Approximate Equivalence in von Neumann Algebras
aa r X i v : . [ m a t h . OA ] A ug Approximate Equivalence in von Neumann Algebras
Qihui Li, Don Hadwin, and Wenjing Liu
Dedicated to Lyra.
Abstract.
Suppose A is a separable unital ASH C*-algebra, R is a sigma-finite II ∞ factor von Neumann algebra, and π, ρ : A → R are unital ∗ -homomorphisms such that, for every a ∈ A , the range projections of π ( a )and ρ ( a ) are Murray von Neuman equivalent in R . We prove that π and ρ areapproximately unitarily equivalent modulo K R , where K R is the norm closedideal generated by the finite projections in R . We also prove a very generalresult concerning approximate equivalence in arbitrary finite von Neumannalgebras.
1. Introduction
In 1977 D. Voiculescu [ ] proved a remarkable theorem concerning approxi-mate (unitary) equivalence of representations of a separable unital C*-algebra ona separable Hilbert space. The beauty of the theorem is that the characterizationwas in terms of purely algebraic terms. This was made explicit in the reformulationof Voiculescu’s theorem in [ ] in terms of rank. Theorem . [ ] Suppose B ( H ) is the set of operators on a separable Hilbertspace H and K ( H ) is the ideal of compact operators. Suppose A is a separable unitalC*-algebra, and π, ρ : A → B ( H ) are unital ∗ -homomorphisms. The following areequivalent: (1) There is a sequence { U n } of unitary operators in B ( H ) such that (a) U n π ( a ) U ∗ n − ρ ( a ) ∈ K ( H ) for every n ∈ N and every a ∈ A . (b) k U n π ( a ) U ∗ n − ρ ( a ) k → for every a ∈ A . (2) There is a sequence { U n } of unitary operators in B ( H ) such that, forevery a ∈ A , k U n π ( a ) U ∗ n − ρ ( a ) k → . (3) For every a ∈ A , rank ( π ( a )) = rank ( ρ ( a )) . Mathematics Subject Classification.
This author was partially supported by NSFC(Grant No.11671133).This author was supported by a Collaboration Grant from the Simons Foundation.This author is supported by a grant from the Eric Nordgren Research Felloship Fund.This paper is in final form and no version of it will be submitted for publication elsewhere. If π : A → B ( H ) is a unital ∗ -homomorphism, we will write π ∼ a ρ in B ( H ) tomean that statement (2) in the preceding theorem holds and we will write π ∼ a ρ ( K ( H )) in B ( H ) to indicate statements (1) and (2) hold. When the C*-algebra A is not separable, π ∼ a ρ means that there is a net of unitaries { U λ } such that,for every a ∈ A , k U λ π ( a ) U ∗ λ − ρ ( a ) k → . It was shown in [ ] that π ∼ a ρ if andonly if rank( π ( a )) = rank( ρ ( a )) always holds even when A or H is not separable,where, for T ∈ B ( H ), rank( T ) is the Hilbert-space dimension of the projection R ( T ) onto the closure of the range of T .Later Huiru Ding and the first author [ ] extended the notion of rank to oper-ators in a von Neumann algebra M , i.e., if T ∈ M , then M -rank( T ) is the Murrayvon Neumann equivalence class of the projection R ( T ). If p and q are projectionsin a C*-algebra W , we write p ∼ q in W to mean that there is a partial isometry v ∈ W such that v ∗ v = p and vv ∗ = q . Thus M -rank( T ) = M -rank( S ) if and onlyif R ( S ) ∼ R ( T ). In [ ] they extended Voiculescu’s theorem for representationsof a separable AH C*-algebra into a von Neumann algebra on a separable Hilbertspace, i.e., π ∼ ρ in M if and only if, for every a , M -rank ( π ( a )) = M -rank ( ρ ( a )) .When the algebra A is ASH, their characterization works when the von Neumannalgebra is a II factor [ ] (See Theorem 2.)When M is a σ -finite type II ∞ factor with a faithful normal tracial weight τ ,there are analogues F M and K M of the finite rank operators and compact operators;namely, F M is the ideal generated by the projections P ∈ M with τ ( P ) < ∞ , and K M is the norm closure of F M . It happens that F M is contained in every nonzeroideal of M and K M is the only nontrivial norm closed ideal of M . π, ρ : A → M are unital ∗ -homomorphisms, then π ∼ a ρ ( K M ) if and only if the sequence { U n } of unitary operators in M for which part (2) in Theorem Qihui Li, Junhao Shen,and Rui Shi holds can be chosen so that, for every n ∈ N and every a ∈ A , U n π ( a ) U ∗ n − ρ ( a ) ∈ K M .This naturally leads to the following question: Question.
Suppose M is a σ -finite II factor von Neumann algebra, A is aseparable unital C*-algebra, and π, ρ : A → M are unital ∗ -homomorphisms. If π ∼ a ρ in M , does it follow that π ∼ a ρ ( K M )?In [ ] Rui Shi and the first author gave an affirmative answer to this questionwhen A is commutative. More recently this result has been extended by ShilinWen, Junsheng Fang and Rui Shi [ ? ] to the case when A is an AF C*-algebra, andin [ ] to the case when A is AH. In this paper we extend the results to the casewhen A is a ASH C*-algebra (Theorem 6) . We also extend the results in some ofthe results in [ ] and [ ? ] for arbitrary finite von Neumann algebras and for the newclass of strongly LF-embeddable C*-algebras (Theorem 2 and Theorem 3).The proof of Voiculescu’s theorem (Theorem 1) has two parts. The ’hard” partis showing that if
A ⊂ B (cid:0) ℓ (cid:1) is a separable unital C*-algebra, and π : A → B (cid:0) ℓ (cid:1) is a unital ∗ -homomorphism such that K (cid:0) ℓ (cid:1) ⊂ ker π, then id A ⊕ π ∼ a id A ( K (cid:0) ℓ (cid:1) ).The ”easy part” involves the compact operators. Suppose A is a separable unitalC*-algebra and π : A → B (cid:0) ℓ (cid:1) is a unital ∗ -homomorphism. Then sup (cid:8) R ( π ( a )) : π ( a ) ∈ K (cid:0) ℓ (cid:1)(cid:9) PPROXIMATE EQUIVALENCE 3 reduces π and leads to a decomposition π = π ⊕ π . The ”easy part” says that if π ∼ a ρ , then π and ρ must be unitarily equivalent.In [ ] Qihui Li, Junhao Shen, and Rui Shi prove an analogue of the ”hard part”when A is nuclear for a II ∞ factor. The ”easy part” for von Neumann algebras ismuch more difficult for II ∞ factors because, there is a complete characterizationof C*-subalgebras of K (cid:0) ℓ (cid:1) and their representations which is totally lacking forC*-subalgebras of K M . In fact, the ”easy part” isn’t true for a II ∞ factor. Theanalogue must be π ∼ a ρ ( K M ).We prove a general analogue of the ”easy part” for II ∞ factors when the C*-algebrais ASH (Theorem 5).
2. Finite von Neumann Algebras
A separable C*-algebra is AF if it is a direct limit of finite-dimensional C*-algebras. A separable C*-algebra is homogeneous if it is a finite direct sum ofalgebras of the form M n ( C ( X )) , where X is a compact metric space. A unitalC*-algebra is subhomogeneous if there is an n ∈ N , such that every representationis on a Hilbert space of dimension at most n ; equivalently, if x n = 0 for everynilpotent x ∈ A . Every subhomogeneous algebra is a subalgebra of a homogeneousone. Every subhomogeneous von Neumann algebra is homogeneous; in particular,if A is subhomogeneous, then A is homogeneous. A C*-algebra is approximatelysubhomogeneous (ASH) if it is a direct limit of subhomogeneous C*algebras.There has been a lot of work determining which separable C*-algebras areAF-embeddable. A (possibly nonseparable) C*-algebra B is LF if, for every finitesubset F ⊂ B and every ε > D of B suchthat, for every b ∈ F , dist( b, D ) < ε . Every separable unital C*-subalgebra of a LFC*-algebra is contained in a separable AF subalgebra.We are interested in a more general property. We say that a unital C*-algebra A is strongly LF-embeddable if there is an LF C*-algebra B such that A ⊂ B ⊂ A .It is easily shown that an ASH algebra is strongly LF-embeddable, i.e., if {A λ } is anincreasingly directed family of subhomogeneous C*-algebas and A = ( ∪ λ A λ ) −kk ,then A ⊂ (cid:16) ∪ λ A λ (cid:17) −kk ⊂ A . Lemma . Suppose B is a unital LF C*-algebra and D = M n ( C ) ⊕· · ·⊕ M nk ( C ) and W is a unital C*-algebra. (1) If π, ρ : D → W are unital ∗ -homomorphisms and π ( e ,s ) ∼ ρ ( e ,s ) for ≤ s ≤ k, where { e ij,s } is the system of matrix units for M ns ( C ) , then π and ρ are unitarily equivalent in W . (2) If π, ρ : B → W are unital ∗ -homomorphisms such that π ( p ) ∼ ρ ( p ) in W for every projection p ∈ B , then π ∼ a ρ in W . Proof. (1) Since e ii,s ∼ e ,s in D for 1 ≤ i ≤ n s and 1 ≤ s ≤ k , we see that π ( e ii,s ) ∼ ρ ( e ii,s ) in W for 1 ≤ i ≤ n s and 1 ≤ s ≤ k. It follows from [ , Theorem2] that π and ρ are unitarily equivalent in W .(2) Suppose Λ is the set of all pairs λ = ( F λ , ε λ ) with F λ a finite subset of B and ε λ >
0. Clearly Λ is directed by ( ⊂ , ≥ ). For λ ∈ Λ, we can choose a finite-dimensional algebra D λ ⊂ B such that, for every x ∈ F λ , dist( x, D λ ) < ε λ . It QIHUI LI, DON HADWIN, AND WENJING LIU follows from part (1) that there is a unitary operator U λ ∈ W such that, for every x ∈ F λ , U π ( x ) U ∗ = ρ ( x ). For each a ∈ F λ, we can choose x a ∈ D λ such that k a − x a k < ε λ.. Hence, for every a ∈ F λ k U λ π ( a ) U ∗ λ − ρ ( a ) k = k U λ π ( a − x λ ) U ∗ λ − ρ ( a − x λ ) k < ε λ . It follows that, for every a ∈ A ,lim λ k U λ π ( a ) U ∗ − ρ ( a ) k = 0 . (cid:3) The key property of a finite von Neumann algebra M is that there is a faithfulnormal tracial conditional expectation Φ from M to its center Z ( M ), and that forprojections p and q in M , we have p and q are Murray-von Neumann equivalentif and only if Φ ( p ) = Φ ( q ). Note that in the next lemma and the theorem thatfollows, there is no separability assumption on the C*-algebra A or the dimensionof the Hilbert space on which M acts. Lemma . Suppose A is a (possibly nonunital) C*-algebra, M is a finite vonNeumann algebra with center-valued trace Φ :
M → Z ( M ) . If π, ρ : A → M are ∗ -homomorphisms such that, for every a ∈ A , M -rank ( π ( a )) = M -rank ( ρ ( a )) , then Φ ◦ π = Φ ◦ ρ . Proof.
We can extend π and ρ to weak*-weak* continuous *-homomorphismsˆ π, ˆ ρ : A → M . Suppose x ∈ A and 0 ≤ x ≤
1. Suppose 0 < α < f α : [0 , → [0 ,
1] by f ( t ) = dist ( t, [0 , α ]) . Since f (0) = 0 , we see that f ( x ) ∈ A , and χ ( α, ( x ) = weak*-lim n →∞ f ( x ) /n ∈A , so R ( f ( x )) = χ ( α, ( x ) .It follows that ˆ π (cid:0) χ ( α, ( x ) (cid:1) = R ( π ( f α ( x ))) = χ ( α, ( π ( x ))and ˆ ρ (cid:0) χ ( α, ( x ) (cid:1) = R ( ρ ( f α ( x ))) = χ ( α, ( ρ ( x )) .Hence Φ (cid:0) ˆ π (cid:0) χ ( α, ( x ) (cid:1)(cid:1) = Φ (cid:0) ˆ ρ (cid:0) χ ( α, ( x ) (cid:1)(cid:1) . Suppose 0 < α < β <
1. Since χ ( α,β ] = χ ( α, − χ ( β, , we see thatΦ (cid:0) ˆ π (cid:0) χ ( α,β ] ( x ) (cid:1)(cid:1) = Φ (cid:0) ˆ ρ (cid:0) χ ( α,β ] ( x ) (cid:1)(cid:1) . Thus, for all n ∈ N ,Φ ˆ π n − X k − kn χ ( kn , k +1 n ] ( x ) !! = Φ ˆ ρ n − X k − kn χ ( kn , k +1 n ] ( x ) !! . Since, for every n ∈ N , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x − n − X k − kn χ ( kn , k +1 n ] ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ /n , PPROXIMATE EQUIVALENCE 5 it follows that Φ ( π ( x )) = Φ (ˆ π ( x )) = Φ (ˆ ρ ( x )) = Φ ( ρ ( x )) . Since A is the linear span of its positive contractions, Φ ◦ π = Φ ◦ ρ . (cid:3) Theorem . Suppose A is strongly LF-embeddable, M is a finite von Neumannalgebra with center-valued trace Φ :
M → Z ( M ) . If π, ρ : A → M are unital ∗ -homomorphisms, then the following are equivalent: (1) π ∼ a ρ ( M ) . (2) M -rank ( π ( a )) = M -rank ( ρ ( a )) for every a ∈ A . (3) Φ ◦ π = Φ ◦ ρ. Proof. (3) ⇒ (1). We can extend π and ρ to weak*-weak* continuous ∗ -homomorphisms ˆ π, ˆ ρ : A → M . Since Φ is weak*-weak* continuous, it followsthat Φ ◦ ˆ π = Φ ◦ ˆ ρ . Since A is strongly LF-embeddable, there is an LF algebra B such that A ⊂ B ⊂ A . For every projection p ∈ B we haveΦ (ˆ π ( p )) = Φ (ˆ ρ ( p )) , which implies that ˆ π ( p ) ∼ ˆ ρ ( p ) . Hence, by Lemma 1, ˆ π | B ∼ a ˆ ρ | B in M . Thus π ∼ a ρ ( M ).(1) ⇒ (3) . Suppose { U λ } is a net of unitaries in M such that, for every a ∈ A , k U λ π ( a ) U ∗ λ − ρ ( a ) k → . Thus, since Φ is tracial and continuous,Φ ( ρ ( a )) = lim λ Φ ( U λ π ( a ) U ∗ λ ) = Φ ( π ( a )) . (3) ⇒ (2) . Assume (3) . Then, for any a ∈ A ,Φ ( R ( π ( a ))) = lim n →∞ Φ (cid:16) π (cid:16) ( aa ∗ ) /n (cid:17)(cid:17) = lim n →∞ Φ (cid:16) ρ (cid:16) ( aa ∗ ) /n (cid:17)(cid:17) = Φ ( R ( π ( a ))) . Hence R ( π ( a )) ∼ R ( ρ ( a )) . Thus M -rank( π ( a )) = M -rank( ρ ( a )).(2) ⇒ (3) . This is Lemma 2. (cid:3)
Remark . It is important to note that the proof of (2) ⇒ (3) holds even when A is not unital. In [ ] it was shown that if A is a separable unital C*-algebra and π and ρ arerepresentations on a separable Hilbert space such that, for every x ∈ A rank π ( x ) ≤ rank ρ ( x ) , then there is a representation σ such that π ⊕ σ ∼ a ρ. In [ ], Rui Shi and the first author proved an analogue for representations of separa-ble abelian C*-algebras into II factor von Neumann algebras. This result was ex-tended by Shilin Wen, Junsheng Fang and Rui Shi [ ? ] to separable AF C*-algebras.We extend this result further, including separable ASH C*-algebras. QIHUI LI, DON HADWIN, AND WENJING LIU
Theorem . Suppose A is a separable strongly LF-embeddable C*-algebra and M is a II factor von Neumann algebra with a faithful normal tracial state τ .Suppose P is a projection in M and π : A → P M P and ρ : A → M are unital ∗ -homomorphisms such that, for every a ∈ A , M -rank ( π ( a )) ≤ M -rank ( ρ ( a )) . Then there is a unital ∗ -homomorphism σ : A → P ⊥ M P ⊥ such that π ⊕ σ ∼ a ρ ( M ) . Proof.
As in the proof of Theorem 2 choose a separable AF C*-algebra B such that A ⊂ B ⊂ A , and extend π and ρ to unital weak*-weak* continuous ∗ -homomorphisms ˆ π and ˆ ρ with domain A . It was shown in [ ] that the conditionon π and ρ is equivalent to: for every a ∈ M with 0 ≤ a , τ ( π ( a )) ≤ τ ( ρ ( a )).It follows from weak* continuity that, for every a ∈ A with 0 ≤ a , τ (ˆ π ( a )) ≤ τ (ˆ ρ ( a )). In particular this holds for 0 ≤ a ∈ B . However, since B is AF, it followsfrom [ ] that there is a unital ∗ -homomorphism γ : B → P ⊥ A P ⊥ such that(ˆ π | B ) ⊕ γ ∼ a ˆ ρ | B ( M ) .If we let σ = γ | A , we see π ⊕ σ ∼ a ρ ( M ). (cid:3)
3. Representations of ASH algebras relative to ideals
We prove a version of Voiculescu’s theorem for representations of a separableASH C*-algebras into sigma-finite type II ∞ factor von Neumann algebras. We firstprove a more general result. We begin with a probably well-known lemma. Lemma . Suppose J is a norm closed two-sided ideal in a von Neumannalgebra M and J is the ideal in M generated by the projections in J . Supposealso that A is a C*-algebra and π, ρ : A → M are unital ∗ -homomorphisms. Then (1) J is the norm closed linear span of the set of projections in J , so J −kk = J , (2) J = { T ∈ M : T = P T P for some projection P ∈ J } , (3) T ∈ J if and only if χ (0 , ∞ ) ( | T | ) = R ( T ) ∈ J , (4) If P and Q are projections in J then P ∨ Q = R ( P + Q ) ∈ J , (5) π − ( J ) −kk = π − ( J ) , (6) If {A i : i ∈ I } is an increasingly directed family of unital C*-subalgebrasof A and A = [ ∪ i ∈ I A i ] −kk , then (cid:2) ∪ i ∈ I A i ∩ π − ( J ) (cid:3) −kk = π − ( J ) . Proof. (1) , (2) , (3) can be found in [ ].(4) . Suppose a ∈ π − ( J ). Then π ( a ) ∈ J , so π ( g ε ( | a | )) = g ε ( | π ( a ) | ) χ ( ε/ , ∞ ) ( | π ( a ) | ) ∈ J , and k a − ag ε ( | a | ) k ≤ ε. (5) . Let η : M → M / J be the quotient map. Suppose a ∈ π − ( J ) and ε > . Then there is an i ∈ I and a b ∈ A i such that k a − b k < ε . Thus k ( η ◦ ( π | A i )) ( b ) k = k ( η ◦ π ) ( b ) k = k ( η ◦ π ) ( b − a ) k ≤ ε, PPROXIMATE EQUIVALENCE 7 so there is a w ∈ A i so that k w k = k ( η ◦ ( π | A i )) ( w ) k = k ( η ◦ ( π | A i )) ( b ) k ≤ ε.z = b − w ∈ ker ( η ◦ ( π | A i )) = π − ( J ) ∩ A i , and k b − z k = k w k < ε . It followsfrom part (2) that there is a v ∈ π − ( J ) ∩ A i such that k z − v k ≤ ε. Hence k a − v k ≤ k a − b k + k b − z k + k z − v k ≤ ε. (6) . Let η : M → M / J be the quotient map. Suppose a ∈ π − ( J ) and ε > . Then there is an i ∈ I and a b ∈ A i such that k a − b k < ε . Thus k ( η ◦ ( π | A i )) ( b ) k = k ( η ◦ π ) ( b ) k = k ( η ◦ π ) ( b − a ) k ≤ ε, so there is a w ∈ A i so that k w k = k ( η ◦ ( π | A i )) ( w ) k = k ( η ◦ ( π | A i )) ( b ) k ≤ ε.z = b − w ∈ ker ( η ◦ ( π | A i )) = π − ( J ) ∩ A i , and k b − z k = k w k < ε . It followsfrom part (5) that there is a v ∈ π − ( J ) ∩ A i such that k z − v k ≤ ε. Hence k a − v k ≤ k a − b k + k b − z k + k z − v k ≤ ε . (cid:3) Suppose A is a unital C*-algebra, M ⊂ B ( H ) is a von Neumann algebra witha norm-closed ideal J and π : A → M is a unital ∗ -homomorphism. We define H π, J = sp −kk ( ∪ { ran π ( a ) : a ∈ A and π ( a ) ∈ J } ) .It is clear that H π, J is a reducing subspace for π and we call the summand π ( · ) | H π, J = π J .In Voiculescu’s theorem, where π, ρ : A → B ( H ) and A and H are separable,we write π = π K ( H ) ⊕ π , ρ = ρ K ( H ) ⊕ ρ . The proof of Voiculescu’s theorem involves showing π ∼ a π ⊕ ρ = π K ( H ) ⊕ π ⊕ ρ , and ρ ∼ a ρ ⊕ π ⋍ ρ K ( H ) ⊕ π ⊕ ρ , which was the hard part. Using descriptions of C*-algebras of compact operatorsand their representations, it is not too hard to show that the equality of rankconditions imply that π K ( H ) and ρ K ( H ) are unitarily equivalent. When B ( H ) isreplaced with a sigma-finite type II ∞ factor von Neumann algebra M and K ( H )is replaced with the closed ideal K M generated by the finite projections, the hardpart is harder (and unsolved) and the easy part is not true.In a deep and beautiful paper [ ] of Qihui Li, Junhao Shen, and Rui Shi provedthe best-to-date attack of the hard part. Theorem . [ ] Suppose A is a separable nuclear C*-algebra, M is a sigma-finite type II ∞ factor von Neumann algebra and π, σ : A → M are unital ∗ -homomorphisms such that σ | π − ( K M ) = 0 . Then π ∼ a π ⊕ σ ( K M ) . QIHUI LI, DON HADWIN, AND WENJING LIU
The following is a fairly general version of the analogue of the ”easy part” ofthe proof of Voiculescu’s theorem when the C*-algebra is ASH. In particular, thereis no assumption that the von Neumann algebra M is a sigma-finite or acts on aseparable Hilbert space. Theorem . Suppose A is a separable unital ASH C*-algebra, M ⊂ B ( H ) is avon Neumann algebra with a norm closed two-sided ideal J . Suppose π, ρ : A → M are unital ∗ -homomorphisms such that (1) Every projection in J is finite, (2) M -rank ( π ( a )) = M -rank ( ρ ( a )) for every a ∈ A .Then there is a sequence { W n } of partial isometries in M such that W ∗ n W n is the projection onto H π, J and W n W ∗ n is the projection onto H ρ, J , , W n π J ( a ) W ∗ n − ρ J ( a ) ∈ J for every n ∈ N and every a ∈ A ,
5. lim n →∞ k W n π J ( a ) W ∗ n − ρ J ( a ) k = 0 for every a ∈ A . Proof.
First, suppose x ∈ A and x = x ∗ . It follows from [ ] that there is asequence { U n } such that k U n π ( x ) U ∗ n − ρ ( x ) k → . It follows that π ( x ) ∈ J if and only if ρ ( x ) ∈ J when x = x ∗ . However, for any a ∈ A , we get π ( a ) ∈ J if and only if π ( | a | ) ∈ J . Hence π − ( J ) = ρ − ( J ). Also, π ( a ) ∈ J if and only if R ( π ( a )) ∈ J . Since R ( π ( a )) and R ( ρ ( a )) are Murrayvon Neumann equivalent (from (2)), we see that π ( a ) ∈ J if and only if ρ ( a ) ∈ J .It follows that π − ( J ) ∩ A n = ρ − ( J ) ∩ A n for each n ∈ N , and, from Lemma 3, " ∞ [ n =1 π − ( J ) ∩ A n −kk = " ∞ [ n =1 ρ − ( J ) ∩ A n −kk = π − ( J ) = ρ − ( J ) .Since A is an ASH algebra, we can assume that there is a sequence A ⊂ A ⊂ · · · of subalgebras of A such that ∪ ∞ n =1 A n is norm dense in A such that, for each n ∈ N , A n = M k ( n, (cid:0) C ( X n, ) ⊕ · · · ⊕ M k ( n,s n ) ( X n,s n ) (cid:1) with X n, , . . . , X n,s n compact Hausdorff spaces.Suppose T = ( f ij ) ∈ M k ( C ( X )) is a k × k matrix of functions. We define T z = diag( f, f, . . . , f ) where f = P ki,j =1 | f ij | . If { e ij : 1 ≤ i, j ≤ n } is the systemof matrix units for M n ( C ) , then T = P ni,j =1 f ij e ij . It is clear that if T ≥ , then R ( T ) ≤ R (cid:0) T z (cid:1) . Since f ij e ss = e si T e js , we have | f ij | e ss = ( e si T e js ) ∗ ( e si T e js ) = e sj T ∗ e is e si T e js = e ∗ js T ∗ e ii T e js . Thus T z = g X s =1 k X i,j =1 | f ij | e ss = g X s =1 k X i,j =1 e ∗ js T ∗ e ii T e js . Suppose A = A ⊕ · · · ⊕ A s n ∈ A n . We define ∆ n : A n → Z (cid:0) A n (cid:1) by∆ n ( A ) = A z ⊕ · · · ⊕ A z s n . PPROXIMATE EQUIVALENCE 9
Thus if A ∈ A n , then ∆ n ( A ) has the form∆ n ( A ) = m X k =1 B k AC k , with B , C , . . . , B m , C m ∈ A n .It is clear thata. ∆ n (cid:0) A n (cid:1) is contained in the center Z (cid:0) A n (cid:1) of A n , andb. If A ≥ , then R ( A ) ≤ R (∆ n ( A )) ∈ Z (cid:0) A n (cid:1) .We call a projection Q ∈ A n good ifc. ˆ π ( Q ) , ˆ ρ ( Q ) ∈ J d. Q ∈ (cid:2) A n ∩ π − ( J ) (cid:3) − weak* e. For all T ∈ Q A Q , M -rank(ˆ π ( T )) = M -rank(ˆ ρ ( T )) . Our proof is based on four claims.
Claim 0:
Suppose Q , Q ∈ A n are good projections and Q ⊥ Q . Then Q = Q + Q is a good projection. It is clear that Q satisfies ( c ) and ( d ). Let P = ˆ π ( Q ) ∨ ˆ ρ ( Q ) ∈ J . Thus P is a finite projection in M , so P M P is a finitevon Neumann algebra. Let Φ P : P M P → Z ( P M P ) be the center-valued trace.Since Q and Q are good, we know from Lemma 2 thatΦ P ◦ ˆ π | Q k A Q k = Φ P ◦ ˆ ρ | Q k A Q k for k = 1 , . Since Q ⊥ Q , we know ˆ π ( Q ) ⊥ ˆ π ( Q ) and ˆ ρ ( Q ) ⊥ ˆ ρ ( Q ). Thusif 1 ≤ i = j ≤
2, then, since Φ P is tracial, if A ∈ A , then,Φ P (ˆ π ( Q i AQ j )) = Φ P (cid:16) ˆ π ( Q i ) ˆ π ( A ) ˆ π ( Q j ) (cid:17) = Φ P (ˆ π ( Q j ) ˆ π ( Q i ) ˆ π ( A ) ˆ π ( Q j )) = 0 . Similarly, Φ P (ˆ ρ ( Q i AQ j )) = 0 . Thus Φ P (ˆ π ( QAQ )) = Φ P (ˆ π ( Q AQ )) + Φ P (ˆ π ( Q AQ ))= Φ P (ˆ ρ ( Q AQ )) + Φ P (ˆ ρ ( Q AQ )) .Thus, by Lemma 2, Q satisfies ( e ). Hence Q is a good projection. This proves theclaim. A simple induction proof implies that the sum of a finite pairwise orthogonalfamily of good projections is good. Claim 1: If Q ∈ A n is a good projection, then there is a good projection P ∈ Z (cid:0) A n (cid:1) such that Q ≤ P. Proof:
Suppose Q ∈ A n is a good projection. Choose B , C , . . . , B k , C k in A n such that E = def m X k =1 B k QC k = ∆ n ( Q ) ∈ Z (cid:0) A n (cid:1) . Since R ( E ) ∈ Z (cid:0) A n (cid:1) and E ≥
0, we see that E = R ( E ) E R ( E ) = m X k =1 [ R ( E ) B k R ( E )] Q [ R ( E ) C k R ( E )] . Hence we can assume, for 1 ≤ k ≤ m , that B k , C k ∈ R ( E ) A R ( E ) . Since ˆ π ( Q ), ˆ ρ ( Q ) ∈ J , we see that ˆ π ( E ) and ˆ ρ ( E ) ∈ J , which, in turn,implies ˆ π ( R ( E )) and ˆ ρ ( R ( E )) ∈ J . Then F = ˆ π ( R ( E )) ∨ ˆ ρ ( R ( E )) ∈ J is afinite projection. Thus F M F is a finite von Neumann algebra. Also, since, for 1 ≤ k ≤ m , B k , C k ∈ R ( E ) A n R ( E ) , we see that ˆ π ( B k QC k ) , ˆ ρ ( B k QC k ) ∈ F M F .Let Φ F be the center-valued trace on F M F . Since Q is a good projection and in E A E , we know from Lemma 2, that for every A ∈ A ,Φ F (ˆ π ( QAQ )) = Φ F (ˆ ρ ( QAQ )) . Now ˆ π, ˆ ρ : E A E → F M F are ∗ -homomorphisms, and, since Φ F is tracial, wesee for A ∈ A , Φ F (ˆ π ( EAE )) == m X j,k =1 Φ F ([ˆ π ( B k ) ˆ π ( Q )] [ˆ π ( Q ) ˆ π ( C k ) ˆ π ( A ) ˆ π ( B j ) ˆ π ( Q ) ˆ π ( C j )])= m X j,k =1 Φ F ([ˆ π ( Q ) ˆ π ( C k ) ˆ π ( A ) ˆ π ( B j ) ˆ π ( Q ) ˆ π ( C j )] [ˆ π ( B k ) ˆ π ( Q )])= m X j,k =1 Φ F (ˆ π ( QC k AB j QC j B k Q )) = m X j,k =1 Φ F (ˆ ρ ( QC k AB j QC j B k Q ))= m X j,k =1 Φ F ([ˆ ρ ( Q ) ˆ ρ ( C k ) ˆ ρ ( A ) ˆ ρ ( B j ) ˆ ρ ( Q ) ˆ ρ ( C j )] [ˆ ρ ( B k ) ˆ ρ ( Q )])= Φ F (ˆ ρ ( EAE )) .Thus Φ F ◦ ˆ π = Φ F ◦ ˆ ρ on E A E , and since ˆ π, ˆ ρ, and Φ F are weak* continuous,we have Φ F ◦ ˆ π = Φ F ◦ ˆ ρ on (cid:0) E A E (cid:1) − weak* = R ( E ) A R ( E ).Finally, since (cid:2) A n ∩ π − ( J ) (cid:3) − weak* is a weak* closed ∗ -algebra, and an idealfor A n , we see that E = ∆ n ( Q ) = m X k =1 B k QC k ∈ (cid:2) A n ∩ π − ( J ) (cid:3) − weak* , so P = R ( E ) ∈ (cid:2) A n ∩ π − ( J ) (cid:3) − weak* . Thus P = R ( E ) ∈ Z (cid:0) A n (cid:1) is a goodprojection and Q ≤ P . Claim 2: If Q , Q ∈ A n are good projections, then there is a good projec-tion Q ∈ Z (cid:0) A n (cid:1) such that Q , Q ≤ Q . Proof:
By Claim 1 we can choose good projections P , P ∈ Z (cid:0) A n (cid:1) suchthat Q ≤ P and Q ≤ P . Since P and P commute and P (1 − P ) ≤ P , P P ≤ P and (1 − P ) P ≤ P , we see that { P (1 − P ) , P P , (1 − P ) P } is anorthogonal family of good projections. Thus, by Case 0, Q = P ∨ P = P (1 − P ) + P P + (1 − P ) P is a good projection in Z (cid:0) A (cid:1) . Thus Claim 2 is proved. Claim 3:
If 0 ≤ x ∈ A n ∩ π − ( J ), then R (∆ n ( x )) ∈ Z (cid:0) A n (cid:1) is good. Proof : We know that ˆ π ( R ( x )) and ˆ ρ ( R ( x )) are Murray von Neumann equiv-alent and M -rank( π ( x )) and M -rank( ρ ( x )) are equal. Since π ( x ) ∈ J , weknow ˆ π ( R ( x )),ˆ ρ ( R ( x )) ∈ J . Arguing as in the proof of Claim 1, we see that F = ˆ π ( R ( x )) ∨ ˆ ρ ( R ( x )) ∈ J and thatˆ π, ˆ ρ : [ x A x ] −kk → F M F PPROXIMATE EQUIVALENCE 11 satisfy Φ F ◦ ˆ π = Φ F ◦ ˆ ρ . Thus Φ F ◦ ˆ π = Φ F ◦ ˆ ρ on [ x A x ] − weak* = R ( x ) A R ( x ).Thus R ( x ) is a good projection. This proves Claim 3.We can choose a countable dense set { b , b , . . . } of ∪ ∞ n =1 (cid:0) A n ∩ π − ( J ) (cid:1) whoseclosure is π − ( J ).We now want to define a sequence 0 = P ≤ P ≤ P ≤ · · · of good projectionssuch that(1) P n ∈ Z (cid:0) A n (cid:1) for all n ∈ N ,(2) If 1 ≤ k ≤ n and b k ∈ A n , then R ( b k ) ≤ P n , i.e., b k = P n b k Define P = 0. Suppose n ∈ N and P k has been defined for 0 ≤ k ≤ n . We let x n = P k ≤ n +1 ,b k ∈A n +1 b k b ∗ k ∈ A n +1 ∩ π − ( J ). Thus, by Claim 3, P n and R (∆ n +1 ( x n )) are good projections in A n , and they commute since R (∆ n +1 ( x n )) ∈ Z (cid:16) A n +1 (cid:17) . By Claim 2, there is a good projection P n +1 ∈Z (cid:16) A n +1 (cid:17) such that P n ≤ P n +1 and R (∆ n +1 ( x n )) ≤ P n +1 . Clearly, if 1 ≤ k ≤ n and b k ∈ A n , we have R ( b k ) = R ( b k b ∗ k ) ≤ R ( x n ) ≤ P n +1 .Since P n is a good projection, P n ∈ (cid:2) A n ∩ π − ( J ) (cid:3) − weak* . Thus P n ≤ sup (cid:8) R ( x ) : x ∈ A n ∩ π − ( J ) (cid:9) ∈ A n . Thus ˆ π ( P n ) ≤ P π, J (the projection onto H π, J ) and ˆ ρ ( P n ) ≤ P ρ, J (the projectiononto H ρ, J ). Let P e = lim n →∞ P n (weak*). Thus ˆ π ( P e ) ≤ P π, J and ˆ ρ ( P e ) ≤ P ρ, J .On the other hand, for every k ∈ N ,lim n →∞ k b k − P n b k k = 0 . This implies P e b = b for every b ∈ (cid:2) π − ( J ) (cid:3) −kk .Thus ˆ π ( P e ) = P π, J and ˆ ρ ( P e ) = P ρ, J . Thus P π, J and P ρ, J are Murray vonNeumann equivalent.Since P n ∈ A ′ n for each n ∈ N , we have of every A ∈ ∪ ∞ k =1 A k ,lim n →∞ k AP n − P n A k = 0 . Hence, lim n →∞ k AP n − P n A k = 0holds for every A ∈ A .Choose a dense subset { A , A , . . . } of A . Suppose and m ∈ N . It follows thatwe can choose a subsequence { P n k } of { P n } such that, for all 1 ≤ n < ∞ , ∞ X k =1 k A n P n k − P n k A n k < ∞ ,and, for 1 ≤ n ≤ m , ∞ X k =1 k A n P n k − P n k A n k < m .Define e k = P n k − P n k − (with P n = 0) and define ϕ : A → P ⊕ ≤ k< ∞ e k A e k by ϕ ( T ) = ∞ X k =1 e k T e k . It follows from [ , page 903] that the above conditions on k A n P n k − P n k A n k that,for all k ∈ N , A k − ϕ ( A k ) ∈ ˆ π − ( J ) ∩ ˆ ρ − ( J )and k P e A n − ϕ ( A n ) k < m .for 1 ≤ n ≤ m .Suppose k ∈ N . For each n ≥ n k , e k A n e k ⊂ A n , which is homogeneous.Hence C ∗ ( e k A n e k ) is subhomogeneous. Thus C ∗ ( e k A e k ) is ASH. If we let E k = ˆ π ( e k ) ∨ ˆ ρ ( e k ) for each k ∈ N , we have E k is a finite projection, E k M E k is a finite von Neumann algebra,ˆ π, ˆ ρ : C ∗ ( e k A e k ) → E k M E k , and, if Let Φ E k is the center-valued trace on E k M E k , thenΦ E k ◦ (cid:0) ˆ π | C ∗ ( e k A e k ) (cid:1) = Φ E k ◦ (cid:0) ˆ ρ | C ∗ ( e k A e k ) (cid:1) , and C ∗ ( e k A e k ) is ASH, it follows from Theorem 2 thatˆ π | C ∗ ( e k A e k ) ∼ a ˆ ρ | C ∗ ( e k A e k ) ( E k M E k ) . Since ˆ π ( e k ) and ˆ ρ ( e k ) are projections, then by [ , Proposition 5.2.6], any unitarythat conjugates ˆ π ( e k ) to a projection that is really close to ˆ ρ ( e k ) is close to aunitary that conjugates ˆ π ( e k ) exactly to ˆ ρ ( e k ). We can therefore, for each k ∈ N ,choose a unitary U k ∈ E k M E k such that k U k ˆ π ( e k a n e k ) U ∗ k − ˆ ρ ( e k a n e k ) k < km when 1 ≤ n ≤ k + m < ∞ , and such that U k ˆ π ( e k ) U ∗ k = ρ ( e k ) . For each k ∈ N , let V k = U k ˆ π ( e k ). Then V k is a partial isometry whose initialprojection is ˆ π ( e k ) = V ∗ k V k and final projection is ˆ ρ ( e k ) = V k V ∗ k . Also k V k ˆ π ( e k ) π ( a n ) ˆ π ( e k ) V ∗ k − ˆ ρ ( e k ) ρ ( a n ) ˆ ρ ( e k ) k < km for 1 ≤ n ≤ k + m < ∞ . Then W m = P ∞ k =1 V k is a partial isometry in M withinitial projection ˆ π ( P e ) = P π, J and final projection ˆ ρ ( P e ) = P ρ, J . Moreover, W m ˆ π ( ϕ ( a n )) W ∗ m = ⊕ X ≤ k< ∞ V k ˆ π ( e k a n e k ) V ∗ k , and ˆ ρ ( ϕ ( a n )) = ⊕ X ≤ k< ∞ ˆ ρ ( e k a n e k ) . Since V k ˆ π ( e k a n e k ) V ∗ k , ˆ ρ ( e k a n e k ) ∈ J for each n, k ∈ N and sincelim k →∞ k V k ˆ π ( e k a n e k ) V ∗ k − ˆ ρ ( e k a n e k ) k = 0 , we see that W m ˆ π ( ϕ ( a n )) W ∗ m − ˆ ρ ( ϕ ( a n )) ∈ J for every n ∈ N . Also, k W m ˆ π ( ϕ ( a n )) W ∗ m − ˆ ρ ( ϕ ( a n )) k < m PPROXIMATE EQUIVALENCE 13 for 1 ≤ n ≤ m .Also ˆ π ( ϕ ( a n )) − π ( a n ) = ˆ π ( ϕ ( a n ) − a n ) ∈ J and ˆ π ( ϕ ( a n )) − ρ ( a n ) = ˆ ρ ( ϕ ( a n ) − a n ) ∈ J for every n ∈ N and k ˆ π ( ϕ ( a n )) − π ( a n ) k < m and k ˆ ρ ( ϕ ( a n )) − ρ ( a n ) k < m for 1 ≤ n ≤ m .For each n ∈ N , W m π ( a n ) W ∗ m − ρ ( a n )= [ W m ( π ( a n ) − ˆ π ( ϕ ( a n ))) W ∗ m ] + [ W m ˆ π ( ϕ ( a n )) W ∗ m − ˆ ρ ( ϕ ( a n ))]+ ˆ ρ ( ϕ ( a n )) − ρ ( a n ) . Thus, for every n ∈ N , W m π ( a n ) W ∗ m − ρ ( a n ) ∈ J . Also, for 1 ≤ n ≤ m , k W m π ( a n ) W ∗ m − ρ ( a n ) k < m .It follows, for every a ∈ A , that W m ˆ π ( ϕ ( a )) W ∗ m − ˆ ρ ( ϕ ( a )) ∈ J and lim m →∞ k W m π ( a ) W ∗ m − ρ ( a ) k = 0. (cid:3) Remark . In two cases, namely, when H π, J = H ρ, J = H, or when π ( · ) | H ⊥ π, J and ρ ( · ) | H ⊥ ρ, J are unitarily equivalent, the conclusion in Theorem 5 becomes π ∼ a ρ ( J ) . When A is a separable ASH C*-algebra and M is a sigma-finite II ∞ factor vonNeumann algebra, we can use Theorems 5 and 4 to have both parts of Voiculescu’stheorem, including an extension of results in [ ]. Corollary . Suppose A is a separable ASH C*-algebra, M is a sigma-finitetype II ∞ factor von Neumann algebra on a Hilbert space H , and τ is a faithfulnormal tracial weight on M . Suppose π, ρ : A → M are unital ∗ -homomorphismssuch that, for every a ∈ AM -rank ( π ( a )) = M -rank ( ρ ( a )) .Then π ∼ a ρ ( K M ) . Theorem . Suppose
M ⊂ B ( H ) is a semifinite von Neumann algebra withno finite summands, H is separable, and A is a separable unital ASH C*-algebra.Also suppose π, ρ : A → M are unital ∗ -homomorphisms such that, for every a ∈ AM -rank ( π ( a )) = M -rank ( ρ ( a )) .Then π ∼ a ρ ( K M ) . Proof.
We can write π = π K M ⊕ π and ρ = ρ K M ⊕ ρ . It follows fromTheorem 4 that π ∼ a π K M ⊕ π ⊕ ρ ( K M ) and ρ ∼ a ρ K M ⊕ π ⊕ ρ ( K M ) .It follows from Theorem 5 that π ∼ a ρ ( K M ). (cid:3) References [1] H. Ding and D. Hadwin, Approximate equivalence in von Neumann algebras. Sci. China Ser.A 48 (2005), no. 2, 239–247.[2] S. Wen, J. Fang and R. Shi, Approximate equivalence of representations of AF algebras intosemifinite von Neumann algebras, Oper. Matrices 13 (2019), no. 3, 777–795.[3] D. Hadwin, Nonseparable approximate equivalence. Trans. Amer. Math. Soc. 266 (1981), no.1, 203–231.[4] D. Hadwin and Rui Shi, A note on the Voiculescu’s theorem for commutative C*-algebras insemifinite von Neumann algebras, arXiv:1801.02510[math.OA].[5] P. R. Halmos, Ten problems in Hilbert space. Bull. Amer. Math. Soc. 76 1970 887–933.[6] R. V. Kadison and J. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 2:Advanced Theory (Graduate Studies in Mathematics, Vol. 16), Academic Press, 1983.[7] Q. Li, J. Shen, R. Shi, A generalization of the Voiculescu theorem for normal operators insemifinite von Neumann algebras, arXiv:1706.09522 [math.OA].[8] D. Sherman, Unitary orbits of normal operators in von Neumann algebras, J. Reine Angew.Math. 605 (2007), 95–132.[9] R. Shi and Junhao Shen, Approximate equivalence of representations of AH algebras intosemifinite von Neumann algebras, arXiv:1805.07236, 2018.[10] D. V. Voiculescu, A non-commutative Weyl-von Neumann theorem. Rev. Roumaine Math.Pures Appl. 21 (1976), no. 1, 97–113.[11] N. E. Wegge-Olsen, K-theory and C*-algebras, A friendly approach, Oxford Science Publica-tions. The Clarendon Press, Oxford University Press, New York, 1993.
East China University of Science and Technology, Shanghai, China
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