aa r X i v : . [ m a t h . OA ] A ug Path-connected Closures of Unitary Orbits
Don Hadwin and Wenjing Liu
Dedicated to Lyra
Abstract. If A and B are unital C*-algebras and π : A → B is a unital ∗ -homomorphism, then U B ( π ) − is the set of all ∗ -homomorphisms from A to B that are approximately (unitarily) equivalent to π. We address the questionof when U B ( π ) − is path-connected with respect to the topology of pointewsenorm convergence. When A is singly generated and B = B (cid:0) ℓ (cid:1) , an affirmativeanswer was given in [ ]; we extend this to the case when A is separable. Wealso give an affirmative answer when A is AF and B is a von Neumann algebra, A is ASH and B is a finite von Neumann algebra, or when A is homogeneousand B is an arbitrary von Neumann algebra.
1. Introduction
In [ ] D. Hadwin proved that the norm closure of the unitary orbit of anoperator in B (cid:0) ℓ (cid:1) is path connected. In this paper we address the problem ofextending this result to representations of separable C*-algebras.Suppose A and B are unital C*-algebras and A is separable. We define Rep( A , B )as the set of all unital ∗ -homomorphisms from A to B with the topology of point-wise norm convergence. Suppose { a , a , . . . } is a norm dense subset of the closedunit ball of A . We define a metric d = d A , B by d ( π, ρ ) = ∞ X m,n =1 m + n k π ( a n ) − ρ ( a n ) k .Clearly, d makes Rep( A , B ) into a complete metric space. When B is finite-dimensional, Rep( A , B ) is compact.Let U B denote the group of unitary elements of B . If π ∈ Rep( A , B ), we definethe unitary orbit U B ( π ) of π by U B ( π ) = { U ∗ π ( · ) U : U ∈ U B } .If T ∈ B we define the unitary orbit U B ( T ) of T by U B ( T ) = { U ∗ T U : U ∈ U B } . Mathematics Subject Classification.
Primary 46L05 ; Secondary 47C15.
Key words and phrases.
C*-algebra, representation, unitary orbit.
It is clear that U B ( T ) corresponds to U B ( π ) when π is the identity representationof the identity representation of C ∗ ( T ).In this paper we address the problem of when U B ( π ) − is path-connected inRep( A , B ). In Section 2 we discuss special paths in U B ( π ) − . In Section 3 we providean affirmative answer (Theorem 3) for the case when B = B (cid:0) ℓ (cid:1) . We reduce theseparable case to the singly generated case by tensoring with the algebra K (cid:0) ℓ (cid:1) ofcompact operators on ℓ . In Section 4 we give an affirmative answer (Theorem 5)when A is AF and B has the property that U p B p is connected for every projection p ∈ B . We also give an affirmative answer (Theorem 6) when there is an LF C*-algebra D such that A ⊂ D ⊂ A , and B is an arbitrary finite von Neumannalgebra. In section 5 we give an affirmative answer (Theorem 7) when A is abelian(or homogeneous) and B is an arbitrary von Neumann algebra.
2. Connectedness of U B and special paths An internal path in U B ( π ) − joining π to ρ is a continuous map γ : [0 , →U B ( π ) − such that γ (0) = π, γ (1) = ρ and γ ( t ) ∈ U B ( π ) whenever 0 ≤ t <
1. A strong internal path from π to ρ ∈ U B ( π ) − is a continuous map γ : [0 , → U B such that lim t → − γ ( t ) ∗ π () γ ( t ) = ρ .In [ , Theorem 3.9] the first author proved that U B ( T ) − is always path con-nected when B = B (cid:0) ℓ (cid:1) . Actually a slightly stronger result was proved. Theorem . [ , Theorem 3.9] Suppose X ∈ B (cid:0) ℓ (cid:1) and Y ∈ U B ( ℓ ) ( X ) − .Then there is a W such that (1) W is unitarily equivalent to W ⊕ W ⊕ · · · , (2) X ⊕ W is unitarily equivalent to Y ⊕ W , (3) If C ∈ B (cid:0) ℓ (cid:1) is unitarily equivalent to X ⊕ W , then (a) C ∈ U B ( ℓ ) ( X ) − = U B ( ℓ ) ( Y ) − , (b) there is a strong internal path in U B ( ℓ ) ( X ) − from X to C , and (c) there is a strong internal path in U B ( ℓ ) ( Y ) − from Y to C . There is no reason, a priori, that U B ( π ) is even connected. It is well-known thatif P and Q are projections in a unital C*-algebra B and k P − Q k <
1, then P and Q are unitarily equivalent [ ]. It was proved in [ ] that two unital representations π, ρ of a finite-dimensional C*-algebras A are unitarily equivalent if and only if π ( p ) is unitarily equivalent to ρ ( p ) for every minimal projection p ∈ A .If U B is connected, then every U B ( π ) must be connected. If x ∈ B and k − x k < −∞ , − ∩ σ ( x ) = ∅ , so A ( x ) = − log ( x ) ∈ B , A ( x ) = A ( x ) ∗ , and x = e iA ( x ) . Since t e i (1 − t ) A ( x ) is a path in U B from x to 1, we see that { x ∈ U B : k − x k < } is contained in the path component W of 1in U B . Since W = ∪ uW such that u ∈ W , we see that W is open in U B . Thus U B is connectedif and only if it is path-connected. This means that if U B is connected, then U B ( π )is path-connected. Lemma . If A is finite-dimensional, then for every B and every π ∈ Rep ( A , B ) , U B ( π ) is closed Proof.
It follows from [ , Theorem 2 (4)] that if ρ ∈ U B ( π ) − , then ρ ∈U B ( π ). (cid:3) ATH-CONNECTED CLOSURE OF UNITARY ORBITS 3
Example . B. Blackadar [ ] showed that in B = M (cid:0) C (cid:0) S (cid:1)(cid:1) there are twoprojections P, Q that are unitarily equivalent, but are not homotopy equivalent. Thus U B ( P ) = U B ( P ) − is not path-connected. This implies that U B is not connected. We say that a unital C*-algebra B has property UC if U B is connected. Thealgebra B has property HUC if, for every projection P ∈ B , P B P has propertyUC. We say that B is matricially stable if and only if, for every n ∈ N , B isisomorphic to M n ( B ). Lemma . The following are true: (1)
Every von Neumann algebra has property HUC. (2)
A direct limit of unital C*-algebras with property HUC has property HUC. (3)
Every unital AF algebra has property HUC. (4) If A is a unital C*-algebra and, for every n ∈ N , M n ( C ) has propertyUC, then K ( A ) = 0 . (5) If B is matricially stable, then B has property UC if and only if K ( B ) = 0 . Proof. (1). In a von Neumann algebra A every unitary U can be written U = e iA with A = A ∗ , and the path g ( t ) = e (1 − t ) iA connects U to 1 in U A . Thus A has property UC. But P A P is a von Neumann algebra for every projection P ∈ A .Thus A has property HUC.(2). Suppose {A λ : λ ∈ Λ } is an increasingly directed family of unital C*-subalgebras of a unital C*-subalgebras A with property UC, and A = [ ∪ λ ∈ Λ A λ ] − . Let K be the connected component of U A that contains 1. Suppose U ∈ U A and ε >
0. Then there is a λ ∈ Λ and a unitary V ∈ A λ such that k U − V k < ε .However, if E λ is the connected component of U A λ , we have V ∈ E λ ⊂ E . Since ε > U ∈ E − = E . Next suppose each A λ has property HUC and P ∈ A is a projection. Then there is a λ ∈ Λ and a projection Q ∈ A λ such that k P − Q k <
1, which implies there is a unitary W ∈ A such that P = W ∗ QW .Hence P A P = W ∗ QW A W ∗ QW = W ∗ ( Q A Q ) . Thus P A P is isomorphic to Q A Q = [ ∪ λ ≥ λ Q A λ Q ] − . Since each Q A λ Q has property UC when Q ∈ A λ , we see that P A P has propertyUC. Thus A has property HUC.(3). This follows from (1) and (2).(4). This follows from the definition of K ( A ).(5). This follows from (4). (cid:3) B (cid:0) ℓ (cid:1) In this section we extend Theorem 1 to the case where the single operator isreplaced with a representation of a separable C*-algebra. The key idea is a resultof C. Olsen and W. Zame [ ] that if A is a separable C*-algebra, then A ⊗ K (cid:0) ℓ (cid:1) issingly generated. This gives us a general technique for relating the separable caseto the singly generated case.Suppose A is a unital C*-algebra. Let A † denote the unitization of A ⊗ K (cid:0) ℓ (cid:1) .If π ∈ Rep( A , B ) we define π † : A † → B † by π † ( λ a ij )) = λ π ( a ij )) . DON HADWIN AND WENJING LIU
Let B z be the C*-algebra generated by B † and { diag ( a, a, . . . ) : a ∈ A} . Theorem . Suppose A and B are unital C*-algebras and π, ρ ∈ Rep ( A , B ) .Then (1) The map ρ ρ † from Rep ( A , B ) to Rep (cid:0) A † , B † (cid:1) is continuous. (2) If π, ρ ∈ Rep ( A , B ) , then ρ ∈ U B ( π ) − if and only if ρ † ∈ U B † (cid:0) π † (cid:1) − . (3) If ρ ∈ U B ( π ) − and there is an internal path in U ( π ) − joining π to ρ , thenthere is an internal path in U B z (cid:0) π † (cid:1) − joining π † to ρ † . (4) If (a) B † ⊂ E is a C*-algebra with e E e = e B † e , (b) ρ ∈ U E (cid:0) π † (cid:1) − , (c) For every a ∈ A , ρ ( diag ( a, , , . . . )) = diag ( ρ ( a ) , , , . . . )(d) U B is connected, (e) B † ⊂ E is a C*-algebra with e E e = e B † e , and (f) there is a strong internal path in U E (cid:0) π † (cid:1) − from π † to ρ ,then there is a strong internal path in U B ( π ) − from π to ρ . Proof. (1). This is obvious.(2). Suppose ρ ∈ U B ( π ) − . Then there is a sequence { U n } in U B such that, forevery a ∈ A , lim n →∞ k U n π ( a ) U ∗ n − ρ ( a ) k = 0.For each positive integer n, let W n = diag ( U n , . . . , U n , , , , . . . ) in B † (with U n repeated n times). Since n T ∈ A † : lim n →∞ k W n π ( a ) W ∗ n − ρ ( a ) k = 0 o is a unital subalgebra containing the operators ( A ij ) ∈ A † such that, { ( i, j ) ∈ N × N : ( i, j ) = (0 , } is finite, we see that ρ † ∈ U B † (cid:0) π † (cid:1) − .Conversely, suppose ρ † ∈ U B † (cid:0) π † (cid:1) − . Then there is a sequence { V n } in B † suchthat, for every T ∈ A † , lim n →∞ (cid:13)(cid:13) V n π † ( T ) V ∗ n − ρ † ( T ) (cid:13)(cid:13) = 0.Since π † ( e ) = ρ † ( e ) = e , we see thatlim n →∞ k V n e − e V n k = lim n →∞ (cid:13)(cid:13) V n π † ( e ) V ∗ n − ρ † ( e ) (cid:13)(cid:13) = 0,and there is a sequence { W n } in U A such thatlim n →∞ (cid:13)(cid:13) W n − e V n e | ran( e ) (cid:13)(cid:13) = 0 . Hence, for every a ∈ A , lim n →∞ k W n π ( a ) W ∗ n − ρ ( a ) k = 0.Thus ρ ∈ U B ( π ) − . ATH-CONNECTED CLOSURE OF UNITARY ORBITS 5 (3). Suppose there is an internal path γ : [0 , → U ( π ) − joining π to ρ .For 0 ≤ t < γ ( t ) = U t π () U ∗ t with U t ∈ U B . For each 0 ≤ t < V t = diag ( U t , U t , . . . ) ∈ U B z and let Γ ( t ) = V t π † () V ∗ t . Then, for every T ∈ A † ,lim t → − (cid:13)(cid:13) V t π † ( T ) V ∗ t − ρ † ( T ) (cid:13)(cid:13) = 0.(4). Suppose Γ : [0 , → U E is continuous, and, for every T ∈ A † ,lim t → − (cid:13)(cid:13) Γ ( t ) π † ( T ) Γ ( t ) ∗ − ρ ( T ) (cid:13)(cid:13) = 0.Since ρ ( e ) = π † ( e ) = e , we conclude thatlim t → − k Γ ( t ) e − e Γ ( t ) k =lim t → − (cid:13)(cid:13) Γ ( t ) π † ( e ) Γ ( t ) ∗ − ρ ( e ) (cid:13)(cid:13) = 0.Since Γ ( t ) is unitary, there is a t ∈ [0 ,
1) such that, whenever t ≤ t < C t = e Γ ( t ) e is invertible in A and if U t = C t [ C ∗ t C t ] − / , then U t ∈ U A and lim t → − k C t − U t k = 0 . Since U A is connected, there is a continuous map t U t ∈ U A for 0 ≤ t ≤ t sothat U = 1 . If, for every a ∈ A , we consider T a = diag ( a, , , . . . ), it is easily seenthat lim t → − k U t π ( a ) U ∗ t − ρ ( a ) k = 0 . (cid:3) Theorem . Suppose A is a separable unital C*-algebra and π ∈ Rep (cid:0) A , B (cid:0) ℓ (cid:1)(cid:1) .Then U B ( ℓ ) ( π ) − is path-connected. Proof.
Suppose ρ ∈ U B ( ℓ ) ( π ) − . Then, by Theorem 2, ρ † ∈ U B ( ℓ ) † (cid:0) π † (cid:1) − . But B (cid:0) ℓ (cid:1) † ⊂ B (cid:0) ℓ ⊕ ℓ ⊕ · · · (cid:1) = E . Also, by [ ] there is an operator T ∈ A † such that A † = C ∗ ( T ). Thus ρ ( T ) ∈ U E ( π ( T )) − . We know from Theorem 1with X = π † ( T ) and Y = ρ † ( T ), that there is a W in E such C ∈ U E ( π ( T )) − and a strong internal path from π ( T ) to C in U E ( π ( T )) − and a strong internalpath in U E ( ρ ( T )) − from ρ ( T ) to C . There is a representation δ of C ∗ ( T ) suchthat δ ( T ) = W and if δ ( A ) = A ⊕ δ ( A ), we have δ ( T ) = T ⊕ W . Since e and δ ( e ) = e ⊕ δ ( e ) are projections with infinite rank and infinite corank,there is a unitary operator V such that V ∗ δ ( e ) V = e and V ∗ T V ∈ E . Let C = V ∗ δ ( T ) V and ρ () = V ∗ δ () V . It follows that there is a σ ∈ Rep (cid:0) A , B (cid:0) ℓ (cid:1)(cid:1) such that, for every a ∈ A , ρ ( diag ( a, , , · · · )) = ( σ ( a ) , , , · · · ) .Since there is an internal path in U E (cid:0) π † ( T ) (cid:1) − from π † ( T ) to ρ ( T ), there is astrong internal path in U E (cid:0) π † (cid:1) − from π † to ρ . It follows from part (4) of Theorem2 that there is a strong internal path in U B ( ℓ ) ( π ) − from π to σ. Similarly, thereis a strong internal path in U B ( ℓ ) ( ρ ) − from ρ to σ . Thus there is a path in U B ( ℓ ) ( π ) − = U B ( ℓ ) ( ρ ) − from π to ρ . (cid:3) DON HADWIN AND WENJING LIU
4. AF algebras
Lemma . Suppose ∈ A ⊂ D are separable unital C*-algebras, B is a unitalC*-algebra and π, ρ ∈ Rep ( D , B ) , and suppose V, W ∈ U B such that (1) for every x ∈ D , W ∗ ρ ( x ) W = π ( x ) , (2) for every x ∈ A , V ∗ ρ ( x ) V = π ( x ) , (3) U B∩ ρ ( A ) ′ is connected.Then there is a path t U t of unitary operators in B such that U = V , U = W , and for every t ∈ [0 , and every x ∈ A , U ∗ t ρ ( x ) U t = π ( x ) . Proof.
We know that, for every x ∈ A , W ∗ ρ ( x ) W = V ∗ ρ ( x ) V. Thus
V W ∗ = X ∈ ρ ( A ) ′ ∩ B . Thus W = U ∗ V . Since U ρ ( A ) ′ ∩B is path connected,there is a path t X t of unitary elements in ρ ( A ) ′ ∩ B such that X = 1 and X = X . For t ∈ [0 ,
1] let U t = X ∗ t V . Then U t is a path in U B , U = V and U = X ∗ V = W . Moreover, for each t ∈ [0 ,
1] and each x ∈ A , U ∗ t ρ ( x ) U t = V ∗ X t ρ ( x ) X ∗ t V = V ∗ ρ ( x ) V = π ( x ) . (cid:3) Theorem . Suppose A ⊂ A ⊂ · · · ⊂ A and A = [ ∪ n ∈ N A n ] − is separable.Suppose π, ρ ∈ Rep ( A , B ) such that, for every n ∈ N , (1) ρ | A n ∈ U B ( π | A n ) , (2) U ρ ( A n ) ′ ∩B is connected.Then there is a strong internal path from π to ρ . Proof.
For each n ∈ N , choose U n ∈ U B such that, for every a ∈ A n , U ∗ n ρ ( a ) U n = π ( a ) .It follows from Lemma 3 that we can define a path t U t from [ n, n + 1] so thatfor n ≤ t ≤ n + 1 and a ∈ A n , we have U ∗ t ρ ( a ) U t = π ( a ) .Thus the map t U t is continuous, and, for every a ∈ ∪ n ∈ N A n we havelim t → + ∞ k U ∗ t ρ ( a ) U t − π ( a ) k = 0 . Hence, if we define π t ( · ) = U ∗ t ρ ( · ) U t for t ∈ [0 , ∞ ) and π ∞ = ρ , we have a stronginternal path in U B ( π ) − from π to ρ . (cid:3) Theorem . Suppose A is a separable unital AF C*-algebra, B is a C*-algebrawith property HUC, and π ∈ Rep ( A , B ) . Then U B ( π ) − is path-connected. ATH-CONNECTED CLOSURE OF UNITARY ORBITS 7
Proof.
We can assume that ker π = 0 , since A / ker ρ is a separable unitalAF algebra. Since A is unital and AF, there is a sequence {A n } of unital finite-dimensional C*-subalgebras 1 ∈ A ⊂ A ⊂ · · · such that " ∞ [ n =1 A n − = A .Suppose ρ ∈ U M ( π ) − . Since each A n is finite-dimensional, where approximateequivalence is the same as unitary equivalence, we have ρ | A n ∈ U B ( π | A n ) for each n ∈ N .Fix n ∈ N and write A n as M s ( C ) ⊕ · · · ⊕ M s t ( C ) and, for 1 ≤ k ≤ t , let { e ij,k : 1 ≤ i, j ≤ s k } be the system of matrix units for M s k ( C ) . It is easily seenthat ρ ( A n ) ′ ∩ B is the set of all t X k =1 s k X j =1 ρ ( e j , k ) ρ ( e ,k ) xρ ( e ,k ) ρ ( e ij,k )for x ∈ B . It follows that ρ ( A n ) ′ ∩ B is isomorphic to ⊕ X ≤ k ≤ t ρ ( e ,k ) B ρ ( e ,k ) . Since B has property HUC, we see that ρ ( A n ) ′ ∩ B has property UC. The desiredconclusion now follows from Theorem 4. (cid:3) Corollary . If A is a separable unital AF C*-algebra and B is either an AF C*-algebra or a von Neumann algebra, then, for every ρ ∈ Rep ( A , B ) , U B ( ρ ) − is path-connected. A separable C*-algebra is homogeneous if it is a finite direct sum of algebrasof the form M n ( C ( X )) , where X is a compact metric space. A unital C*-algebrais subhomogeneous if it is a unital subalgebra of a homogeneous C*-algebra. Ev-ery subhomogeneous von Neumann algebra is homogeneous; in particular, if A issubhomogeneous, then the second dual A of A is homogeneous. A C*-algebrais approximately subhomogeneous (ASH) if it is a direct limit of subhomogeneousC*-algebras.A (possibly nonseparable) C*-algebra B is LF if, for every finite subset F ⊂ B and every ε > D of B such that, for every b ∈ F , dist( b, D ) < ε . Every separable unital C*-subalgebra of a LF C*-algebra iscontained in a separable AF subalgebra. See [ ] for details.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 D such that A ⊂ D ⊂ 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*-algebras and A = ( ∪ λ A λ ) −kk ,then A ⊂ (cid:16) ∪ λ A λ (cid:17) −kk ⊂ A . The proof of the next theorem relies on results in[ ]. DON HADWIN AND WENJING LIU
Theorem . Suppose A is a separable strongly LF embeddable C*-algebra and M is a finite von Neumann algebra. Then, for every π ∈ Rep ( A , M ) , U M ( π ) − ispath connected. Proof.
Suppose ρ ∈ U M ( π ) − . It follows that there are weak*-weak* contin-uous unital ∗ -homomorphisms ˆ π, ˆ ρ : A → M such that ˆ π | A = π and ˆ ρ | A = ρ .Since A is strongly LF embeddable, there is a separable unital AF C*-algebra D such that A ⊂ D ⊂ A .It follows from [ , Theorem 2] that ˆ ρ | D ∈ U M (ˆ π | D ) − . We know from Theorem 5that U M (ˆ π | D ) − is path connected. Thus there is a path in U M (ˆ π | D ) − from ˆ π | D toˆ ρ | D . Restricting to A , we obtain a path in U M ( π ) − from π to ρ . (cid:3)
5. Abelian algebras
Lemma . Suppose N is a countably generated von Neumann algebra. Then N is isomorphic to a direct sum P ⊕ i ∈ I N i so that each N i acts on a separable Hilbertspace. In particular, each N i is σ -finite. Proof.
Suppose N ⊂ B ( H ) and e ∈ H with k e k = 1 . Let P be the orthogonalprojection onto ( N e ) − . Thus P ∈ N ′ . Let P e ∈ Z ( N ) be the central coverof P. Then the map T T | P ( H ) is a normal isomorphism between N | P e ( H ) and N | P ( H ) . Since N is countably generated, P ( H ) is separable. Thus N | P e ( H ) is adirect summand of N that is isomorphic to a von Neumann algebra on a separableHilbert space. The rest of the proof follows from this idea and Zorn’s lemma. (cid:3) Suppose M is a von Neumann algebra and T ∈ M . In [ ] H. Ding andD. Hadwin defined M -rank( T ) to be the Murray von Neumann equivalence classof the orthogonal projection R ( T ) onto the closure of the range of T . We say M -rank( S ) ≤ M -rank( T ) if and only if there is a projection P ∈ M such that P ≤ R ( T ) and P is Murray von Neumann equivalent to R ( S ). They proved thatif a separable unital C*-algebra is a direct limit of homogeneous algebras, and M acts on a separable Hilbert space, then for all π, ρ ∈ Rep( A , M ), ρ ∈ U M ( π ) − ifand only if, for every x ∈ A , M -rank ( π ( x )) = M -rank ( ρ ( x )) .A key ingredient of the proof of this result was a sequential semicontinuity of M -rank with respect to the *-SOT that was proved when M is a von Neumann algebraacting on a separable Hilbert space [ , Theorem 1]. We extend this to the generalcase. Lemma . Suppose M is a von Neumann algebra, A, B ∈ M and, for each n ∈ N , B n ∈ M and M -rank ( B n ) ≤ M -rank ( A ) . If B n → B is the ∗ -SOT, then M -rank ( B ) ≤ M -rank ( A ) . Proof.
Let P n = R ( B n ) , Q = R ( A ) , and, for each n ∈ N , choose a partialisometry V n ∈ M such that V ∗ n V n = P n and V n V ∗ n ≤ Q . Let N = W ∗ ( { A, B, B , V , B , V , . . . } ) . Clearly, we have, for every n ∈ N , that N -rank ( B n ) ≤ N -rank ( A ) . ATH-CONNECTED CLOSURE OF UNITARY ORBITS 9
By Lemma 4, we can write N = ⊕ X i ∈ I N i with each N i acting on a separable Hilbert space.Write A = ⊕ X i ∈ I A i , B = ⊕ X i ∈ I B i , B n = ⊕ X i ∈ I B n,i , V n = ⊕ X i ∈ I V n,i . Since R ( A ) = P ⊕ i ∈ I R ( A i ) and R ( B ) = P ⊕ i ∈ I R ( B n,i ) , for each i ∈ I , N i -rank( B n,i ) ≤ N i -rank( A i ) and the limit in the ∗ - SOT of B n,i is B i . Thus, by [ ,Theorem 1], for each i ∈ I , N i -rank ( B i ) ≤ N i -rank ( A i ) .Thus, for each i ∈ I , there is a partial isometry W i ∈ N i such that W ∗ i W i = R ( B i ) and W i W ∗ i ≤ R ( A i ) . Then W = P ⊕ i ∈ I W i is a partial isometry in N such that W ∗ W = R ( B ) and W W ∗ ≤ R ( A ) . Since we also have W ∈ M , we conclude M -rank( B ) ≤ M -rank( A ). (cid:3) Corollary . If A is a unital C*-algebra, M is a von Neumann algebra and π ∈ Rep ( A , M ) and ρ ∈ U M ( π ) − , then, for every a ∈ A , M -rank ( π ( a )) = M -rank ( ρ ( a )) . Proof.
Suppose a ∈ A . There is a sequence { U n } in U M such thatlim n →∞ k U ∗ n π ( A ) U n − ρ ( A ) k = lim n →∞ k π ( a ) − U n ρ ( a ) U ∗ n k = 0 . Also M -rank( U ∗ n π ( a ) U n ) = M -rank( π ( a )) and M -rank( U n ρ ( a ) U ∗ n ) = M -rank( ρ ( a ))for each n ∈ N . Thus, by Lemma 5, M -rank ( ρ ( a )) ≤ M -rank ( π ( a )) and M -rank ( π ( a )) ≤ M -rank ( ρ ( a )) . (cid:3) Suppose A is a unital C*-algebra and M is a von Neumann algebra and π : A → M is a unital ∗ -homomorphism. Then there is a unique ∗ -homomorphismˆ π : A → M that is weak*-weak* continuous (see [ ]). Lemma . Suppose ( X, d ) is a compact metric space, M is a sigma-finite vonNeumann algebra, and π, ρ : C ( X ) → M , ρ ∈ U M ( π ) − . Then there is a sequence F , F , . . . of finite disjoint collections of nonempty Borel sets such that (1) P E ∈F n ˆ π ( χ E ) = P E ∈F n ˆ ρ ( χ E ) = 1 , (2) { ˆ π ( χ E ) : E ∈ F n } ⊂ sp ( { ˆ π ( χ F ) : F ∈ F n +1 } ) and { ˆ ρ ( χ E ) : E ∈ F n } ⊂ sp ( { ˆ ρ ( χ F ) : F ∈ F n +1 } ) , (3) For every E ∈ F n , and diam ( E ) < /n . (4) For every E ∈ ∪ n ∈ N F n ˆ π ( χ E ) and ˆ ρ ( χ E ) are Murray von Neumannequivalent. Proof.
Let Bor( X ) be the C*-algebra with the supremum norm. We thenhave C ( X ) ⊂ Bor ( X ) ⊂ C ( X ) and ˆ π | Bor( X ) , ˆ ρ | Bor( X ) are unital ∗ -homomorphisms.Let Σ = (cid:8) U ⊂ X : U is open and ˆ π (cid:0) χ ¯ U \ U (cid:1) = ˆ ρ (cid:0) χ ¯ U \ U (cid:1) = 0 (cid:9) . It is easily shownthat if U, V ∈ Σ, then U \ ¯ V , U ∪ V , U ∩ V ∈ Σ. Moreover, if a ∈ X and S ( a, r ) = { x ∈ X : d ( a, x ) = r } for all r >
0, it follows from the fact that M is σ -finite that if E a = (cid:8) r ∈ (0 , ∞ ) : ˆ π (cid:0) χ S ( a,r ) (cid:1) = ˆ ρ (cid:0) χ S ( a,r ) (cid:1) = 0 (cid:9) , then (0 , ∞ ) \ E a is countable.We can assume that diam ( X ) < F = { X } .Suppose n ∈ N and F n has been defined.For each a ∈ X, there is an r a ∈ E a ∩ (cid:16) , n +1) (cid:17) . Since X is compact and { ball ( a, r a ) : a ∈ X } is an open cover with sets in Σ, there is a finite subcover { U , . . . , U s } . We let V = U , and V k = U k \ ∪ ≤ j 1] suchthat f ( x ) = 0 if and only if x ∈ X \ U . Thus the sequence f /n ↑ χ U , which means f /n → χ U weak* in C ( X ) . Thus π ( f ) /n ↑ ˆ π ( χ U ) and ρ ( f ) /n ↑ ˆ ρ ( χ U ) in the weak*topology. Thus ˆ π ( χ U ) is the projection onto the closure of the range of π ( f ) andˆ ρ ( χ U ) is the projection onto the closure of the range of ρ ( f ). It follows fromCorollary 2 that ˆ π ( χ U ) and ˆ ρ ( χ U ) are Murray von Neumann equivalent. (cid:3) Theorem . Suppose A is a separable unital commutative C*-algebra and M is a von Neumann algebra. If π ∈ Rep ( A , M ) then U B ( π ) − is path-connected. Infact, for every ρ ∈ U M ( π ) − there is a strong internal path from π to ρ . Proof. Suppose ρ ∈ U M ( π ) − . Since A is separable, there is a sequence { U n } ∈ U M such that, for every a ∈ A ,lim n →∞ k U ∗ n π ( a ) U n − ρ ( a ) k = 0 . Let N = W ∗ ( π ( A ) ∪ ρ ( A ) ∪ { U , U , . . . } ). Then N is a countably generated vonNeumann algebra, and π, ρ : A → N . Hence we can write N = ⊕ X i ∈ I N i , where each N i acts on a separable Hilbert space, and we can write π = ⊕ X i ∈ I π i and ρ = ⊕ X i ∈ I ρ i . ATH-CONNECTED CLOSURE OF UNITARY ORBITS 11 We also have ˆ π = ⊕ X i ∈ I ˆ π i and ˆ ρ = ⊕ X i ∈ I ˆ ρ i . For each i ∈ I , we can choose a sequence F n,i of families of nonempty opensubsets as in Lemma 6. Since, for each i ∈ I and each n ∈ N and each E ∈ F n,i we know ˆ π i ( χ E ) and ˆ ρ i ( χ E ) are Murray von Neumann equivalent in N i and since X E ∈F n ˆ π i ( χ E ) = X E ∈F n ˆ ρ i ( χ E ) = 1 , there is a unitary U n,i ∈ N i such that U ∗ n,i ˆ π i ( χ E ) U n,i = ˆ ρ i ( χ E )for every E ∈ F n,i . For each n ∈ N , let U n = P ⊕ i ∈ I U n,i for each i ∈ I , and let D n = P ⊕ i ∈ I sp( { ˆ π i ( χ E ) : E ∈ F n,i } ). Since U n U ∗ n +1 ∈ D ′ n , we know from the proofof Lemma 3 that the map n U n on N extends to a continuous map t U t = P ⊕ i ∈ I U t,i such that U = 1 , and such that, for every n ∈ N , for every i ∈ I , every n ≤ t < ∞ , and every E ∈ F n,i U ∗ t,i ˆ π i ( χ E ) U t,i = U ∗ n,i ˆ π i ( χ E ) U n,i = ˆ ρ ( χ E ) . Suppose f ∈ C ( X ) and ε > 0. Since f is uniformly continuous, there is a positiveinteger n such that, if x, y ∈ X and d ( x, y ) < /n , then | f ( x ) − f ( y ) | < ε/ i ∈ I and all E ∈ F n ,i we choose x i,n ,E ∈ E . Since diam( E ) < /n ,we then have k [ f − f ( x n ,i,E )] χ E k < ε/ , so (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) π i ( f ) − X E ∈F n ,i f ( x n ,i,E ) ˆ π i ( χ E ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε/ , and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρ i ( f ) − X E ∈F n ,i f ( x n ,i,E ) ˆ ρ i ( χ E ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε/ t ≥ n , we have k U ∗ t π ( f ) U t − ρ ( f ) k =sup i ∈ I (cid:13)(cid:13) U ∗ t,i π i ( f ) U t,i − ρ i ( f ) (cid:13)(cid:13) ≤ sup i ∈ I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) U ∗ t,i π i ( f ) − X E ∈F n ,i f ( x n ,i,E ) ˆ π i ( χ E ) U t,i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + sup i ∈ I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X E ∈F n ,i f ( x n ,i,E ) (cid:2) U ∗ t,i ˆ π i ( χ E ) U t,i − ˆ ρ i ( χ E ) (cid:3)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + sup i ∈ I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X E ∈F n ,i f ( x n ,i,E ) ˆ ρ i ( χ E ) − ρ i ( f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε/ ε/ ε . Thus, the map t U t is continuous on [1 , ∞ ), and, for every f ∈ C ( X ),lim t →∞ k U t π ( f ) U ∗ t − ρ ( f ) k = 0. (cid:3) Corollary . Suppose A is a separable unital homogeneous C*-algebra and M is a von Neumann algebra. If π ∈ Rep ( A , M ) then U B ( π ) − is path-connected.In fact, for every ρ ∈ U M ( π ) − there is a strong internal path from π to ρ . References [1] B. Blackadar, K-theory for operator algebras. Second edition. Mathematical Sciences ResearchInstitute Publications, 5. Cambridge University Press, Cambridge, 1998.[2] K. R. Davidson, C*-algebras by example, Fields Institute Monographs, 6. American Mathe-matical Society, Providence, RI, 1996.[3] H. Ding and D. Hadwin, Approximate equivalence in von Neumann algebras,Sci. China Ser. A 48 (2005), no. 2, 239–247.[4] D. Hadwin, An operator-valued spectrum. Indiana Univ. Math. J. 26 (1977), no. 2, 329–340.[5] D. Hadwin and W. Liu, Approximate Equivalence in von Neumann Algebras, preprint, 2020.[6] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, Vol. II,Advanced theory, Corrected reprint of the 1986 original. Graduate Studies in Mathematics,16. American Mathematical Society, Providence, RI, 1997.[7] C. Olsen and W. Zame, Some C*-algebras with a single generator, Trans. Amer. Math. Soc.215 (1976), 205–217.[8] M. Rørdam, F. Larsen and N. Laustsen, An introduction to K-theory for C*-algebras, LondonMathematical Society Student Texts, 49, Cambridge University Press, Cambridge, 2000. University of New Hampshire E-mail address : [email protected] Current address : University of Massachusetts Boston E-mail address ::