Stability of group relations under small Hilbert-Schmidt perturbations
aa r X i v : . [ m a t h . OA ] M a r STABILITY OF GROUP RELATIONS UNDER SMALLHILBERT-SCHMIDT PERTURBATIONS
DON HADWIN AND TATIANA SHULMAN
Abstract.
If matrices almost satisfying a group relation are close to matricesexactly satisfying the relation, then we say that a group is matricially stable.Here ”almost” and ”close” are in terms of the Hilbert-Schmidt norm. Usingtracial 2-norm on II -factors we similarly define II -factor stability for groups.Our main result is that all 1-relator groups with non-trivial center are II -factor stable. Many of them are also matricially stable and RFD. For amenablegroups we give a complete characterization of matricial stability in terms ofthe following approximation property for characters: each character must bea pointwise limit of traces of finite-dimensional representations. This allowsus to prove matricial stability for the discrete Heisenberg group H and for allvirtually abelian groups. For non-amenable groups the same approximationproperty is a necessary condition for being matricially stable. We study thisapproximation property and show that RF groups with character rigidity haveit. Introduction
Given an equation of noncommutative variables one can ask if it is ”stable”,meaning that each of its ”almost” solutions is ”close” to a solution.Examples of stability questions are famous questions about almost commutingmatrices, which ask whether almost commuting matrices are close to commutingones. The answers depend very much on classes of matrices and on the matrixnorm one uses to measure ”almost” and ”close”. For instance for the operatornorm those questions are due to Halmos ([18]). When matrices are two self-adjointcontractions the answer is positive by Lin’s theorem ([19]) and when they are twounitaries or three self-adjoint contractions, the answer is negative ([31], [9]). Forthe normalized Hilbert-Schmidt norm the question was formulated by Rosenthal[26] and has an affirmative answer for almost commuting unitaries, self-adjointcontractions and normal contractions ([12], [11], [16]). In our recent work [17]we studied stability of not only commutator relations, but of general C ∗ -algebraicrelations with respect to the normalized Hilbert-Schmidt norm and similar tracialnorms on tracial C ∗ -algebras, in particular on II -factors. There we obtained farreaching generalizations of all the previous results ([12], [11], [16]).The interest to stability questions with respect to the normalized Hilbert-Schmidtnorm also has appeared recently in group theory, in the context of sofic and hyperlin-ear groups ([12], [13], [2]). In particular one is interested in the question of whetherpermutation matrices almost satisfying a group relation are close to permutationmatrices exactly satisfying the relation. Here ”almost” and ”close” are measured Mathematics Subject Classification.
Primary 20Fxx; Secondary 46Lxx.
Key words and phrases. tracial ultraproduct, almost commuting matrices, 1-relator groups,character rigidity. by the normalized Hamming distance. For relations defining a finitely-generatedabelian group it was answered in the affirmative by Arzhantseva and Paunescu [2](in fact they proved it not only with respect to the normalized Hammng distancebut for arbitrary metrics). Although proving stability for permutations is not thesame as for general unitary matrices and requires different techniques, however, aswas noticed in [2], it has similar flavor because the normalized Hamming distancecan be expressed using the Hilbert-Schmidt distance.In this paper we focus on stability of group relations with respect to the normal-ized Hilbert-Schmidt norm and similar tracial norms.Let G be a finitely presented discrete group, and let G = h S | R i = h g , ..., g s | r , ..., r l i be its presentation with g i being generators and r j = r j ( g , ..., g s ) being relations.We will say that G is matricially stable if for any ǫ > δ >
0, such thatif k ∈ N and U , . . . , U s are unitary k × k matrices satisfying k − r j ( U , . . . , U s ) k ≤ δ for all j = 1 , . . . , l , then there are unitary k × k matrices U ′ , . . . , U ′ s satisfying r j ( U ′ , . . . , U ′ s ) = 1for all j = 1 , . . . , l , and k U i − U ′ i k ≤ ǫ , for all i = 1 , . . . , s .This natural notion of stability can be easily generalized to arbitrary, not nec-essarily finitely presented, discrete groups using tracial ultraproducts (see section2 for the details). It implies in particular that the property of being matriciallystable does not depend on the choice of a generating set and a presentation. Using tracial 2-norm on II -factors we similarly define II -factor stability forgroups (and some other versions of stability, see section 2).For amenable groups we give a complete characterization of matricial stabilityin terms of the following approximation property for characters: each charactermust be a pointwise limit of traces of finite-dimensional representations (Theorem4). This allows us to prove matricial stability for the discrete Heisenberg group H (Theorem 6) and for all virtually abelian groups (Theorem 5). For non-amenablegroups the same approximation property is a necessary condition for being matri-cially stable (Theorem 3). Thus it is very interesting for us to know what groupshave this approximation property. Recall that a group G has character rigidity ifthe only extremal characters of G which are not induced from the center are thetraces of finite-dimensional representations ([25]). In Corollary 1 we prove that RFgroups with character rigidity have the approximation property above.One of the main results of the paper is II -factor stability for a big class of non-amenable groups, namely for all 1-relator groups with non-trivial center (Theorem10). Many of those groups are also matricially stable (Theorem 8) and RFD (The-orem 11). By a group being RFD we mean that its full C ∗ -algebra is residuallyfinite-dimensional (it is not the same as being residually finite (RF)). In the context of sofic groups the fact that the stability of metric approximations (whena normalized bi-invariant metric is fixed on a class of approximating groups) does not dependon the choice of the generators and the presentation of a finitely presented groups is due to G.Arzhantseva and L. Paunescu [2].
TABILITY OF GROUP RELATIONS 3
Acknowledgements.
The first author gratefully acknowledges a CollaborationGrant from the Simons Foundation. The research of the second-named author wassupported by the Polish National Science Centre grant under the contract num-ber DEC- 2012/06/A/ST1/00256, by the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS and from the Eric Nordgren Research Fellowship Fundat the University of New Hampshire.1.
Preliminaries
If a unital C*-algebra B has a tracial state ρ , we define a seminorm k·k = k·k ,ρ on B by k b k = ρ ( b ∗ b ) / . Suppose I is an infinite set and α is an ultrafilter on I . We say α is nontrivial if there is a sequence { E n } in α such that ∩ n E n = ∅ . Suppose α is a nontrivialultrafilter on a set I and, for each i ∈ I , suppose A i is a unital C*-algebra witha tracial state ρ i . By Y i ∈ I A i we will denote the C ∗ -product of the C ∗ -algebras A i ,that is the C ∗ -algebra Y i ∈ I A i = { ( a i ) i ∈ I | a i ∈ A i , sup i ∈ I k a i k < ∞} with the norm given by k ( a i ) i ∈ I k = sup i ∈ I k a i k . Note that sometimes one usesanother notation for that, ⊕ l ∞ A i , see [24].The tracial ultraproduct α Y i ∈ I ( A i , ρ i ) is the C*-product Y i ∈ I A i modulo the ideal J α of all elements { a i } in Y i ∈ I A i for whichlim i → α k a i k ,ρ i = lim i → α ρ i ( a ∗ i a i ) = 0 . We denote the coset of an element { a i } ∈ Y i ∈ I A i by { a i } α .Tracial ultraproducts for factor von Neumann algebras was first introduced byS. Sakai [27] where he proved that a tracial ultraproduct of finite factor von Neu-mann algebras is a finite factor. More recently, it was shown in [16] that a tracialultraproduct α Y i ∈ I ( A i , ρ i ) of C*-algebras is always a von Neumann algebra with afaithful normal tracial state ρ α defined by ρ α ( { a i } α ) = lim i → α ρ i ( a i ) . If there is no confusion, we will denote it just by ρ .The C ∗ -algebra of all complex n by n matrices will be denoted by M n ( C ). It hasa unique tracial state tr n = tr. By τ α we denote the corresponding tracial state onthe tracial ultraproduct Q αn ∈ N ( M n ( C ) , tr n ) . For a unital C ∗ -algebra A , its unitary group will be denoted by U ( A ). DON HADWIN AND TATIANA SHULMAN Stability for groups
Let G be a finitely presented discrete group, and let G = h S | R i = h g , ..., g s | r , ..., r l i be its presentation with g i being generators and r j = r j ( g , ..., g s ) being relations.We assume that the set S = { g , ..., g s } is symmetric, i.e. for every g i it contains g − i too.Let C be a class of C ∗ -algebras and let A ∈ C be unital with a tracial state ρ . Definition 1. f : S → U ( A ) is an ǫ -almost homomorphism if k − r j ( f ( g ) , . . . , f ( g s )) k ,ρ ≤ ǫ for all j = 1 , . . . , l . Definition 2. G is C -tracially stable if for any ǫ > there is δ > such that for anyunital C ∗ -algebra A ∈ C with a tracial state ρ and for any δ -almost homomorphism f : S → U ( A ) there is a homomorphism π : G → U ( A ) such that k π ( g ) − f ( g ) k ,ρ ≤ ǫ for any g ∈ S . This natural notion of stability can easily be generalized for arbitrary discrete,not necessarily finitely presented, groups.
Definition 3. G is C -tracially stable if for any tracial ultraproduct α Y i ∈ N ( A i , ρ i ) ofunital C ∗ -algebras A i ∈ C with a trace ρ i , any homomorphism f : G → U α Y i ∈ N ( A i , ρ i ) ! is liftable, meaning that for each i ∈ N there is a homomorphism f i : G → U ( A i ) such that f ( g ) = { f i ( g ) } α . We will show now that for a finitely presented group these two definitions ofstability coincide. It will imply in particular that in the first definition the propertyof being stable does not depend on the choice of a generating set and a finitepresentation.
Proposition 1.
For a finitely presented group the two definitions of stability abovecoincide.Proof.
Let G = h S | R i = h g , ..., g s | r , ..., r l i . To see that the first definition of stability implies the second one, assume that G is C -stable with respect to the first definition of stability and let f : G →U α Y i ∈ N ( A i , ρ i ) ! be a homomorphism. First of all we notice that any unitary in A similar notion was introduced in [20, 21] with the difference that there the operator normwas involved. One should distinguish ǫ -almost homomorphisms from completely different notionsof group quasi-representations and δ -homomorphisms as in [28, 6], where almost multiplicativityis required on the whole group. TABILITY OF GROUP RELATIONS 5 α Y i ∈ N ( A i , ρ i ) can be lifted to a unitary in Q i ∈ N A i . Indeed let u ∈ α Y i ∈ N ( A i , ρ i ) be aunitary. Let T be the unit circle. Since C ( T ) is the universal C ∗ -algebra generatedby one unitary, there is a ∗ -homomorphism φ : C ( T ) → α Y i ∈ N ( A i , ρ i ) such that φ ( z ) = u (here z ∈ C ( T ) is the identity function). By [[17], Th.5.3] applied to T , φ can be lifted to a ∗ -homomorphism ψ : C ( T ) → Q i ∈ N A i . Then ψ ( z ) is a unitarylift of u .Thus for each 1 ≤ k ≤ s , we can write f ( g k ) = { g k ( i ) } α , for some g k ( i ) ∈ U ( A i ) , i ∈ N . We then have, for each j ≤ l ,0 = k f ( r j ( g , . . . g s )) − k ,ρ α = k r j ( f ( g ) , . . . f ( g s )) − k ,ρ α =lim i → α k r j ( g ( i ) , . . . , g s ( i )) − k ,ρ i . Since α is a nontrivial ultrafilter on N , there is a decreasing sequence E ⊃ E ⊃ · · · in α such that ∩ k ∈ N E k = ∅ . Since G is C -stable with respect to the first definition,for each positive integer m there is a number δ m > k r j ( g ( i ) , . . . , g s ( i )) − k ,ρ i < δ m ,j ≤ l , there is a homomorphism γ m,i : G →U ( A i ) such thatmax ≤ k ≤ s k g k ( i ) − γ m,i ( g k ) k ,ρ i < /m. Since lim i → α k r j ( g ( i ) , . . . , g s ( i )) − k ,ρ i = 0 we can find a decreasing sequence { A n } in α with A n ⊂ E n such that, for every i ∈ A n k r j ( g ( i ) , . . . , g s ( i )) − k ,ρ i ≤ δ n . For i ∈ A n \ A n +1 we define f i = γ n,i . For i ∈ N \ A we define f i arbitrarily. Wethen have that { f i } i ∈ N is a lifting of f .On the other hand, if G is not C -stable with respect to the first definition ofstability, then there is an ε > n there is aunital C*-algebra A n with a trace ρ n and g ( n ) , . . . , g s ( n ) ∈ U ( A n ) such that k r j ( g ( n ) , . . . , g s ( n )) − k ,ρ n < /n, but for every homomorphism γ : G → U ( A n )max ≤ k ≤ s k g k ( n ) − γ ( g k ) k ,ρ n ≥ ε. If we let α be any free ultrafilter on N , we have that the map f defined by f ( g k ) = { g k ( n ) } α is a homomorphism from G into U α Y n ∈ N ( A n , ρ n ) ! that is not liftable. (cid:3) DON HADWIN AND TATIANA SHULMAN
Given a discrete group G and a C*-algebra A , let π : G → U ( A ) be a unitaryrepresentation of G on U ( A ). Let C G denote the group algebra of G . Then π induces a homomorphism π : C G → A . Recall that the full C ∗ -algebra C ∗ ( G ) isthe completion of C G with respect to the norm k a k := sup {k π ( a ) k : π : G → U ( A ) is a homomorphism } . The C ∗ -algebra C ∗ ( G ) has the following universal property (which determinesit uniquely up to isomorphism of C ∗ -algebras). Given any C ∗ -algebra A and anyunitary representation π : G → U ( A ), there exists a unique ∗ -homomorphism˜ π : C ∗ ( G ) → A that satisfies ˜ π ( δ ( g )) = π ( g ) for every g ∈ G (here δ : G → C G isthe canonical embedding).In [17] we introduced the following definition of C -tracial stability for C ∗ -algebras.We call a C ∗ -algebra A C -tracially stable if for any ultrafilter α on N and any unital C ∗ -algebras A i ∈ C with a trace ρ i , any ∗ -homomorphism φ : A → α Y i ∈ N ( A i , ρ i ) isliftable.Our definition of stability for groups agrees with the definition of tracial stabilityfor C ∗ -algebras in the following sense. Proposition 2.
A group G is C -stable iff its full C ∗ -algebra C ∗ ( G ) is C -traciallystable.Proof. Assume G is C -stable and let φ : C ∗ ( G ) → α Y i ∈ N ( A i , ρ i ) be a ∗ -homomorphism,for some A i ∈ C . Define a unitary representation f : G → U α Y i ∈ N ( A i , ρ i ) ! by f ( g ) = φ ( δ ( g )) . Since G is C -stable, f lifts to a unitary representation f ′ : G →U (cid:0)Q i ∈ N A i (cid:1) . By the universal property of C ∗ ( G ) there exists a ∗ -homomorphism˜ f ′ : C ∗ ( G ) → Q i ∈ N A i such that ˜ f ′ ( δ ( g )) = f ′ ( g ), for all g ∈ G . It implies that forany a ∈ C G , ˜ f ′ ( a ) is a lift of φ ( a ). Since C G is dense in C ∗ ( G ), it implies that ˜ f ′ is a lift of φ .Now assume C ∗ ( G ) is C -tracially stable and let f : G → U α Y i ∈ N ( A i , ρ i ) ! bea homomorphism, for some A i ∈ C . By the universal property of C ∗ ( G ) thereexists a ∗ -homomorphism ˜ f : C ∗ ( G ) → α Y i ∈ N ( A i , ρ i ) such that ˜ f ( δ ( g )) = f ( g ), forall g ∈ G . Since C ∗ ( G ) is C -tracially stable, we can lift ˜ f to a ∗ -homomorphism ψ : C ∗ ( G ) → Q i ∈ N A i . Then a homomorphism f ′ : G → U (cid:0)Q i ∈ N A i (cid:1) defined by f ′ ( g ) = ψ ( δ ( g )) will be a lift of f . (cid:3) Recall that a C ∗ -algebra has real rank zero (RR0) if each self-adjoint elementcan be approximated by self-adjoint elements with finite spectra.In this paper the role of the class C will be played by the class of all matrix C ∗ -algebras, the class of all II -factors, the class of all von Neumann factors andthe class of all C ∗ -algebras of real rank zero.Thus we will address matricial stability, II -factor stability, W ∗ -factor stabilityand RR TABILITY OF GROUP RELATIONS 7 real rank zero ([5]), RR W ∗ -factor stability, and of course W ∗ -stability implies both matricial and II -factor stability.From now on let G be a discrete countable group. Theorem 1.
The classes of matricially stable groups, II -factor stable groups, W ∗ -factor stable groups, and RR -stable groups are closed under finite free products andunder the direct product with an abelian group.Proof. This follows from [Th. 2.7 and Prop. 2.9 in [17]]. (In fact Th.2.7 in [17] isproved for the class of C -tracially stable C ∗ -algebras, where the class C ⊆ RR C is closed only under unital corners, and thus the theorem applies for matricial, II -factor and W ∗ -factor stability too). (cid:3) Of course besides W ∗ -factor stability one also can introduce W ∗ -stability mean-ing liftings from tracial ultraproducts of (not necessarily factorial) von Neumann al-gebras. In general we don’t know if W ∗ -factor stability coincides with W ∗ -stability.However if a group is finitely presented, then they coincide as we show below. Allnecessary information about direct integrals and measurable cross-sections can befound in [1]. Theorem 2.
Let G be a finitely presented group. Then G is W ∗ -factor stable ifand only if it is W ∗ -stable.Proof. We will give a proof for a group presented by one relation, because for finitelymany relations it is absolutely similar. So let G = h x , . . . , x s | φ ( x , . . . , x s ) = 1 i .The ”if” part is obvious, so let us assume that G is W ∗ -factor stable. Then for any ε > δ > M , τ ), for all y , . . . , y s ∈ U ( M )we have that if k ϕ ( y , . . . , y s ) − k ,τ < δ , there is a homomorphism π : G →U ( M ) such that(2.1) s X k =1 k y k − π ( x k ) k ,τ < ε/ . We are going to prove that then for any ε > δ := r δ ǫ s such that, for all von Neumann algebras ( M , τ ), for all y , . . . , y s ∈ U ( M ) we havethat if k ϕ ( y , . . . , y s ) − k ,τ < δ, there is a homomorphism π : G → U ( M ) suchthat s X k =1 k y k − π ( x k ) k ,τ < ε. So let ( M , τ ) be a von Neumann algebra, y , . . . , y s ∈ U ( M ),(2.3) k ϕ ( y , . . . , y s ) − k ,τ < δ. Without loss of generality we can assume that τ is faithful and also we can replace M with W ∗ ( y , . . . , y s ), so we can assume M = W ∗ ( y , . . . , y s ). Then M actsfaithfully on L ( W ∗ ( y , . . . , y s ) , τ ), which is a separable Hilbert space. Thus wecan write M = Z ⊕ Ω M ω dµ ( ω ) DON HADWIN AND TATIANA SHULMAN for some probability space (Ω , µ ), where each M ω is a factor von Neumann al-gebra with a unique faithful normal tracial state τ ω , and such that, for every y = R ⊕ Ω y ( ω ) dµ ( ω ) ∈ M , we have τ ( y ) = Z Ω τ ω ( y ( ω )) dµ ( ω ) .Hence k y k ,τ = τ ( y ∗ y ) = Z Ω k y ( ω ) k ,τ ω dµ ( ω ) . Let E = n ω ∈ Ω : k ϕ ( y ( ω ) , . . . , y s ( ω )) − k ,τ ω ≥ δ o . Then k ϕ ( y , . . . , y s ) − k ,τ = Z Ω k ϕ ( y ( ω ) , . . . , y s ( ω )) − k ,τ ω dµ ( w ) ≥ Z E k ϕ ( y ( ω ) , . . . , y s ( ω )) − k ,τ ω dµ ( w ) ≥ δ µ ( E ) . Using (2.1) and (2.2), it follows that µ ( E ) ≤ δ k ϕ ( y , . . . , y s ) − k ,τ < ε s . For each ω ∈ E , we define π ω : G → U ( M ω ) by π ω ( g ) = 1. Then, for ω ∈ E , s X k =1 k y k ( ω ) − π ω ( x k ) k ,τ ω ≤ s X k =1 s. Hence Z E s X k =1 k y k ( ω ) − π ω ( x k ) k ,τ ω ≤ sµ ( E ) < ε . By W ∗ -factor stability of G , (2.1), for each ω ∈ Ω \ E, there is a representation π ω : G → U ( M ω ) so that s X k =1 k y k − π ω ( x k ) k ,τ ω < ε/ π ω so that, forevery g ∈ G , the map g π ω ( g ) is weak* measurable. Define a representation π : G → U ( M ) by π ( g ) = Z ⊕ Ω π ω ( g ) dµ ( ω ) . Then s X k =1 k y k − π ( x k ) k ,τ = s X k =1 Z Ω k y k ( ω ) − π ω ( x k ) k ,τ = s X k =1 Z E k y k ( ω ) − π ω ( x k ) k ,τ + s X k =1 Z Ω \ E k y k ( ω ) − π ω ( x k ) k ,τ ≤ ε/
37 + Z Ω \ E ε/ dµ ( ω ) < ε. (cid:3) TABILITY OF GROUP RELATIONS 9
Remark.
Using noncommutative continuous functions [14], one can rewrite thisproof to show that any finitely generated C ∗ -algebra which has a unital 1-dimensionalrepresentation is W ∗ -factor tracially stable if and only if it is W ∗ tracially stable. Inparticular Theorem 2 holds for any finitely generated group, not necessarily finitelypresented.3. A necessary condition for matricial stability and acharacterization of matricial stability for amenable groups
Recall that a character of a group G is a positive definite function on G whichis constant on conjugacy classes and takes value 1 at the unit.We will say that a character τ is embeddable if it factorizes through a homomor-phism to a tracial ultraproduct of matrices, that is if there is a non-trivial ultrafilter α on N and a homomorphism f : G → U α Y n ∈ N ( M n ( C ) , tr n ) ! such that τ α ◦ f = τ .This definition is analogous to the definition of embeddable trace on a C ∗ -algebra(see [17]). On an amenable group every character is embeddable. If Connes’ em-bedding conjecture holds, then on any group every character is embeddable.The following easy statement gives a necessary condition for matricial stability. Theorem 3. If G is matricially stable, then each embeddable character of G is apointwise limit of traces of finite-dimensional representations.Proof. Let τ be an embeddable character on G . Then there is a non-trivial ultra-filter α on N and a homomorphism f : G → U α Y n ∈ N ( M n ( C ) , tr n ) ! such that(3.1) τ α ◦ f = τ. By matricial stability of G , there exists homomorphisms f n : G → U ( M n ( C )) suchthat f ( g ) = { f n ( g ) } α . Together with (3.1) it implies that(3.2) τ ( g ) = lim α tr n ( f n ( g )) , for all g ∈ G . It easily implies that there is a subsequence n j such that(3.3) τ ( g ) = lim j →∞ tr n j ( f n j ( g )) , for all g ∈ G . Indeed, since G is countable, we list all its elements as g , g , . . . andthen by (3.2) the set { n ∈ N | | τ ( g ) − tr n ( f n ( g )) | < / } is in α and hence is notempty. So there is n such that | τ ( g ) − tr n ( f n ( g )) | < / . We continue inductively. Suppose n < n < . . . < n k − such that | τ ( g i ) − tr n l ( f n l ( g i )) | < l ,i = 1 , . . . , l , l = 2 , . . . , k −
1, are already found. The set { n ∈ N | n > n k − , | τ ( g i ) − tr n ( f n ( g i )) | < k , i = 1 , . . . , k } = { n ∈ N | n > n k − } \ \ i ≤ k { n ∈ N | | τ ( g i ) − tr n ( f n ( g i )) | < k } is in α and hence is not empty. Thus there is n k > n k − such that | τ ( g i ) − tr n k ( f n k ( g i )) | < k ,i = 1 , . . . , k .Now the statement follows from (3.3). (cid:3) The next 2 statements are corollaries of our results in [17]. The first of themgives a complete characterization of matricial stability and of W ∗ -factor stabilityfor amenable groups. II -factor stability is automatic for amenable groups. Theorem 4.
Let G be an amenable group. The following are equivalent: (1) G is matricially stable (2) G is W ∗ -factor stable. (3) Each character of G is a pointwise limit of traces of finite-dimensionalrepresentations.Proof. As is well known, a positive definite function on G extends in unique way to astate on C ∗ ( G ) (see e.g. [8], p.188), and it is obvious that a positive definite functionis constant on conjugacy classes if and only if the corresponding state is a trace.Thus (embeddable) characters of G are in 1-to-1 correspondence with (embeddable)tracial states on C ∗ ( G ) and the condition (3) is equivalent to the condition thatfor each tracial state τ on C ∗ ( G ) there are finite-dimensional representations π n of C ∗ ( G ) such that τ ( a ) = lim n →∞ trπ n ( a ) , for each a ∈ C ∗ ( G ). Since for any group G , C ∗ ( G ) has a one-dimensional represen-tation, the statement follows from [Theorem 3.8, [17]]. (cid:3) Theorem 5.
The class of W ∗ -factor stable groups contains all virtually abeliangroups.Proof. As is well known, G is virtually abelian if and only if C ∗ ( G ) is GCR ([29],[30]). Since C ∗ ( G ) has a 1-dimensional representation, the statement follows from[Corollary 3.9, [17]]. (cid:3) We will use Theorem 4 to prove that the discrete Heisenberg group is W ∗ -factorstable. Recall that the discrete Heisenberg group H is the group generated by u, v with the relations that u and v commute with uvu − v − . It is known that H isamenable. Lemma 1.
If each extreme character is a pointwise limit of traces of finite-dimensionalrepresentations, then so is any character.Proof.
Let τ be a character, ǫ > g , . . . , g n ∈ G . Since the set of all charactersof a group is convex and compact in ∗ -weak topology, there are rational numbers s /m, . . . , s l /m with s + . . . + s l = m , and extreme characters σ , . . . , σ l such that(3.4) | τ ( g k ) − l X i =1 s i m σ i ( g k ) | ≤ ǫ,k = 1 , . . . , n . By the assumption, there exist representations π i : G → M n i ( C ) suchthat(3.5) | σ i ( g k ) − trπ i ( g k ) | ≤ ǫ. TABILITY OF GROUP RELATIONS 11
Let L ∈ N be such that s i n i L is an integer, for all 1 ≤ i ≤ l . Let π = ⊕ li =1 π (cid:16) sini L (cid:17) i (here π (cid:16) sini L (cid:17) i denotes a direct sum of s i n i L copies of π i ). It is easy to check that(3.6) trπ ( g k ) = l X i =1 s i m trπ i ( g k ) ,k = 1 , . . . , n . By (3.4), (3.5), (3.6), | τ ( g k ) − trπ ( g k ) | ≤ ǫ. (cid:3) Theorem 6. H is W ∗ -factor stable.Proof. Suppose τ is an extreme point in the set of characters of H . Then it extendsto an extreme tracial state on C ∗ ( H ), i.e. a factor tracial state on C ∗ ( H ). We willdenote it also by τ . Let π : C ∗ ( H ) → B ( H ) be the GNS representation for τ . Let U = π ( u ) and V = π ( v ). Since π ( A ) ′′ is a factor and U V U − V − = π (cid:0) uvu − v − (cid:1) is in its center, there is a real number θ such that U V U − = e πiθ V and V − U V = e πiθ U First suppose θ is rational, then there is a positive integer n such that nθ ∈ Z . Inthis case we have U n V U − n = V and V − n U V n = U, which implies U n = α and V n = β for scalars α and β . For every positive integer m there is a positive integer k such that m < kn . Thus U − m = U kn − m U kn = α k U kn − m and V − m = β k V kn − m . Since
U V = e πiθ V U , every monomial in
U, V, U − , V − can be written as ascalar times U a V b for integers a, b with 0 ≤ a, b < n . Hence C ∗ ( U, V ) is finite-dimensional, which means C ∗ ( U, V ) = C ∗ ( U, V ) ′′ is isomorphic to M k ( C ) for some k ∈ N . Hence τ is a matricial tracial state.Next suppose θ is irrational. Then U, V give a representation of the irrational ro-tation C ∗ -algebra A θ . Since A θ is simple, C ∗ ( U, V ) is isomorphic to A θ and hencehas a unique tracial state. In this case we can choose a sequence { θ k } of rationalnumbers such that θ k → θ , and find finite-dimensional irreducible representations π k : H → M n k ( C ) such that π k (cid:0) uvu − v − (cid:1) = e πiθ k . Let α be a non-trivial unl-trafilter on N , then in the tracial ultraproduct Q αi ∈ N M n i ( C ) we get ˆ U = { π k ( u ) } α and ˆ V = { π k ( v ) } α satisfy ˆ U ˆ V ˆ U − ˆ V − = e πiθ . Thus C* (cid:16) ˆ U , ˆ V (cid:17) is also isomorphicto A θ and hence has a unique tracial state which has to coincide with τ . Hence,for every a ∈ C ∗ ( H ) τ ( a ) = lim k → α tr ( π k ( a )) . It follows from Lemma 1 and Theorem 4 that H is W ∗ -factor stable. (cid:3) Remark.
It would be interesting to know if our characterization of matricial sta-bility for amenable groups can be reformulated in terms of ”separation properties”of groups. By this we mean properties like residual finiteness (which means that agroup has a separating family of homomorphisms into finite groups), the propertyof being maximally almost periodic (which means that a group has a separat-ing family of finite-dimensional representations), the property of being conjugacyseparable (which means that homomorphisms to finite groups separate conjugacyclasses), the property that finite-dimensional representations separate conjugacyclasses, etc. For example, it is easy to see that for an amenable group matricialstability implies that the group is maximally almost periodic. We don’t know if itis also a sufficient condition, and we believe that it is not. Otherwise for C ∗ ( G ) tobe nuclear and matricially tracially stable would be equivalent to be nuclear RFD(since an amenable group G is maximally almost periodic iff C ∗ ( G ) is RFD by [4])and in in [17] we constructed an example of nuclear RFD C ∗ -algebra which is notmatricially tracially stable. This makes us think that for an amenable group beingmaximally almost periodic is probably not sufficient for matricial stability. Separa-tion properties for conjugacy classes seem to us to be more relevant. For instanceif a group is conjugacy separable, then the Stone-Weierstrass theorem leads to aneasy proof that each character of G is a pointwise limit of linear combinations oftwo traces of finite-dimensional representations (which is close to the condition 3)in Theorem 4). In the opposite direction, by Theorem 4 the property that finite-dimensional representations separate conjugacy classes would be necessary if thecharacters separate conjugacy classes. Question : Let G be an amenable maximally almost periodic group. Do itscharacters separate conjugacy classes?4. Character rigidity and the approximation property ( ∗ ). Below we will say that a group G has the approximation property ( ∗ ) if anyembeddable character of G is a pointwise limit of traces of finite-dimensional rep-resentations. Thus by Theorem 3, the approximation property ( ∗ ) is necessary forbeing matricially stable, and by Theorem 4, if a group is amenable, then it is alsosufficient.Following [25] (also [3]) we will say that a character is induced from the center ifit vanishes outside the center.An example of a character induced from the center is a character δ e defined by δ e ( g ) = (cid:26) g = e g = e. Proposition 3.
Let G be a maximally almost periodic group. Then δ e is a pointwiselimit of traces of some finite-dimensional representations of G .Proof. Let ǫ > g , . . . , g n ∈ G , g i = 1 for all i = 1 , . . . , n . Since G is maximallyalmost periodic, we can find a finite-dimensional representation π such that π ( g i ) =1 for all i = 1 , . . . , n . Let χ : G → C be the trivial representation, ˜ π = π ⊕ χ . Then | tr ˜ π ( g i ) | = (cid:12)(cid:12)(cid:12)(cid:12) ( dimπ ) trπ ( g i ) + 1 dimπ + 1 (cid:12)(cid:12)(cid:12)(cid:12) < , since this is absolute value of the average of numbers of absolute value not largerthan 1, not all of which are equal. TABILITY OF GROUP RELATIONS 13
Let ˜ π ⊗ N be the N-th tensor power of the representation ˜ π . Then tr ˜ π ⊗ N ( g i ) = ( tr ˜ π ( g i )) N < ǫ if N is big enough. Thus | tr ˜ π ⊗ N ( g i ) − δ e ( g i ) | < ǫ for i = 1 , . . . , n . (cid:3) Below we will show that when a group is residually finite (RF), the approximationproperty above holds not only for δ e but for all characters induced from the center. Lemma 2.
Let G be a RF group and suppose its center Z ( G ) is finitely generated.Let g , . . . , g N ∈ Z ( G ) , g i = g j when i = j . Let H be the subgroup generated by g , . . . , g N and χ be a 1-dimensional representation of H . Let g ′ , . . . , g ′ m / ∈ Z ( G ) and let ǫ > . Then there exists a finite group G , a surjective homomorphism f : G → G and a 1-dimensional representation ˜ χ of f ( H ) such that | ˜ χ ( f ( g i )) − χ ( g i ) | < ǫ, for i = 1 , . . . , N and f ( g ′ i ) / ∈ f ( Z ( G )) , for i = 1 , . . . , N ′ .Proof. Since H is a finitely generated abelian group, it can be written as H = Z s × Γ , where s ∈ N and Γ is a finite abelian group. So we can write g j = ( n j , n j , . . . , n js , t )with n ji ∈ Z , t ∈ Γ, j ≤ N . Let Z ( i ) denote the i-th copy of Z in H . For each i ≤ s there is θ i such that(4.1) χ | Z ( i ) ( n ) = e πinθ i . Let L ( i ) = max j ≤ N | n ji | ,i = 1 , . . . , s . For each i ≤ s there exists k ,i such that for any k ≥ k ,i , the k-throots of unity form an ǫs ( L ( i ) +1) -net in the unit circle.Since g ′ , . . . , g ′ m / ∈ Z ( G ), there exist g ′′ , . . . , g ′′ m ∈ G such that g ′ i g ′′ i = g ′′ i g ′ i , i =1 , . . . , m . Since G is RF, there is a finite group G and a surjective homomorphism f : G → G such that(4.2) f ( g ′ i g ′′ i ) = f ( g ′′ i g ′ i ) , for i = 1 , . . . , m and(4.3) f ( n , . . . , n s , t ) = f ( n ′ , . . . , n ′ s , t ′ ) , when t ∈ Γ, n i , n ′ i ≤ k ,i and the tuples ( n , . . . , n s , t ) and ( n ′ , . . . , n ′ s , t ′ ) do notcoincide. It follows from (4.2) that(4.4) f ( g ′ i ) / ∈ f ( Z ( G )) ,i = 1 , . . . , m .It is easy to see that f ( H ) ∼ = ( Q i ≤ s f ( Z ( i ) )) × f (Γ). Hence f ( H ) = Z k × . . . × Z k s × ˜Γ , for some k , . . . , k s ∈ N and some finite abelian group ˜Γ. It follows from (4.3) that k i ≥ k ,i and that | ˜Γ | ≥ | Γ | . Since ˜Γ is a homomorphic image of Γ, the latter implies that ˜Γ ∼ = Γ. The first inequality, k i ≥ k ,i , implies that there is l i < k i suchthat(4.5) | e πil i /k i − e πiθ i | ≤ ǫs ( L ( i ) + 1) . Define a 1-dimensional representation ˜ χ i of Z k i by˜ χ i ( m ) = e πiml i /k i , for each m ∈ Z k i . Using (4.5), for any m ≤ L ( i ) we easily obtain by induction that | ˜ χ i ( m ) − χ | Z ( i ) ( m ) | = | e πi ( m mod k i ) l i /k i − e πimθ i | = | e πiml i /k i − e πimθ i | ≤ ǫ ( m + 1) s ( L ( i ) + 1) . In particular for any m ≤ L ( i ) we obtain(4.6) | ˜ χ i ( m ) − χ | Z ( i ) ( m ) | ≤ ǫs . Define a 1-dimensional representation ˜ χ of f ( H ) by˜ χ ( f ( n , . . . , n s , t )) = ˜ χ ( n ) . . . ˜ χ s ( n s ) χ ( t ) , for all n i ∈ Z , t ∈ Γ. From (4.6) we deduce ( estimating | a . . . a s − b . . . b s | in astandard way) that for any n i ≤ L ( i ) , t ∈ Γ | ˜ χ ( f ( n , . . . , n s , t )) − χ ( n , . . . , n s , t ) | ≤ ǫ. Hence | ˜ χ ( f ( g i )) − χ ( g i ) | < ǫ, for i = 1 , . . . , N . This, together with (4.4), completes the proof. (cid:3) Theorem 7.
Suppose G is RF. Then each character of G induced from the centerof G is a pointwise limit of traces of finite-dimensional representations.Proof. By Lemma 1 it will be sufficient to prove that each extreme point of theset of all characters induced from Z ( G ) is a pointwise limit of traces of finite-dimensional representations. Since an extreme point of the set of characters of anabelian group is a 1-dimensional representation, we should prove that if χ | Z ( G ) is a 1-dimensional representation and χ vanishes outside Z ( G ), then χ is a pointwise limitof traces of finite-dimensional representations. Let g , . . . , g N ∈ Z ( G ), g ′ , . . . , g ′ m / ∈ Z ( G ), ǫ >
0. We need to find a finite-dimensional representation π of G such that | χ ( g i ) − tr ( π ( g i )) | ≤ ǫ , i = 1 , . . . , N , and | χ ( g ′ i ) − tr ( π ( g ′ i )) | ≤ ǫ , i = 1 , . . . , m . Let H be the subgroup generated by g , . . . , g N . By Lemma 2 there is a finite group G , a surjective homomorphism f : G → G and a 1-dimensional representation ˜ χ of f ( H ) such that(4.7) | ˜ χ ( f ( g i )) − χ ( g i ) | ≤ ǫ,i = 1 , . . . , N , and(4.8) f ( g ′ i ) / ∈ f ( Z ( G )) ,i = 1 , . . . , m . Let ˜ π be the representation of G induced from the 1-dimensionalrepresentation ˜ χ of f ( H ). By Frobenius formula tr ˜ π ( f ( g )) = X x ∈ G /f ( H ) ˆ χ ( x − f ( g ) x ) , TABILITY OF GROUP RELATIONS 15 where(4.9) ˆ χ ( k ) = ˜ χ ( k ) ; k ∈ f ( H )0 ; k / ∈ f ( H ) . Since f ( H ) is a central subgroup of G , it implies easily that(4.10) tr ˜ π ( f ( g )) = ˜ χ ( f ( g )) ; f ( g ) ∈ f ( H )0 ; f ( g ) / ∈ f ( H ) . Let π = ˜ π ◦ f . Then, by (4.7), (4.8) and (4.10), for each i ≤ N | χ ( g i ) − tr ( π ( g i )) | = | χ ( g i ) − ˜ χ ( f ( g i )) | ≤ ǫ, and for each i ≤ m | χ ( g ′ i ) − tr ( π ( g ′ i )) | = 0 . (cid:3) A group G has character rigidity if the only extremal characters of G which arenot induced from the center of G are the traces of finite-dimensional representations([25]). Corollary 1. If G is RF and has character rigidity, then G has the approximationproperty ( ∗ ). As was proved by Bekka [3] SL ( Z ) has character rigidity. Thus, by Corollary 1,the necessary condition for matricial stability from Theorem 3 holds. Since SL ( Z )is non-amenable, we don’t know if it is also sufficient. Question : Is SL ( Z ) matricially stable?5. One-relator groups with center.
Recall that a one-relator group is a group G with a presentation G = h S | R i where the generating set S is finite and R is a single word on S ± . All 1-relatorgroups but the Baumslag-Solitar groups BS (1 , m ) are non-amenable ([7]).We are going to prove that any one-relator group with a non-trivial center is II -factor stable. All such groups are known to be residually finite ([10]).It was shown in [23] that every such non-cyclic group is presentable in one oftwo ways: as(5.1) G = D x , . . . , x n | x a = x b , x a = x b , . . . , x a n − n − = x b n − n E where a i , b i ≥ a i , b j ) = 1 for i > j (when the commutator quotient group isnot free abelian of rank two); or as(5.2) G = D u, x , . . . , x m | ux u − = x m , x a = x b , x a = x b , . . . , x a m − m − = x b m − m E where a i , b i ≥ a . . . a m − = b . . . b m − , ( a i , b j ) = 1 for i > j (when the commu-tator quotient group is free abelian of rank two).Since cyclic groups are II -factor (even RR
0) stable, we are left with the twocases above. We are not going to use anywhere that ( a i , b j ) = 1. We will need a lemma from [17] adjusted for the case of full group C ∗ -algebras.It states that pointwise k k -limits of liftable homomorphisms are liftable. Lemma 3. ( [17] , Lemma 2.2) Suppose G is a group, { ( A i , ρ i ) : i ∈ N } is a family oftracial C*-algebras, α is a nontrivial ultrafilter on N , and π : G → U α Y i ∈ N ( A i , ρ i ) ! is a homomorphism such that, for each g ∈ G , π ( g ) = { g ( i ) } α . The following are equivalent: (1) π is liftable (2) For every ε > and every finite subset F ⊂ G , there is a set E ∈ α andfor every i ∈ E there is a homomorphism π i : G → U ( A i ) such that, forevery g ∈ F and every i ∈ E , k π i ( g ) − g ( i ) k ,ρ i < ε. Groups of the form (5.1).Lemma 4.
Suppose { ( A n , ρ n ) } is a sequence of tracial C ∗ -algebras of real rank zero, α is a non-trivial ultrafilter on N , and r , . . . , r N , q ∈ α Y n ∈ N ( A n , ρ n ) are projectionssuch that P Ni =1 r i = q . Suppose projections Q n ∈ A n , n ∈ N are such that { Q n } α = q . Then there exist projections R i,n ∈ A n , n ∈ N , i = 1 , . . . , N , such that { R i,n } α = r i , i = 1 , . . . , N , and P Ni =1 R i,n = Q n . Proof.
All r i ’s belong to the tracial ultraproduct α Y n ∈ N (cid:16) Q n A n Q n , ρ n ( Q n ) ρ n (cid:17) . q isthe unit element in this ultraproduct. Since projections with sum 1 generate acommutative C ∗ -algebra, hence RR (cid:3) Theorem 8.
Let G be as in (5.1). Then G is RR -stable.Proof. To avoid notational nightmare we will prove RR G = (cid:10) x, y, z | x = y , y = z (cid:11) , and the proof for the general case is absolutelysimilar.Suppose { ( A n , ρ n ) } is a sequence of tracial C ∗ -algebras of real rank zero, α is anon-trivial ultrafilter on N , and X, Y, Z ∈ α Y n ∈ N ( A n , ρ n ) = def ( A , ρ ) are unitary and X = Y , Y = Z . Then X = Y = Z = def W . We can write X = { X n } α , Y = { Y n } α , Z = { Z n } α . Suppose ε >
0. Since X , Y and Z commute with W ,they commute with every spectral projection of W , and since A is a von Neumannalgebra, the spectral projections of W are in A . We can choose an orthogonalfamily of nonzero spectral projections { P , . . . , P s } of W whose sum is 1 and wecan choose λ , . . . , λ s ∈ T such that if Ω = P sk =1 λ k P k , then k W m − Ω m k < ε, for m ∈ { , / , / , / } . Here and below by W / , W / , etc., we mean thenormal operators obtained by applying the Borel functions z / = def | z | / e iArgz ,etc., to W . TABILITY OF GROUP RELATIONS 17
Let X ′ = XW − / , Y ′ = Y W − / , Z ′ = ZW − / . Then X ′ , Y ′ and Z ′ areunitary and(5.3) ( X ′ ) = ( Y ′ ) (5.4) ( Y ′ ) = ( Z ′ ) , (5.5) ( X ′ ) = ( Y ′ ) = ( Z ′ ) = 1 . Moreover,(5.6) k X − X ′ Ω / k < ε, (5.7) k Y − Y ′ Ω / k < ε, (5.8) k Z − Z ′ Ω / k < ε. Clearly T = s X k =1 P k T P k for T ∈ { X, Y, Z, X ′ , Y ′ , Z ′ , W, Ω } . For each n we can find an orthogonal family { P n, , . . . , P n,s } of projections in A n whose sum is 1 such that, for 1 ≤ k ≤ s,P k = { P n,k } α .It is clear that P sk =1 P k A P k is the tracial ultraproduct α Y n ∈ N ( P sk =1 P n,k A n P n,k , ρ n )and that each (cid:16) P k A P k , ρ ( P k ) ρ (cid:17) is the tracial ultraproduct α Y n ∈ N (cid:16) P n,k A n P n,k , ρ n ( P n,k ) ρ n (cid:17) .By (5.5) X ′ , Y ′ , Z ′ can be written in the form X ′ = X j =1 e πij q j , Y ′ = X j =1 e πij r j , Z ′ = X j =1 e πij s j , where { q j } , { r j } , { s j } are families of projections in P sk =1 P k A P k which sum to 1.It is easy to see that (5.3) is equivalent to the system of equations q + q = r + r + r q + q = r + r + r q + q = r + r + r q + q = r + r + r q + q = r + r + r and (5.4) is equivalent to the system of equations r + r + r + r + r = s + s + s + s + s + s + s r + r + r + r + r = s + s + s + s + s + s + s r + r + r + r + r = s + s + s + s + s + s + s . Since C ∗ ( q , . . . , q ) is commutative, it is RR Q = { Q n, } , . . . , Q = { Q n, } ∈ α Y n ∈ N ( P sk =1 P n,k A n P n,k ) with sum 1 such that q = { Q n, } α , . . . , q = { Q n, } α . By Lemma 4 we can find pro-jections R = { R n, } , . . . , R = { R n, } ∈ α Y n ∈ N ( P sk =1 P n,k A n P n,k ) such that r = { R n, } α , . . . , r = { R n, } α and Q + Q = R + R + R Q + Q = R + R + R Q + Q = R + R + R Q + Q = R + R + R Q + Q = R + R + R . Again by Lemma 4 we can find projections S = { S n, } , . . . , S = { S n, } ∈ α Y n ∈ N ( P sk =1 P n,k A n P n,k ) such that s = { S n, } α , . . . , s = { S n, } α and R + R + R + R + R = S + S + S + S + S + S + S R + R + R + R + R = S + S + S + S + S + S + S R + R + R + R + R = S + S + S + S + S + S + S . Let X ′ n = X j =1 e πij Q n,j , Y ′ n = X j =1 e πij R n,j , Z ′ n = X j =1 e πij S n,j . For each n , let Ω n = P sk =1 λ k P n,k . Then Ω = { Ω n } α . For each n ∈ N there is aunital ∗ -homomorphism π n : C ∗ ( G ) → A n such that π n ( x ) = X ′ n Ω / n , π n ( y ) = Y ′ n Ω / n , π n ( z ) = Z ′ n Ω / n . (Here again by Ω / n , etc., we mean the normal operator obtained by applying theBorel function z / = def | z | / e iArgz , etc., to Ω n . Since Ω n has finite spectrum,Ω / n , etc., belong to A n .) Clearly, { π n ( x ) } α = X ′ Ω / , { π n ( y ) } α = Y ′ Ω / , { π n ( z ) } α = Z ′ Ω / . By (5.6), (5.7), (5.8) and Lemma 3, G is RR (cid:3) Groups of the form (5.2).
We will need a few easy lemmas. The first lemmais folklore.
Lemma 5.
Let M be a II -factor, p ∈ M be a projection and ≤ β ≤ τ ( p ) . Thenthere is a projection p ′ ∈ p M p such that τ ( p ′ ) = β .Proof. It follows from folklore fact that in II -factor one can find a projection withprescribed trace. (cid:3) Lemma 6.
Let p ∈ Q α ( M i , ρ i ) be a projection. Then p can be lifted to a projection { P i } ∈ Q M i with ρ i ( P i ) = ρ ( p ) , for all i . TABILITY OF GROUP RELATIONS 19
Proof.
Lift p to a projection { ˜ P i } . Then ρ i ( ˜ P i ) − ρ ( p ) → α . If ρ i ( ˜ P i ) ≥ ρ ( p ), then by Lemma 5 there is a projection Q i ∈ ˜ P i M i ˜ P i such that ρ i ( Q i ) = ρ i ( ˜ P i ) − ρ ( p ) . Let P i = ˜ P i − Q i . If ρ i ( ˜ P i ) ≤ ρ ( p ), then ρ i (1 − ˜ P i ) ≥ ρ (1 − p ). By Lemma 5 there is a projection Q i ∈ (1 − ˜ P i ) M i (1 − ˜ P i ) such that ρ i ( Q i ) = ρ i (1 − ˜ P i ) − ρ ( p ) . In this case let P i = ˜ P i + Q i . Either way P i is a projection and ρ i ( P i ) = ρ ( p ) . We have ρ i ( P i − ˜ P i ) = ρ ( p ) − ρ i ( ˜ P i ) → α { P i } is a lift of p . (cid:3) Lemma 7.
Let p, q , . . . , q n ∈ Q α ( M i , ρ i ) be projections and P nk =1 q k = p . Sup-pose p is lifted to a projection { P i } with ρ i ( P i ) = ρ ( p ) . Then each q k can be liftedto a projection { Q k,i } such that for all i n X k =1 Q k,i = P i and for all i, k ρ i ( Q k,i ) = ρ ( q k ) . Proof.
By Lemma 6 we can lift q to { Q ,i } ∈ Q α ( P i M i P i , ρ i ρ i ( P i ) ) with ρ i ( Q ,i ) = ρ ( q ) . Now, again by Lemma 6, we can lift q to { Q ,i } ∈ Y α (cid:18) ( P i − Q ,i ) M i ( P i − Q ,i ) , ρ i ρ i ( P i − Q ,i ) (cid:19) with ρ i ( Q ,i ) = ρ ( q ) . Then we lift q to { Q ,i } ∈ Y α P i − X k =1 Q k,i ! M i P i − X k =1 Q k,i ! , ρ i ρ i (cid:16) P i − P k =1 Q k,i (cid:17) with ρ i ( Q ,i ) = ρ ( q ) . Continuing this process we obtain pairwisely orthogonal lifts { Q k,i } ∈ Q α ( P i M i P i , ρ i ρ i ( P i ) ) of q k , k ≤ n −
1, such that ρ i ( Q k,i ) = ρ ( q k ) . Now welift q n to the projection { P i − P n − k =1 Q k,i } ∈ Q α ( P i M i P i , ρ i ρ i ( P i ) ). Then for all i ρ i P i − n − X k =1 Q k,i ! = ρ ( p ) − ρ n − X k =1 q k ! = ρ ( q n )and n X k =1 Q k,i = P i . (cid:3) Theorem 9.
Let G be as in (5.2). Then G is II -factor stable. We would like to warn the reader that the notation in the proof below differsslightly from the notation in the proof of Theorem 8: ˜Ω now plays the role of W and Q i ’s play the role of P i ’s. Proof.
Suppose { ( A n , ρ n ) } is a sequence of II -factors, α is a non-trivial ultrafilteron N , and u, x , . . . , x m ∈ α Y n ∈ N ( A n , ρ n ) = def ( A , ρ ) are unitaries satisfying the grouprelations. Then x a ...a m − = x b a ...a m − = . . . = x b ...b m − m = def ˜Ω . Obviously x , . . . , x m commute with ˜Ω. Let N i = b . . . b i − a i . . . a m − . Since N = N m , u ˜Ω u − = ( ux u − ) N = x N m = x N m m = ˜Ω . Thus u, x , . . . , x m commute with ˜Ω and hence with every spectral projection of ˜Ω,and since A is a von Neumann algebra, the spectral projections of ˜Ω are in A .Let ε >
0. We can choose an orthogonal family of nonzero spectral projections { Q , . . . , Q s } of ˜Ω whose sum is 1 and we can choose λ , . . . , λ s ∈ T such that ifΩ = P sk =1 λ k Q k , then (cid:13)(cid:13)(cid:13) ˜Ω i − Ω i (cid:13)(cid:13)(cid:13) < ε, for i ∈ n , N , . . . , N m o . Here and below by Ω t , ˜Ω t etc., we mean the normaloperator obtained by applying the Borel function z t = def | z | t e itArgz , etc., to Ω , ˜Ω,etc. Clearly T = s X k =1 Q k T Q k for T ∈ n u, x , . . . , x m , ˜Ω , Ω o . For each n we can find an orthogonal family { Q n, , . . . , Q n,s } of projections in A n whose sum is 1 such that, for 1 ≤ k ≤ s,Q k = { Q n,k } α .It is clear that P sk =1 Q k A Q k is the tracial ultraproduct α Y n ∈ N ( P sk =1 Q n,k A n Q n,k , ρ n )and that each (cid:16) Q k A Q k , ρ ( Q k ) ρ (cid:17) is the tracial ultraproduct α Y n ∈ N (cid:16) Q n,k A n Q n,k , ρ n ( Q n,k ) ρ n (cid:17) .Let x ′ i = x i ˜Ω − Ni ,i = 1 , . . . , m. Then(5.9) k x i − x ′ i Ω /N i k < ε, and(5.10) ux ′ u − = x ′ m (5.11) ( x ′ ) a = ( x ′ ) b , ( x ′ ) a = ( x ′ ) b , . . . , ( x ′ m − ) a m − = ( x ′ m ) b m − (5.12) ( x ′ ) a ...a m − = . . . = ( x ′ m ) b ...b m − = 1 . We notice also that u, x ′ Ω /a ...a m − , . . . , x ′ m Ω /b ...b m − satisfy the group rela-tions. TABILITY OF GROUP RELATIONS 21
Now we are going to ”lift” the relations (5.10) – (5.12) and we will do it in twosteps.
STEP 1 : To ”lift” the relations (5.11) and (5.12) so that x ′ and x ′ m will belifted to { X ′ (1) n } , { X ′ ( m ) n } ∈ Q ( P sk =1 Q n,k A n Q n,k ) unitarily equivalent to eachother. (Possibly this unitary equivalence won’t be a lift of u .)To do STEP 1 we notice that the relation (5.12) implies that each x ′ i can bewritten as a linear combination of projections x ′ i = N i X k =1 e πikNi p ( i ) k ,i = 1 , . . . , m . In (5.11) each relation( x ′ i ) a i = ( x ′ i +1 ) b i now translates into a system of linear equations with some of p ( i ) k , k = 1 , . . . , N i ,in the left-hand sides and some of p ( i +1) k , k = 1 , . . . , N i +1 , in the right-hand sides.(We don’t write out the details since we did it in the proof of Theorem 8).Since each p ( i ) k is the direct sum of the projections Q j p ( i ) k Q j , this system oflinear equations translates into s systems of linear equations, one for each coor-dinate. Thus for each j = 1 , . . . , s we have a system of linear equations withsome of Q j p ( i ) k Q j , k = 1 , . . . , N i , in the left-hand sides and some of Q j p ( i +1) k Q j , k = 1 , . . . , N i +1 , in the right-hand sides.We notice also that (5.10) implies that up (1) k u − = p ( m ) k for all k = 1 , . . . , N = N = N m . In particular(5.13) ρ (cid:16) Q j p (1) k Q j (cid:17) = ρ (cid:16) Q j p ( m ) k Q j (cid:17) for all k = 1 , . . . , N , j = 1 , . . . , s .By Lemma 6 we can lift each projection Q j p (1) k Q j to a projection { P (1) k,n,j } ∈ Q ( Q n,j A n Q n,j ) of the same trace as Q j p (1) k Q j . By Lemma 7 we can lift each Q j p (2) k Q j to a projection { P (2) k,n,j } ∈ Q ( Q n,k A n Q n,k ) of the same trace as Q j p (2) k Q j and such that the family n { P (1) k,n,j } , { P (2) l,n,j } | k = 1 , . . . , N, l = 1 , . . . , N o would satisfy the same linear re-lations as the family { Q j p (1) k Q j , Q j p (2) l Q j | k = 1 , . . . , N, l = 1 , . . . , N } . We keepdoing this. We end up with projections { P ( i ) k,n,j } , i = 1 , . . . , m , k = 1 , . . . , N i , j = 1 , . . . , s of the same trace as Q j p ( i ) k Q j and satisfying the same system of linearrelations. In particular, by (5.13) we have ρ n ( P (1) k,n,j ) = ρ n ( P ( m ) k,n,j ) ,n ∈ N , k = 1 , . . . , N , j = 1 , . . . , s . Then there is unitary { W n,j } ∈ Q ( Q n,k A n Q n,k )such that(5.14) W n,j P (1) k,n,j W − n,j = P ( m ) k,n,j for all n, k, j . Let for each k, n, i P ( i ) k,n = s X j =1 P ( i ) k,n,j . For each n let W n = s X j =1 W n,j . Then the projections { P ( i ) k,n } , k = 1 , . . . , N i , i = 1 , . . . , m are lifts of p ( i ) k and satisfythe same system of linear equations. We have also(5.15) W n P (1) k,n W − n = P ( m ) k,n for all n, k . Let X ′ ( i ) n = N i X k =1 e πikNi P ( i ) k,n ,i = 1 , . . . , m , k = 1 , . . . , N i , n ∈ N . Then the unitaries { X ′ ( i ) n } , i = 1 , . . . , m , arelifts of x i ’s and satisfy the relations (5.11) and (5.12). It follows from (5.15) that { X ′ (1) n } , { X ′ ( m ) n } ∈ Q ( P sk =1 Q n,k A n Q n,k ) are unitarily equivalent to each other.STEP 1 is done. STEP 2 : Given the lifts { X ′ ( i ) n } of x ′ i , i = 1 , . . . , m , constructed in STEP 1, tofind a lift of u which would conjugate { X ′ (1) n } and { X ′ ( m ) n } .At first we lift u to anything, say { X n } ∈ Q ( P sk =1 Q n,k A n Q n,k ), that is { X n } α = u. Let for each n ˜ X n = N X k =1 P ( m ) k,n X n P (1) k,n , where P ( m ) k,n and P (1) k,n are projections constructed in STEP 1. Then P ( m ) k,n ˜ X n = ˜ X n P (1) k,n for all k, n and { ˜ X n } α = u, because X p ( m ) k up (1) k = X up (1) k p (1) k = X up (1) k = u. We are going to show that the unitary from the polar decomposition of ˜ X n alsowill conjugate P ( m ) k,n and P (1) k,n , for all k ’s.By (5.15), for each n we have(5.16) ˜ X n = W n N X k =1 P (1) k,n W − n X n P (1) k,n . TABILITY OF GROUP RELATIONS 23
For each k, n, j , P (1) k,n,j A n P (1) k,n,j is a II -factor. As is well known, in II -factors apartial isometry in polar decomposition can always be chosen unitary. Since P (1) k,n s X j =1 Q n,j A n Q n,j P (1) k,n = ⊕ sj =1 P (1) k,n,j A n P (1) k,n,j , for each k, n we have P (1) k,n W − n X n P (1) k,n = V k,n (cid:12)(cid:12)(cid:12) P (1) k,n W − n X n P (1) k,n (cid:12)(cid:12)(cid:12) with V k,n ∈ P (1) k,n (cid:16)P sj =1 Q n,j A n Q n,j (cid:17) P (1) k,n being unitary. Let V n = N X k =1 V k,n . It is unitary and(5.17) V n = N X k =1 P (1) k,n V n P (1) k,n . We have, by (5.15) and (5.16),(5.18)˜ X n = W n V n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 P (1) k,n W − n X n P (1) k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = W n V n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 W − n P ( m ) k,n W n W − n X n P (1) k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = W n V n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W − n N X k =1 P ( m ) k,n X n P (1) k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = W n V n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 P ( m ) k,n X n P (1) k,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = W n V n (cid:12)(cid:12)(cid:12) ˜ X n (cid:12)(cid:12)(cid:12) . Let ˜ V n = W n V n . Then ˜ X n = ˜ V n (cid:12)(cid:12)(cid:12) ˜ X n (cid:12)(cid:12)(cid:12) and since { ˜ X n } α = u , we conclude that { ˜ V n } α = u. By (5.17) ˜ V n = W n N X k =1 P (1) k,n V n P (1) k,n = N X k =1 P ( m ) k,n W n V n P (1) k,n . Hence P ( m ) k,n ˜ V n = ˜ V n P (1) k,n which implies that X ′ ( m ) n ˜ V n = ˜ V n X ′ (1) n . STEP 2 is done.For each n , let Ω n = P sk =1 λ k Q n,k . Then Ω = { Ω n } α . Let X ( i ) n = X ′ ( i ) n Ω /N i n . Then by (5.9) k x i − { X ( i ) n } α k ≤ ǫ. Since ˜ V n , X (1) n , . . . , X ( m ) n satisfy the group relations, Lemma 3 completes the proof. (cid:3) Theorems 8 and 9 imply that
Theorem 10.
One-relator groups with a nontrivial center are II -factor stable. RFD.
Recall that a C ∗ -algebra is residually finite-dimensional (RFD) if it hasa separating family of finite-dimensional representations.Though property of being RFD is not directly related to stability, argumentssimilar to ones used in the proof of Theorem 8 can be applied to show that forgroups of the form (5.1), C ∗ ( G ) is RFD. We will need a lemma. Lemma 8.
Suppose
Q, R , . . . , R N ∈ B ( H ) are projections and P Ni =1 R i = Q , P n ∈ B ( H ) are finite-rank projections and P n ↑ , Q n ∈ P n B ( H ) P n and SOT- lim Q n = Q . Then there exists projections R ( i ) n ∈ P n B ( H ) P n such that SOT- lim R ( i ) n = R i and P Ni =1 R ( i ) n = Q n . Proof.
Let ˜ H = Q ( H ) . Then Q is the unit in B ( ˜ H ). Since projections with sum 1generate a commutative C ∗ -algebra, hence RFD, the statement follows from [[15],Th. 11]. (cid:3) Theorem 11.
Let G be of the form (5.1). Then C ∗ ( G ) is RFD.Proof. Again we will do it for the case G = (cid:10) x, y, z | x = y , y = z (cid:11) , and theproof for the general case is analogous. Let 0 = a ∈ C ∗ ( G ) . Then there exists anirreducible representation π of C ∗ ( G ) such that π ( a ) = 0. The representation π must factorize through the C ∗ -algebra C ∗ (cid:0) x, y, z | x = y , y = z , x = y = z = 1 (cid:1) . Indeed, π (cid:0) x (cid:1) = π (cid:0) y (cid:1) = π (cid:0) z (cid:1) ∈ π ( C ∗ ( G )) ′ = C
1. Hence there is λ ∈ T suchthat π ( x ) = π ( y ) = π ( z ) = λ . Then there is an isomorphism π ( C ∗ ( G )) ∼ = C ∗ (cid:0) x, y, z | x = y , y = z , x = y = z = 1 (cid:1) given by x ′ π ( x ) λ − / , y ′ π ( y ) λ − / , z ′ π ( z ) λ − / . Here by λ − / etc. we mean | λ | − / e − iArgλ etc. By arguments used in the proofof Theorem 8, the latter algebra is isomorphic to the universal C ∗ -algebra D of therelations(5.19) q + q = r + r + r q + q = r + r + r q + q = r + r + r q + q = r + r + r q + q = r + r + r r + r + r + r + r = s + s + s + s + s + s + s r + r + r + r + r = s + s + s + s + s + s + s r + r + r + r + r = s + s + s + s + s + s + s , where all q i , r k , s m , i = 1 , . . . , , k = 1 , . . . , , m = 1 , . . . ,
21, are projections.
TABILITY OF GROUP RELATIONS 25
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