. A metric space (X,d) is geodesic if for all x,y € X, there is a segment in X with endpoints x,y. And (X, d) is geodesically linear if for all x, y G X, there is a unique segment in X with endpoints x,y. Segments in Euclidean space Rn are straight lines, so the Euclidean space R™ is geodesically linear. It is easy to see that the closed annulus {z G R2 : 1 < \z\ < 2} is not geodesic in the metric inherited from R2. However it is geodesic in the metric defined by taking the infimum of the Euclidean lengths of piecewise-linear paths between two points. It is not geodesically linear in this metric. The open annulus {z G R2 : 1 < \z\ < 2} is not geodesic in this metric, since there is no segment between the points z and —z. There are several equivalent definitions for hyperbolic spaces; for simplicity, we 31 will only introduce one of them here. Definition 3.4.3. A geodesic metric space (X, d) is called 8-hyperbolic (where 8 > 0 is some positive real number) if whenever A is a triangle with geodesic sides a, (5 and 7 in X then for any p G a, there is q € (3 U 7 with d(p, q) < 8. In this situation we say that A is 8-thin. A geodesic metric space is hyperbolic if it is (^-hyperbolic for some 8 > 0. Example 3.4.4. Every bounded geodesic metric space is hyperbolic. Ifd(x, y) < B for all x and y, then naturally any side of a triangle is contained in the B—neighborhood of the union of the other two sides. Example 3.4.5. In graph theory, a tree is a connected graph with no cycles. Every tree is a hyperbolic metric space. It is obviously a geodesic space, since any two points are joined by a unique path by the properties of trees. Furthermore, any side of a triangle is contained in the union of the other two sides. Therefore, every tree is 0—hyperbolic. To discuss hyperbolic groups, we need to make a group a metric space. Let G be a group, and let S be the set of generators of G. Every element in G can be written as a word by the generators, that is, if g £ G, then g can be expressed as 2 1 g — s^s^ • • • s^ where s1; s2, • • • sn £ S and ai,a2, • • • an = ±1. The natural number n is called the length of the word. We define d$(g, h) to be the length of the shortest word representing g~lh for g,h G G. 32 Proposition 3.4.6. ds(g, h) is a metric on the set G. Proof. By definition, ds(g,h) € N, therefore, ds(g,h) > 0. Moreover, ds(g,h) = 0 if and only if g~xh is represented by the empty word(of length 0), so g~lh = e, where e is the identity or the empty word. Thus, ds(g, h) — 0 iff g = h. Now, let s^1 S22 • • • s^n be a word of minimum length representing g~1h, then h~lg = 1 0 aa 01 (g-'h)- = s- »...S2 sr , so ds(h,g) < ds(g,h). Similarly, ds(g,h) < ds(h,g), so ds(g,h) = ds(h.g). Now, we prove the triangle inequality. Assume that g~lh = l 2 n l h b 2 h s1 S2 • • • s^ and h~ k = t ^t 2 • • • t j both of which are of minimum length. Then s^s^2 • • • s^lt\1t^ • • • th£ is a word(not necessarily minimum length) representing g"xk. Hence ds(g, k) < ds(g, h) + ds(h, k). D Let G be a group and 5 be a subset of G that is closed under taking inverses and does not contain the identity. The Cayley graph is the graph with vertex set G and edge set E = {gh : g~xh G S}. The Cayley graph of the group G with respect to the set S is denoted by X(G, S). It is obvious that two vertices g,h € G are adjacent in X(G,S) if and only if ds(g, h) = 1. And ds(g, h) is precisely the length of a shortest path in X(G, S) from g to h. X(G, S) can be made into a metric space by defining d(x, y) to be the length of the shortest path joining x and y which coincides with the word metric d$ where x and y are vertices, hence each edge has length 1. Definition 3.4.7. A finitely generated group G is said to be word-hyperbolic if there 33 is a finite generating set S of G such that the Cayley graph X(G, S) is hyperbolic with respect to the word metric ds. Example 3.4.8. Every finite group is hyperbolic, since its Cayley graphs are all bounded. Example 3.4.9. Every free group is hyperbolic, since its Cayley graphs are trees which will be illustrated as follows. Every tree is ^-hyperbolic, therefore, free groups are ^-hyperbolic. We say that two geodesic rays ji : [0, oo] —» X and j2 '• [0, oo] —> X are equivalent and write 7J ~ 72 if there is K > 0 such that for any t >0, d(7i(£),72(i)) < K. The boundary of a geodesic metric space is usually defined as the set of equivalence classes of geodesic rays starting at a base-point, equipped with the compact-open topology. For more information on this subject, see [12]. Definition 3.4.10. (Boundary of a hyperbolic group) Let G be a word-hyperbolic group. Then for any finite generating set S of G the Cayley graph X(G, S) is 6—hyperbolic with respect to the word-metric ds- We define the boundary dG of G asdG = dX{G,S). The boundary of free group of rank 2 < r < 00 is homeomorphic to the space of the ends of a regular 2n-degree tree, that is to a Cantor set [12]. If G is finite, then dG is the empty set. If G is infinite cyclic then dG is homeomorphic to the set {0,1} with the discrete topology. The boundary of Z is {—oc, +00}. 34 Definition 3.4.11. An action of G on a compact space X is amenable if there exist continuous functions Zn:X^ l\G) such that 1. ||£n(aOI|2 = 1 for all x G X. 2. For all g G G, ||A5£n(x) — ^n{.gx)\\2 —• 0 uniformly in x G X. In this situation, X is called an amenable compact G-space. A G-space is a topological space X together with a continuous action G x X —> X, (g, x) I—> gx satisfying (gh)x = g{hx) and ex = x for all g,h € G and all x G X, where e G G is the unit element of G. The following theorem shows that the action of the free group on its boundary is amenable. The amenability of this action is essential for the main result of the paper which we will explain in later chapters. Theorem 3.4.12. Let G be a discrete hyperbolic group and dG its boundary, then dG is an amenable compact G-space. Adams proves this theorem in [1] for for every quasi-invariant measure on the boundary. Also there is a simpler proof in [4]. 35 Chapter 4 Group C*-algebras In this chapter, we will show that the group C*-algebra C*(F2) is not isomorphic to the reduced group C*-algebra C*(F2). There are many approaches to show that C*(F2) is not isomorphic to C*(F2). We use here the fact that C*(F2) is simple but C*(F2) is not. A discrete group is said to be exact if and only if its reduced C*-algebra is exact. Free groups are exact. Guichardet proved that if G is amenable then C*(G) is nuclear. Lance proved that G is amenable if and only if C*(G) is nuclear for a discrete group G. Most content of this section can be found in [8]. Definition 4.0.13. Let G be a group. A unitary representation IT of a group G is a homomorphism of G into the unitary group of B(7t) which is continuous in the strong operator topology, meaning that the map taking s € G to 7r(s)£ is continuous for every vector £ £ H, and that 7r(s) is unitary. Recall that if G is a locally compact group then there exists a unique (up to a 36 strictly positive number) left invariant Haar measure ds and modular function Ac- Let Ll{G) denote the Banach space of all integrable functions on G with respect to the Haar measure equipped with norm ll/lli = / l/(*)| ds. for / G L\G), and s G G JG Define a multiplication (convolution) and an involution in LX{G) as follows: (/* Then the space Ll(G) becomes an involutive Banach algebra. Note that the modular function Ac = 1 when G is discrete or abelian or compact which are the only cases we consider in this paper. The algebra Ll{G) is called the Ll group algebra of G. When G is not discrete, Ll{G) does not have a unit. However, it has an approx imate identity, i.e., there is a net {ej} of non-negative functions in Ll(G) such that / ej ds — 1 for all i and \\etf — f\\i —> 0 for all / in Ll{G). l 1 If G is discrete, L (G) can be written as (- {G) whose unit is Se, which is the characteristic function of the identity element e G G. Moreover, the group algebra CG 0 wnere a consists of all finite sums / = XLeG ^' s € C, and 5S is the characteristic function of s G G. The group algebra CG forms a dense subalgebra of Ll(G) — ^(G). 37 Note that the Ll(G) norm is not a C*-algebra norm. To produce a C*-algebra from Ll{G), we need to define some appropriate C*-norms. First we notice that there is a bijective correspondence between a unitary rep resentation of G and a non-degenerate representation of Ll(G), this correspondence preserves direct sums and hence irreducibility [16]. Let 7T be a unitary representation of G. It induces a representation TT of L1 (G) by 7i(f) = J f(t)n(t)dt. It is easy to see that IW)ll Conversely, if re is a nondegenerate representation of Ll{G), then it induces a unique unitary representation of G. Let f\, where A 6 A, be a one-norm approximate identity for Ll(G). Then Um7r(/A)7r(^)a: = Ti{g)x l for every g in L (G) and x in 7i. Therefore the contractions 7T(/A) converge strongly to the identity operator. Define •n{s)^{g)x = Tr{gs)x 1 where gs(t) = gis" ^. Definition 4.0.14. The left regular representation of a locally compact group G 38 on L2(G), the Hilbert space of all square integrable functions on G, is defined by -1 2 Xsg = gs where gs(t) = <7(s i) for s,t E G and g E L (G). Note that this representation is a unitary representation, therefore, it induces a nondegerate representation on Ll{G). Definition 4.0.15. The reduced group C*-algebra of G is defined to be C*{G) = A(Li(G)). Let B(L2(G)) be the C*-algebra of all bounded operators on the Hilbert space L2(G). Then C;{G) is a C*-subalgebra of B(L2(G)). Definition 4.0.16. The group C*-algebra ofG, C*(G), is defined to be the norm- closure of the universal representation of Ll(G). There is another equivalent definition of the group C*-algebra of G, which defines a C*-norm on Ll(G) by ll/H = sup{||7r(/)|| : 7T is a ^representation of Ll(G)}. The family of representations is not empty because each irreducible representation of C*(G) gives an irreducible representation of G up to unitary equivalence from the construction of representations between G and LX(G). These two definitions are equivalent because of the fact that ||x©y|| = max{||x||,||y||}) for x,y G B(H). 39 As stated earlier, if G is a discrete amenable group, then C*(G) = C*(G). To prove this statement we need some definitions and propositions. Definition 4.0.17. A function 4> on a discrete group G is called positive definite if n 1 Y] ctitTj^isJ st) > 0. for all n > 1, a; G C, s* Let V(G) denote the set of all positive definite functions on G such that (e) = 1. Any state $ on C*{G) determines a positive function in V(G) by setting n J 2 1 2 ||/||c-(G) = sup*e5(c,.(G))$(/* * /) / = sup0e7,(G)(^ aiOj^CsJ^j)) / . Moreover, we need the following two propositions [8]. Proposition 4.0.18. If Proposition 4.0.19. Suppose that in V(G) with finite support, then there is a unit vector x in l2(G) such that Proof. Define a linear map on CG by Tf = f*4>. Since 0 has finite support, it is clear that T is bounded by XLeG l^(5)l when CG is endowed with the £2(G) norm. Thus T 40 may extended to a bounded operator on £2(G) by continuity. If / = Y^i=i a*^i e ^G, then n n Ff, /> = E E fitw'^m = E E «^(^) > o. s€G t€G i=l j=l Hence T is positive. Note that X(s)Tf = 5S * (/ * ) = (Ss * /) * 0 = TX(s)f. So T commutes with X(G). Let x = T1/2^. Then l 2 l (\(s)x,x) = {\{s)T ' 5e,T 'He) = (5s,T5e) = (5a,) = (f>{s). Finally, ||x||2 = Now we prove the theorem [8]. Theorem 4.0.20. If G is a discrete amenable group, then C*{G) = C*{G). Proof. We prove this statement by showing that the C*-norms for C*{G) and C*(G) coincide on CG, and using that CG is dense in both. Since G is amenable, we may assume that m is a left invariant mean on £°°(G). We know that (/a(G))* = l°°(G). By Goldstine's Theorem, the unit ball of ^(G) is weak-* 1 dense in the unit ball of £°°(G)*. Since CG is dense in £ {G), there is a net f1, 7 € I\ in the £X(G) unit ball of CG converging weak-* to m. 41 First we show that we may assume that each /7 is positive and satisfies (/7,1) = 1, where 1 is the constant function l(s) = 1. Indeed, K/7.1>l = l£^)l seG But lim7 (/7,1) = m(l) = 1. So it is apparent that each /7 may be replaced with the function Since m is left translation invariant, for each s G G, w* — lim7(<5s * g-y — g7) = Ss * m — m = 0. Thus 8S * g1 — g1 converges to 0 weakly in ^(G). By the Hahn-Banach Theorem, for any finite set Si, • • • ,sn in G, the norm closed convex hull of 1 in £ (G)* contains 0. Therefore we obtain a sequence gn in the convex hull of the g7's such that lim \\SS * gn — ^n||i = 0 for each s € G. n—>oo Then /in = gin' are positive functions in ^(G) with ||/in||2 = 1- Note the simple 2 2 2 inequality \a — b\ < \a — b\\a + b\ = \a — b \ for all a, b > 0. Therefore, since hn is 42 positive, 2 2 lim \\X(s)hn - hn\\ 2 = lim V \hn(st) - hn(t)\ n—>oo n—»oo ' » teG < lim y^\gn(st) -gn(t)\ n—>oo *—' teG = lim ||<5s*p„ - gn\\i = 0 n—>oo for every s in G. Since (X(s)hn, hn) is real, we obtain lim (X(s)hn, hn) = lim (\{s)hn, hn) + -\\\{s)hn - hn\\\ n—>oo n—>oo / = - lim(||A(s)/in-/in||| + ||/in||l) = l. In conclusion, n(s) = (X(s)hn,hn) are positive definite functions in V(G). These functions have finite support because each hn has finite support; and they converge pointwise to the constant function 1. Now let (p belong to V(G), then by the previous proposition, (fxj)n belongs to V(G) and has finite support. Furthermore, they converge pointwise to 2 2.2, there are unit vectors xn in £ (G) such that (f)(s)4>n(s) = (X(s)xn,xn) for all s G G. 43 Thus if / = XXi aiSSi e CG, then seG n—>oo < ' s€G = lim (A(/* *f)xn,xn) n—>oo H|A(/)||2. Taking the supremum over all positive definite functions yields ll/llc.(G) = ||A(/)|| for all f ECG. D Now we show that C*(F2) is simple and has a unique trace [8, 20]. Notation: let St denote the characteristic function of t in F2 as an element of 2 2 £ (¥2). These vectors form an orthonormal basis for £ (¥2) X(s)5t(r) = Stis-'r) = Sst(r) for r, s, t G F2 Thus X(s)5t = Sst. An expectation of a C*-algebra onto a subalgebra is a completely positive, unital and idempotent map. A trace on a C*-algebra A is a state r such that r(xy) — r(yx) for all x and y in A. 44 Proposition 4.0.21. The map T(A) = (ASe,5e) is a faithful trace on C*(F2). Thus the map $(A) = T(A)1 is a faithful expectation of C*(F2) onto the scalars. a s an( = s s Proof. Since CF2 is dense in ^(G), we consider / = ^SGF2 ^ ^ 9 SseF2 @ ^ - Then r(X(f)) = (X(f)Se,5e) = ^Pas (5s,5e) s = Oie Similarly, r(A(g)) = fie. Then, for s,t in F2, a X st 6 r(Kf * s t s = r(A(455 */)) So r is a trace on C*(F2) by continuity. Now suppose that o in C*(F2) is positive and r(a) = 0. Then (a5s,Ss) = (a\(s)8e, X(s)Se) 1 = (\{S- )aX(s)6e, Se) = r(A(s-1)aA(s)) = T(O) = 0. Hence,by Cauchy-Schwarz inequality, 11/2 /„* X \ i/z1/2 I (a5s, 5t) | < {a6a, 5S) " {a5t, 5t) = 0. Hence a = 0 and r is faithful. Clearly, $ is a faithful expectation onto the scalars. • ( \ 0 * Lemma 4.0.22. Let H — M © M"1. Suppose that B has the form and Ui \ * * / ( \ * * are unitary operators such that UiUj have the form when i ^ j. Then V* V ,1 " , 2„ l-^TUiBWW < -\\B n ^-^ n Proof. First suppose that B and C have the forms ( \ B H , and C = * * i * * (o 0) Then \B + Cf = ||(5* + C*)(B + C)\\ = \\B*B + C*C\\ < \\B\\2 + \\C\ 46 Notice that when i^j, the operator (UiU*)*B(UiU*) has the form * * O * * * * I * U/ \* */ \* U/ \U0 0U / Hence || J2 UiBUtW2 = WU^B + YlWUjBiUiUWUW i=l i=2 n 2 <\\B\\ + \\Y,(U*lUl)B{U1U*l)f i=2 n = \\B\\2 + \\Y^UiBU*\\2. i=2 2 2 By induction, it follows that || £"=1 £/i£[/;|| < n||£|| . Therefore, \\(l/n)f2UiBU;\\<(l/V^)\\B\\. j=i O The same result holds when S has the form For the general case, V° V split B as a sum of two terms in the form / \ /. \ 00 0 * I * * / This yields the desired result. Corollary 4.0.23. For s G F2, X(s) if s = ukfor some k G Z. lim -J]AK)A(S)A(Ui = 0 otherwise 47 l l Proof. An element of F2 is a product of powers of u, v, u~ and v~ . We know that it is reduced when no adjacent terms cancel. If s is not a power of u, then it has some non-zero power of v in its reduced form. Thus there are integers kQ and £0 such that k l ± ± s = u °sQv °, where s0 has the form v tv . Let ± M. = span {5S : s = 1 or s = u i in reduced form}. Then x ± .M = span = {Ss : s = y i in reduced form}. Q e L Let f = M, then / = £) A> A(s0)/ = ]CasA(s0)<^ = X)aA0s •^" - Hence A(s )A/I is contained in Mx\ while A(wfe)A/("L is contained in M for k ^ 0. Hence 0 O with respect to this decomposition, A(SQ) = and A(ir)A(V*) has the form i * * / ko io when £ ^ k. By the previous lemma, for s = u sQu , v* °/ lim {IIn) V A(ui)A(s)A(«-i) n—>oo ^—' i=l n fe i 1 io A(u °)( lim (1/n) V X(u )\(s0)\(u- ))X(u ) = 0. n—»oo ^——*• On the other hand, if s = uk, then evidently (l/«)5>K)A(s)A(0 = A(S) t=i for all n > 1. • 48 Corollary 4.0.24. For all A e Cr*(F2) 1 m n lim lim -TVXMv^AXUrhi^) = T(A)1 m-»+oo n-^+oo mn z—' ^—' i=l j=l a s m < 2 rev us Proof. Consider / = X^seF2 ^ ^ - By *he P i° corollary, lim (l/n)T\(vi)\(f)\(v-i) = T^^(vk) = /o- n—>oo *—*• *—' Hence the desired limit equals m i i lim (1/m) V A(« )A(/0)A(n- ) = ae/ = r(A)/ = $(A). m—>oo ^ * i=l By continuity, this identity extends to every element of C*(F2). A routine norm estimate shows that this limit exists as m and n tend to infinity independently of each other. • Theorem 4.0.25. Cr*(F2) is simple. Proof. If A ± 0, then 0 ^ A*A is positive. Hence r(a*a) > 0. Then the ideal generated by A contains -. m n lim lim — ryA(«V)AMA(t)V) = r(^)l which is invertible. Hence this ideal is all of C*(F2). • Since C*(F2) is not simple, C*(F2) and C*(¥2) are not isomorphic. Therefore, F2 is not amenable by Theorem 2.3. 49 Theorem 4.0.26. C*(F2) has unique trace. Proof. If r' is another trace on C*(F2), then by linearity and trace property, -. m n T'(a) = T'( lim lim VVAfuV'l^rt-')) m—>+oo r»—>+oo Tflfl ^-^ ^-^ i=l j=l m n = lim lim r'( V V AfwVUAfiT^O) m-»+oo n~++oo mn Z—' ^—' i=l j=l = lim lim T'(T(A)1) m—>+oo n—»+oo = r(A). which shows that C*(F2) has unique trace. D Now we know that the reduced C*-algebra of the free group on two generators is simple with unique trace. We may ask that what kind of groups for which the reduced C*-algebras are simple with unique trace [20]. The following theorem gives a more general example. Theorem 4.0.27. (Akemann and Lee) Let G be a discrete group which contains a normal free nonabelian subgroup with trivial centralizer. Then the reduced C*-algebra of G is simple with unique trace. Moreover, both C*(F2) and Cr*(F2) are projectionless [6, 8]. C*(F2) has a faithful irreducible representation which is a primitive C*-algebra. We need to point out that this result for C*(F2) and C*(F2) can be extend to all free groups of rank 2 < r < oo. Moreover, there are many interesting topics about full group C*-algebras such as 50 primitivity conditions for full group C*-algebras due to G.J. Murphy [18]. In his paper, he gave a number of examples for which the full group C*-algebras are primitive for discrete groups. Furthermore, we need to know that the reduced C*-algebra C*(G) is exact if G is an hyperbolic group [4]. Therefore, C*(¥r) is exact where 2 < r < oo. 51 Chapter 5 Crossed Products A C*-dynamical system is a locally compact group G acting by automorphisms on a C*-algebra. The crossed product is a C*-algebra built out of a dynamical system which has physical significance. The representation of the crossed product encodes the covariant representation of the dynamical system [25]. Definition 5.0.28. A C* -dynamical system (21, G, a) is a triple consisting of a C*-algebra 21, a locally compact group G and a homomorphism a of G into the automorphism group Aut(%) of%. We denote the automorphism for s in G by as. Definition 5.0.29. Given a C*-dynamical system (21, G, a), a covariant repre sentation is a pair (77, U) where it is a *-representation of % on a Hilbert space Ji and s —* Us is a unitary representation of G in the same space such that USTT(A)U* = 7r(as(A)), for allAe%se G. 52 Example 5.0.30. Let h G //omeo(T) be given by h(z) = e2niez. Let (C(T),Z,a) be the associated dynamical system where an(f)(z) = f(e-2m6z). Let M : C(T) —> 5(L2(T)) 6e the representation given by pointwise multiplication M(f)h(z) = f(z)h(z). Let U : Z —> t/(L2(T)) 6e i/ie unitary representation given by 2 Unh(z) = h{ It is easy to check that (M,U) is a covariant representation of (C(T),Z, a) [25]. We consider the case where G is discrete. In this case, the space of continuous compact supported 2l-valued functions on G, CC(G, 2t), is just the algebra 2IG of all finite sums / = ^2t€G Att with coefficients At in 2l. The multiplication and adjoint are defined as follows, if g = YlueG ^uu *s an°ther finite sum, then fg — /J(/J Atat(Bt-is))s, for all s,t,u 6 G and tu = s. seG t&G r = j2MAU)t. t£G Furthermore, there is a one-to-one correspondence between a covariant representation (ir, U) of (2l, G, a) and a ^representation of 2IG [8]. Given (TT, U), we define tSG 53 It is easily verified that a is a ^representation of 21G. Conversely, when 21 is unital, a ^representation of 21G gives a covariant representation of (21, G, a) by setting TT(A) = cr(Ae), Us = a(s) for A G 21 and s,eeG. When 21 is not unital, let En be an approximate unit, and let Us = limn^00a(Ens). This bijection preserves direct sums, and therefore irreducibility [16]. Definition 5.0.31. The crossed product 21 xa G is the enveloping C*-algebra o/2lG. That is, one defines a C*-norm by \\f\\ = supja(f)\\ as a runs over all ^-representations o/2lG. It is easy to see that the supremum is bounded by ||/||i = Y2t<=G 11^*11- This crossed product has the following universal property [8]: if (IT, U) is any covariant representation of (21, G, a), then there is a representation of 21 xa G into G*(TT(21), U(G)) obtained by setting "(/) = 5>0W for / = J^Att e 21G, t€G t€G and extending by continuity. In general, if G is an arbitrary locally compact group, let L2(G, 21) be the Ba- nach *-algebra of all (equivalence classes of) 2l-valued integrable functions on G with 54 multiplication, involution and norm defined respectively by 1 (f9)(t) = Jf(s)aa(g(s- t))ds, f*(t) = Aitywnr1))*, \\fh = J\\f(t)\\dt for each /, g € Ll(G, 21) and t £ G, where A is the modular function of G. Then the l crossed product 21 xa G is the enveloping C*-algebra of L {G, 21). We need to know that covariant representations always exist. The following propo sition [16] gives a method to construct covariant representations. Proposition 5.0.32. Let (21, G, a) be a C*-dynamical system, and suppose one has a state u on% which is G-invariant in the sense that uj(as(A)) =UJ(A) for all s € G and A £ 21. Let TT^ be the representation constructed from the GNS construction with the Hilbert space H^, and cyclic vector Q,^. For s £ G, define an operator Us on the dense subalgebra ixUJ{'riw)Q,Uj ofHu by Ua7ru(A)nu = iru(aa{A))£lu. This operator is well-defined and is a unitary representation of G on TC. Note that there is a bijective correspondence between non-degenerate representa tions n of the crossed product 21 xa G and covariant representations (TT, U) which is 55 given as above. This correspondence preserves direct sums, and therefore irreducibil- ity. Let TT be any ^representation of Sj on a Hilbert space H. For the Hilbert space L2{G,7i) of all square integrable 7i-valued functions on G. Then define a covariant representation (ff, A) of (21, G, H) on L2(G, H) = L2{G) ®Was follows: (Z(A)f)(S) = ir(a;\A))(f(s)) (Atf)(s) = f^s) for all A £ 21, / £ L2(G,H) and s,t £ G. Indeed, it is a covariant representation, since 1 (Att(A)K)(f(s)) = (^^(/(r *)) 1 1 = 7r(a- x.(>l))A:(/(r S)) 1 = 7r(a; at(A))(/(S)) = 7r(at(A))(/(a)) The completion of ^(GyTi) with respect to this restricted class of representations is called the reduced crossed product. That is, we denote the covariant representa tion (-7T, A) by Ind^V which is called the regular representation. One defines ||/||r = sup{||Inde 7r(/)|| : 7r is a representation of 21} l for / £ L (G,'H). The reduced crossed product is denoted by 21 xQr G. 56 If 21 = C, and a = id, then 21 xQir G = C*(G). Indeed, the only regular rep resentation of L1(G,H) is the regular representation A, and C*(G) is the closure in 2 B(L (G)) with respect to this norm. In particular, ||/||r = ||A(/)||. Similarly, when G is amenable, the reduced crossed product is equal to the full crossed product [25]. Let B be a C*-subalgebra of A. A bounded linear map P : A —» B is called a conditional expectation if P has the following properties[26]: 1. P is an onto projection of norm one, that is , P2 = P and ||F|| = 1; 2. P is positive, that is , for any x 6 A, P(x*x) > 0; 3. for any x E A, y,z £ B, P(yxz) = yP(x)z; 4. for any x e A, P(x*)P(x) < P(x*x). The following corollary shows that a conditional expectation is completely posi tive. Proposition 5.0.33. (Nakamura, Takesaki and Umegaki) Let A, B and P be as above. Then, for each positive integer n and each x\ • • • xn £ A, M = [P(x*Xj)] is a positive element of Mn(B), that is, P is a completely positive map. Now we look at an example of conditional expectations [8]. Theorem 5.0.34. Let (21, Z, a) be a C*-dynamical system. Then there is a faithful conditional expectation 57 n Proof. Let a € Aut($l), which gives rise to an action of the integers by an = a . We can assume that 21 is unital so that there is a unitary u in the crossed product 21 xQZ such that uAu* = a(A) for every A in 21. For every scalar A of modulus 1, Xu also determines a unitary representation of Z by n —> Xnun such that {Xu)nA(Xu)-n = unAu~n = an{A). Thus (id,\u) is another covariant representation of (21, Z, a). Clearly, 21 and Ait generate 21 xa Z. From the universal property of the crossed product, there is a homomorphism p\ of 21 xQ Z onto itself such that p\(A) = A for all A in 21 and P\(u) — Xu. Evidently p\ is an automorphism. It is easy to check that for X £ 21 xQ Z, the function f(t) = pe2i,u(X) is norm continuous. Indeed, one may verify it on the dense subalgebra 21Z, and extend it to the closure by a simple approximation argument. Then define a map $ on 21 xa Z by $(X) = / Pe2«u(X)dt. Jo Notice that since each /9^ is a faithful (completely) positive isometric map, it follows that $ is a faithful (completely) positive contraction. Next, consider the effect on AXB for A, B e 21. $(AXB)= J pe2.u(AXB)dx = A f pe™t(X)dxB = A$(X)B. Jo Jo So $ is an 2l-bimodule map. In particular, $(-A) = A for all A in 21. Also, fc k 2nikt k $(u ) = /" Pe2„it[u ) dx= f e u dt = 0 for k ^ 0. Jo Jo 58 n Hence on an element of 21Z such as a finite sum J^n Anu . It follows that n n This lies in 21. By the density of 21Z and the continuity of $, it follows that the range of $ lies in 21. Since $ is the identity map on 21, it follows that $ is an expectation. • The following theorem is due to Zeller-Meier [26]. Theorem 5.0.35. Let (21, G, a) be a C*-dynamical system with G discrete. Then 1. there exists a conditional expectation o/2l xr G onto 21, and 2. there exists a conditional expectation o/2t x G onto 21. 59 Chapter 6 Injective Envelopes Cohen proved that the injective envelope exists and is unique for a Banach space [7]. In [11], Hamana showed that any unital C*-algebra has a unique injective enve lope. To understand these topics we need some necessary definitions and theorems which can be found in [11, 22]. If S is a subset of a C*-algebra A, then we set S* = {a : a* E S}, and we call S self-adjoint when S = S*. If A has a unit 1 and S is self-adjoint subspace of A containing 1, then we call S an operator system. Let B be a C*~ algebra and : S —» B a linear map then 4> is called a positive map if it maps positive elements of S to positive elements of B. Note that if cj) is positive linear functional on an operator system S, then |||| = 0(1). Furthermore, we define 60 positive if 4> is n-positive for all n. We call is finite. Let \\(f>\\Cb = sup ||0n||- If each (f>n is isometric then (f> is called completely isometric. If \\ space on a Hilbert space 7i is defined to be a linear subspace of B(Tt). B(H) is an operator space by the identification Mn(B(7i)) = B(Hn). Each C*-algebra A is an operator space if Mn(A) is equipped with its unique C*-norm. Definition 6.0.36. Injectivity is a categorical concept. Suppose that we are given some category C consisting of objects and morphisms. Then an object I is called infective in C provided that for every pair of objects E C F and every morphism 4> : E —+ I, there exists a morphism ip : F —> I that extends (f>, i.e., such that ip(e) = 0(e) for every e G E. Theorem 6.0.37. (Stinespring's dilation theorem) Let A be a unital C*-algebra, and let a unital *-homomorphism pi : A —> B{K), and a bounded operator V : H —> /C with ||0(1)|| = \\V\\2 such that ^(a) = V*ir(a)V. The proof of this theorem can be found in [22]. Theorem 6.0.38. Let A, B and C be C*-algebras with unit, and suppose that C is contained in both A and B, with \Q = I A o,nd lc = Is. A linear map : A —> B 61 is called a C-bimodule map if §{c\ac2) — C\ac2 for all ci,C2 E C. Let (j> : A —> B be completely positive, then 1. If (f)(1) = 1, then (j) is a C-bimodule map if and only if (f)(c) = c for all c E C. 2. In general, (j) is a C-bimodule map if and only if cf>(c) = c • 0(1) for all c E C. Moreover, in this case 0(1) commutes with C. Proof. We prove part (2) first. Assume that 0 is a C-bimodule map. We have 2 0(C) = 0(cl )=c0(l)l = C0(l). Similarly 0(c) = 0(l)c. Conversely, assume that 0(c) = c • 4>(1) for all c E C. Taking adjoints gives that 0(c*) = 0(l)c*. Since C is self-adjoint, 0(c) = 0(l)c for all c E C. Now we assume that B = B{7i) for a Hilbert space Ti. Let 7r be a Stinespring dilation of 0 on a Hilbert space /C. Let V : ~H —* /C be the associated linear operator. Note that C is a C*-subalgebra of B(H). Let h EH and note that Vc(h) = 1 ® c(/i) + jV, where M ={uE A®H\{u,u) = 0} = {uE A®H\{u,v) = 0 for all v E A®H], which is a subspace of A ® H. n(c)V(h) = 7r(c)[a 62 show that 1 (g> c(h) — c £g> h G N then the equation Vc = TT(C)V follows. (1 <8> c(/i) - c (g) /i, 1 - (c ® /i, 1 (8) c/i) - (1 (8) c(/i), c = ^(l)c(/i),c(/l)) + (^(c*c)/l,/l) = ((l)c(h),c(h)) + ( - (4>(l)c(h),c(h)) ~ (4>(Vcc(h),h) = 0 By taking adjoints we have cV* — V*TT(C) for all c E C. Now, = \/*7r(ci)7r(a)7r(c2)y = ClV*Tx{a)Vc2 = c1(/)(a)c2 Part (1) is just a special case of part (2). • Now we let O denote the category whose objects consist of operator spaces and whose morphisms are the completely contractive maps. Let 6 denote the category whose objects are operator systems and whose morphisms are the completely positive maps. Similarly, let &x denote the category consisting of operator systems and unital 63 completely positive maps. Then we have the following important theorem [22]: Theorem 6.0.39. Let S C B(H) be an operator system. Then the following are equivalent: 1. S is injective in O; 2. S is injective in &; 3. S is injective in &L; 4- there exists a completely positive projection cf> : B(7i) —> S onto S. Proof. Assume (1), then the identity map from S to S extends to a completely con tractive map 4> : B{7i) —> S. Since cf> extends the identity map, 0 is a projection onto S. Since Conversely, assume (4), ans suppose we are given operator spaces E C F and a com pletely contractive map 7 : E —> S. Then 7 has a completely contractive extension •0 : F —* B(7i), and (j) o ijj : F —> S is the desired completely contractive extension of 7 into S. The rest proof is similar. D The following theorem due to Choi and Effros shows that every injective operator system is in an appropriate sense a C*-algebra [22]. To prove this theorem, we need a lemma. 64 <0 ^ + Lemma 6.0.40. Suppose that T = e M2(A) , then b = 0. \»" V Proof. Since T is positive, we can write T = X*X, where in x12 ' X = ^21 X22 I Then / «.* ~* I / \ Xll ^21 x x X*nXn + X*2lX2l X*nXU+X21X22 T = n 12 a '12 ^22 «21 ^22 \ J Therefore, 0 = x\xxu + a^i^i — ^li^n > OJ which implies that x\\ = 0. Same argument holds for x2X — 0. Hence b = x*nXi2 + xWx22 = 0. • Theorem 6.0.41. (Choi-Effros) Let S C £?(H) &e an injective operator system, and let ^-operation is a C*-algebra. Moreover, the identity map from S to the C*-algebra (S, o) is a unital complete order isomorphism. Proof. It is evident that aob € S for a, b £ S. Distributivity is clear and aol = loa = a. It remains to show associativity, ao(poc) = (aofe)oc. i.e., (a We claim that for any x E B(H) and a £ S that 4>{4>{x)a) — (f>(xa) and tfi(a 4>{ax). Assume that claim, we have that 4>(a(f>(bc)) — 65 and thus associativity follows. Now we prove the claim. Recall the Schwarz inequality, i)(y*y) ~ iJ{y)*i,{y) > 0 for any unital completely positive map tp. Applying this to /. \ ip = 4>^ and a x gives / \ \ Applying (f)^ to this inequality yields / \ 0 4>(ax) — (p(a Since the matrix is positive, 4>(ax) — 4>(a,(fi(x)) = 0 by the previous lemma. Since Now we verify that \\a* o a\\ = ||a||2. Since \\a* o a\\ = \\ By the Schwarz inequality, cf>(a*a) > \\a*a\\ — ||a||2. Hence, (5,o) is a C*-algebra. Clearly the identity map from S to (S, o) is an isometry. Consider Mn(S) C Mn(B(H.)) n Y B(HW) and 0< ) : B(^W) -> Mn(S). By the above, Mn(5) is a C *-algebra with n product A on B = 0( )(,4 • £?), and this C*-algebra is isometrically isomorphic to the operator system Mn{S). But for A = (a^), i? = (bij), we have n n n {n) A on B = 4> (^2aikbkj) = (^0(aifc6fcj)) = (^aifc o bkj.) fe=i fc=i fc=i 66 Thus, (Mn(i.9),on) is the C*-algebra tensor product of Mn and (5,o), and hence by the uniqueness of C*-norm on Mn((S,o)), the identity map from S to (S,o) is an n-isometry for all n. Since the identity map is a unital complete isometry, it is a complete order isomorphism. • We want to construct the injective envelope of an operator space. Definition 6.0.42. Given an operator space F, we say that (S, K) is an injective envelope of F provided that [22] 1. E is injective in O; 2. K : F —> E is a complete isometry; 3. if Ei is injective with K(F) C EI C E, then E\ — E. Assume that F C B(7i). We call a map (j> o We define a partial order on F-projections by setting ip ~< (f> provided that ipocfr = ip — cf) o ijj. Given an F-map By invoking Zorn's lemma, we can easily show the next proposition. 67 Proposition 6.0.43. Let F C B(%) be an operator space. Then there exist minimal F-seminorms on B{7i). Proof. Let 4>x : B(H) —•> B(H) be F-maps such that p^A is a decreasing chain of F-seminorms. Let CBX(B{H),B(H)) = {L e B(B(H),B(H)) : ||L|U < 1}, then CBi(B(7i), B(7i)) is J3W-compact. Thus {(f>\} has a subnet { ing to 4>. Clearly, 0 is an F-map, and since \(4>(x)h, k)\ = liiali\((j)xli(x)h,k)\ < liminfM ||0^(x)||||/i||||A;||, it follows that p^ < p(j>x for all A. Thus, every decreasing chain of F-seminorms has a lower bound, and it follows by Zorn's lemma that minimal F-seminorms exist. • The following theorem gives us the construction of the injective envelope of an operator space. Theorem 6.0.44. Let F C B(H) be an operator space. If F-map such that p^ is a minimal F-seminorm, then the range Proof. We begin be proving that is a F-projection. Since ip o cp is also an F-map and ||^((x))|| < ||<^(x)||, we must have by minimality that \\(f> o (n) ^n(x) = [ 68 too. Hence, U(x)-^o = \\i>n{x- ^2|Ms)|| ;Q ~ n Thus, \\(f>(x) — 0o 4>(x)\\ = 0, and it follows that 4> is an F-projection. Now suppose that ip is a F-projection with I/J -< 0, so that ipo \\ip(x)\\ = \\^(4>(x))\\ < \\ U(x) - 4,(x)\\ = ||0(0(.T) - ^(.x))|| = U(4>(x) - #x))|| = !l#*0 - 4(x)\\ = 0, hence 0(x) = VK2-) for all x. Thus, 0 is a minimal F-projection. Since B(H) is injective in O, and 0 is a completely contractive projection, it follows readily that (f>(B(7i)) with Fj injective in 0. Then the identity map from Fi to E\ extends to a completely contractive projection 7 from B(H) to Fj. Since 7 o 0 is an F-map and ||7(0(x))|| < ||0(x)||, we have that ||7 o (f)(x)\\ = ||0(x)|| by minimality of the seminorm p^. Since 7 is an isometry on c/)(B(7i)) and since 7(^(3;) —70 4>(x)) = 0, we have 4>(x) = j( To show the uniqueness, we need the following lemma [22]. 69 Lemma 6.0.45. Let F C B(H) be an operator space, with 4> '• B(H) —> B{H) a F- map such thatp^ is a minimal F-seminorm. Ifj : 0 : B{7i) —» 0 : B(TC) is completely contractive and j(x) = x for all x G F, then j(4>(x)) — (j>(x) for all x £ B{7i). Proof. Since ||7(0(x))|| < ||^(s)||, we have that ||7((a;))|| = 110(^)11 by minimality of pfi. Hence 7 is an isometry and, in particular, one-to-one. Since p10 'jo(j) = 'yo(f)o~fO(f) = ~fO'yo(j), because 7 o (0 — 7 o (j)) = 0. But since 7 is one-to-one, ^ = 7 o 0 and the result follows. • We now know the existence of an injective envelope, the following theorem shows its uniqueness [22]. Theorem 6.0.46. Let (E\, K,\) and (E2, K2) be two injective envelopes of the operator space F. Then the map i : «i(F) —> K2(F) given by i(Ki(m)) = K2(m) extends uniquely to a completely isometric isomorphism of E\ onto E2. Proof. Let F C B{H) with : B(H) —• B(7i) be a F-map such that p^ is a minimal F-seminorm. If we can prove that the map K\ : F —> E\ has a unique extension to a completely isometric isomorphism 71 : (f)(B(7i)) —> F1; then the result will follow, l because ~f2 07! will be the desired map between E\ and E2. By injectivity of Fi, a completely contractive map 71 : c/)(B(Tl)) —> Fi extending «i exists, and since (p(B(7i)) is injective, there is a completely contractive map /? : 70 Ex -* (f>(B(Ti)) with /?(KI(.T)) = x for all .x in F. Now /3 o «: : Since /? and K\ are both completely contractive, it follows that K\ must be a completely isometry. But the range of K\ will then be an injective operator subspace of E\, and so by the minimality of E\, K\ must be onto. Hence, K\ is completely isometric isomorphism. • Now we are ready to show the corollary mentioned at the beginning of the section. Corollary 6.0.47. Let E be a C*-algebra and let I be an injective C*-algebra such that E C I, then E C 1(E) C / where 1(E) is the injective envelope of E. Proof. Since an injective envelope 1(E) of E is a minimal injective containing E, we only need to show that 1(E) C /. Because / is injective, then, by Theorem 4.1, there exists a completely contractive projection $ : B(H) -> B(H) such that I = §(B(H)). Since E C I, $(x) = x for all x G E. Then $ is an F-projection. By Proposition 4.3, there exists a minimal F-seminorm on B(7i). There are two cases. Case 1, p$ is a minimal F-seminorm. By Theorem 4.4, $ is a minimal F-projection and 1(E) = <5>(B(H)), that is, 1(E) = /. Case 2, p$ is not a minimal F-seminorm, then we assume that p^ is a minimal F- seminorm. Then ^ is a minimal F-projection and 1(E) = ^(B(7i)). Hence, \I> < <£>, 71 that is, $ o 7(E) = *(B(W)) C $(B(W)) = /. • An inclusion of operator spaces V C W is rigid if for each complete contraction W. we have that 0|y = idy implies that (j> — idw [9]. We say that V The rigidity property can be used to characterize injective envelope [22]. Corollary 6.0.48. Let (E, K) be an injective envelope of F, and let : E —> E be completely contractive with ^{K{X)) = K(X) for all x € F. Then ip(e) = e for all eE E. Proof. By the lemma 6.0.45, it is true for (j){B{7i)) and E are completely isometrically isomorphic. This property is preserved by completely isometric isomorphisms. • 72 Chapter 7 Main Result N. Ozawa conjectures that, in general, for any separable exact C*-algebra A there exists a nuclear C*-algebra B between A and its injective envelop 1(A) and B is the unique nuclear C*-algebra that contains A rigidly, which means that the only completely positive map on 1(A) that restricts to the identity of A is the identity map. We say a C*-algebra A is nuclear if, for each C*-algebra B, there is only one C*-norm on the tensor product of A and B. In the paper published in 2007, Ozawa proves that, for a free group T, the reduced crossed product C*(F,dT),which is separable and nuclear, sits between the reduced group C*-algebra C*(F), which is separable and exact, and its injective envelope I(C*(T)). From the discussion above, we know that the boundary dT of the free group T is compact equipped with the product topology, since it is homeomorphic to the Cantor space. Moreover, the action of T on its boundary dT is amenable, therefore, C*(T) is 73 exact, C*(F,dF) is nuclear and the crossed product F x L°°(dF,fi) is injective. Let G be a group acting on the probability spaces (Xi,//i) and (X2,^2)- Let (X\ x X2,Hi <8> ^2) be the product of Xj and X2 with the product measure /^i The diagonal action of G on (Xi x X2, /ii <8> ^2) is defined by fl" Oi,^) = {g-x1,g-x2) for 9 G G, 21 € Ii, and :r2 € X2. A subset A C X is called G-invariant if for each i€i and g £ G, one has gx £ A. A probability measure /i on X is G-invariant if, for every g £ G and Borel subset J4 C X, we have u(gA) = fi(A). If // is G-invariant, then the action of G on the space (X,u) is said to be ergodic if for every G-invariant Borel subset /I C X, either //(^4) = 0 or u(A) — 1. In this case, we shall also say that /i is an ergodic probability measure. Let /A be a quasi-invariant and doubly-ergodic measure on dF. We call // a quasi- invariant measure if for any measurable subset X of dF, one has u(sX) = 0 if and only if /i(X) = 0 for any s G I\ A measure \i on <9r is called doubly-ergodic if the diagonal action of F on (<9r2,/i®2) is ergodic. Furthermore, G*(r, dF) is the G*-algebra generated by L°°(dF, /z) and A(r) where A is the unitary representation of I\ C*(F) is the G*-algebra generated by A(F), therefore, we have the following inclusion: c;(r) c Gr*(r,dr) c r x L°°(di». 74 Proposition 7.0.49. Let ^ be a finite Borel measure on dF, b ET. Ifb-fi = fi, then b = e. Proof. Note that if 6 • // = //, then b~l •// = //. Indeed, for any measurable A, /i(A)=M6-(6_1-A))=/i(6"1-A). Let H be the set of words in d*-/ that do not start with b or b^1. The set H is /i-measurable because H = <9r \ |){words starting with 6™} and each set in the union is closed. Now consider the sets {bnH : n 6 Z}. These n sets are all disjoint, and cT = ^nb H. But by hypothesis, they all have the same measure, and this would force n(dT) to be infinite. • Proposition 7.0.50. Let F be the free group of rank 2 < r < oo, let dF be its boundary and let fi be a quasi-invariant and doubly-ergodic measure on dF. If (f>:C(dF)^ L°°{dT,n) is a unital positive F-equivariant map, then 4> = id. Proof. Let M.(dF) be the space of all finite Borel measures on dF and let M<2{dF) be its subset of measures which are supported on at most two points. In [2], Adams proved that M.>z{3F) = M.{dF) \ M.<2(dF) is a tame T-space. From this and the 75 double ergodicity of //, it follows that any T-equivariant Borel map from dT2 into M>3(dT) has image in a single orbit //®2-almost everywhere. Let 7 : dT2 —> ,M>3(dr) be a T-equivariant Borel map. Then there exists M C <9r2 with ^®2{M) = 0 and j(dT2 \ M) = {a • -y(x,y) : a £ T} for certain x,y £ dF. Now consider the set A = l~l{{l{x,y)}). By the T-equivariance of 7, b • A — l~1({'l(bx,by)}). If a,b £ T and a ^ b, then a • A n b • A = 0. Indeed, suppose that z £ a • A n b • A. Then a • j(z) = b • 7(2), that is b~la • 7(2;) = 7(2;). By Proposition 7.0.49, b~la — e, that is a = b. If n®2(A) = 0, then the quasi-invariance of yi®2 implies that fj,02(b • A) = 0. Thus the sets b • A are all disjoint when b runs through T. And so 2 2 2 1 1 = ^ {dT \ M) = /x« (7- ({6 • l{x, y):b£ 7})) 2 1 2 = ^ (7- (U6er{6 • j(x, y)})) = ^ (Ub€r({b • 7(x, y)})) 2 82 = ^ (Uber6-A) = J> (&./1) fcer = 0 The contradiction shows that /x®2(A) > 0. And since T is a group and the image of 7 an orbit, we could have started with any b • A, so in fact we have shown that /x®2(6 • A) > 0 for every b £ T. Also, YlbV®2^ • A) = 1 (this is what we just proved), so 0 76 Now fix b G F other than the identity. Consider the set B=[jbn-A. As r is nonabelian, there exists a G V such that a • An B = 0. So from the previous paragraph, it is clear that 0 < /x®2(i3) < 1. Also, b • B — B. so the set b contradicts the ergodicity of ^®2. The contradiction shows that the map 7 cannot exist. Thus any 2 2 T-equivariant Borel map form dT into M.(dY) has image in M.<2(dT) /i® - almost everywhere. Fix a dense T-invariant *-subalgebra C in C(dT) which is algebraically (over C) gen erated by a countable set. Then there exists a T-equivariant Borel map such that for ^-a.e. £ G T, we have //fa)#Sfa) = «K/)(0>for/eC. We consider the T=equivariant Borel map 2 (C, r?) G sr -.Ai(dr) 9 0« + ^ + 5r;, where ^ is the Dirac measure on £ G <9I\ By the previous result, this map has image in M<2(dT) ii®2~ almost everywhere. Since the diagonal subset of OF2 is fj,®2-nul\ 77 and £ and n are independent, we conclude that 4>\ for /i- a.e. £ 6 <9r, which means that Theorem 7.0.51. Lei T be the free group of rank 2 < r < oo, H <9r 6e rfs boundary and let [i be a quasi-invariant and doubly-ergodic measure on dF. If 0:c;(r,3r)-^rxL°°(dr» is a completely positive map with 9\c*(v) == ^c*(v), then 9 = id. Proof. By the definition and universal property of crossed products, we know that the crossed product von Neumann algebra T x L°°(dT,fM) is generated by L°°(<9r,//) and A(r) where A is the unitary representation of F. Moreover, the covariant property holds, i.e., \(s)f\(s)* = s • f for s e F and / G L°°(9r,^). There exists a normal faithful conditional expectation E from F x L°°(dF,^i) to L°°(dF,fj,) given by / for 5 = 1 { 0 for s ^ 1 Let V — Sser Xs^(s) € T x L°°(dF,fi) where x.s is the characteristic function, then 78 E(y) = E(£aerXaKs)) = Xe- Therefore, E(\(t)y\(ty) = E(J2Ht)Xs\(s)\(ty) = E(^Kt)x.KnKt)Hs)Kt)') ser = E(Y,t • XsKtsr1)) ser = t-Xe = t • E(y) Let 9 : C*(T,dT) —> T x L°°(c?r,//) be a completely positive map such that 0|c;(r) = idc*(r)- Consider the unital positive map cf> = E o 9\c(or)- Since 9\c*(r) = ^c;(r), the completely positive map 6 is automatically a C*(T)-bimodule map, by Theorem 4.1. Hence, for any s £ F and / G C(<9r), we have 4>{s • f) = E(9(\(s)f\(s)*)) = E(X(s)9(f)X(sY) = s • E(9(f)) = s • Hf) 79 By Proposition 5.1, 0 < E((9(f) - f)2) = E(9(f)2 + f-2Ree(f)f) = f2 + f2-2ReE(e(f))f = 0 This implies that 6\c(ar) = idc{3Y)- Therefore, the conclusion follows. D Corollary 7.0.52. Let T and dT be as above. Then, the nuclear C* -algebra C*(T, dT) sitsfas a C*-algebra) between C*(T) and its injective envelope I(C*(T)); that is, we have natural inclusions c;(r)cc;(r,ar)c/(c;(r)) of C*-algebras. Proof. Since T x L°°(dr,iJ,) is injective, by Corollary 4.6, we have the following in clusion c;(r) c i(c*r{r)) c r x L°°(OT». By Theorem 5.2, a completely positive projection 6 from T x L°°(9r, JJL) onto I(C*(T)) acts identically on C*(T,dT), therefore, c;(r,dT)ci(c;(r)). 80 From Choi-Effros theorem, we know that I(C*(T)) is a C*-algebra under the multi plication defined in Choi-Effros theorem. • 81 Chapter 8 Conclusion N. Ozawa conjectures that for any separable exact C*-algebra A there exists a nuclear C*-algebra B between A and its injective envelop 1(A) and B is the unique nuclear C*-algebra that contains A rigidly. In [21], Ozawa proved that this result holds for the free groups of finite generators, i.e., C*(T) C C*(T, dT) C I(C*(T)) and this embedding is rigid. 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