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ON EQUIVALENCE FOR REPRESENTATIONS OF TOEPLITZ ALGEBRAS 6 1 PHILIPM.GIPSON 0 2 Abstract. Two new notions of equivalence for representations of a Toeplitz t c algebraEn,n<∞,onacommonHilbertspacearedefined. Ourmainresults O apply to C∗-dynamics and the conjugacy of certain ∗-endomorphisms. One particular caseoftherelations isshowntocoincidewiththemultiplicityofa 7 representation. PreviouslyknownresultsduetoLacaandEnomoto-Watatani arerecoveredasspecialcases. ] A O First introduced by Cuntz in [2], the Toeplitz algebra E is the universal C∗- n . h algebra generated by n ≤ ∞ isometries with pairwise orthogonal ranges. The t Toeplitz algebras and their representations have been surprisingly pervasive in the a m theory of operator algebras; appearing in contexts as diverse as crossed products and K-theory. In this paper we will introduce two families of equivalence relations [ (Definitions 2.1 and 2.4) on the set of nondegenerate ∗-representations of En on 2 a Hilbert space H. These families are both indexed by the unital C∗-subalgebras v of B(H). We give applications of the relations to dynamical systems and prove a 4 relationship between conjugacy of endomorphisms and equivalence of representa- 6 1 tionsofToeplitz algebras(Theorems4.3and4.4). Lastly,wedemonstratethatour 2 resultsrecover,asspecialcases,previouslyknownresultsofLaca[6]andEnomoto- 0 Watatani [3] concerning endomorphisms of type I∞ and II1 factors. . 1 0 6 1. Preliminaries 1 : Our primary objects of study will be right Hilbert C∗-modules over unital C∗- v algebras. Recall that an A-module homomorphism φ : X → Y is adjointable if i X there is another A-module homomorphism φ∗ : Y → X such that hφ(x),yi = Y r hx,φ∗(y)iX for all x ∈ X and y ∈ Y. We’ll denote the space of adjointable homo- a morphisms between A-modules X and Y by L(X,Y) and set L(X):=L(X,X). A homomorphism φ ∈ L(X,Y) is unitary if φ◦φ∗ = id and φ∗ ◦φ = id ; in this Y X case we say X and Y are unitarily equivalent and write X ≃Y. A subset {x : i ∈ I} ⊂ X is orthonormal if A is unital and hx ,x i = δ 1 i i j ij A for all i,j ∈ I. An orthonormal basis is a right A-module basis which is also an orthonormalset. It is routine to check that if {f ,...,f } is any finite orthonormal 1 n n basis for X then we have the decomposition x= f hf ,xi for all x∈X. i=1 i i The free (right, Hilbert) A-module of rank n (where A is a unital C∗-algebra) is the finite direct sumAn =A⊕...⊕A (n summanPds) with the coordinatewiseright 2000 Mathematics Subject Classification. 46L05,46L08,46L55. Key words and phrases. Toeplitz Algebras, Equivalence Relations, Endomorphisms, Hilbert Modules. 1 2 PHILIPM.GIPSON A-module structure and A-valued inner product given by n h(a ),(b )i:= a∗b ∈A. i i i i i=1 X The distinguished vectors e ,...,e ∈ An (where e is the vector which has 1 in 1 n i A the i-th coordinate and zeros elsewhere) form an orthonormal basis for An which we will term the standard basis. We can make a natural identification of L(An) with the matrix algebra M (A), n wherethe homomorphismsθ :x7→e he ,xiplaytheroleofmatrixunits. Thus ei,ej i j V = [v ] ∈ M (A) has Vx = n n e v he ,xi for all x ∈ An. Because of ij n j=1 i=1 j ji i this it is straightforward to check that U = [u ] ∈ M (A) is unitary if and only jk n if n u u∗ = δ 1 and nPu∗uP = δ 1 for every j,k = 1,...,n. That is i=1 ij ik jk A i=1 ji li jk A to say that matrix unitaries are precisely the unitary homomorphisms, and vice- verPsa. If {f ,...,f } is any oPrthonormal basis of An then letting u := he ,f i for 1 n ij i j each i,j =1,...,n we have that U :=[u ] is a unitary matrix in M (A) such that ij n Ue =f for each i=1,...,n. i i Acovariantrepresentation ofAnonaHilbertspaceKisapair(σ,π)consistingof a linear map σ :An →B(K) and a nondegenerate ∗-representationπ :A→B(K) which together satisfy the covariance condition σ(xa) = σ(x)π(a) for all x ∈ An and a ∈ A. A covariant representation (σ,π) is Toeplitz if σ(x)∗σ(y) = π(hx,yi) for all x,y ∈An. Note. Henceforth we will only consider the case when A is concretely represented, i.e. A⊂B(H)forsomeH,andeverycovariantrepresentationisoftheform(σ,id) for some linear map σ :An →B(H). Consequently we will abuse the notation and write σ alone instead of (σ,id). Suppose that σ is a Toeplitz representation (i.e. (σ,id) is a Toeplitz representation) then σ(e )∗σ(e ) = he ,e i = δ I for all i,j = 1,...,n. Thus {σ(e ) : i = i j i j ij i 1,...,n}isafamily ofn isometrieswithpairwiseorthogonalranges. We willalways use v ,...,v to denote the universal generators of E and so E := C∗({v ,...,v : 1 n n n 1 n v∗v =δ I}). By the universal nature of the Toeplitz algebra E , the assignments i j ij n v 7→σ(e ) extend uniquely to a ∗-representation of E on H which we will denote i i n by ω . σ Conversely, given a nondegenerate ∗-representation ω : E → B(H) we may n define for all x∈An n σ (x):= ω(v )he ,xi. ω i i i=1 X It is easily checked that σ :An →B(H) is a linear map which, together with the ω identity representationof A, satisfies the covariancecondition; hence is a covariant representation of E . By construction we have that σ is Toeplitz. Note that n ω σ (e )=ω(v ) and this uniquely determines σ . Naturally, σ =σ and ω =ω. ω i i ω ωσ σω 2. Equivalences Given a unital C∗-algebra A ⊂ B(H), we have seen that the Toeplitz covari- antrepresentationsof An coincide with the nondegenerate ∗-representationsof the Toeplitz algebra E . We will now use this relationship to define two families of n equivalence relations for representations of E . n EQUIVALENT REPRESENTATIONS OF TOEPLITZ ALGEBRAS 3 2.1. Free Equivalence. Our first notion of equivalence is inspired from the ob- servation that if σ is a Toeplitz covariant representation of An and U ∈ M (A) a n unitary then σ◦U is also a Toeplitz covariant representation of An. Definition 2.1. Let A ⊂ B(H) be a unital C∗-subalgebra. Two nondegenerate representations ω and τ of E (on H) are A-free equivalent if there is a unitary n homomorphism U ∈L(An) such that σ =σ ◦U. ω τ Proposition 2.2. A-free equivalence is an equivalence relation. Proof. Reflexivity is obvious. Whenever σ (x) = σ (Ux) for some unitary U ∈ L(An) and all x ∈ An we ω τ have σ (U∗x) = σ (UU∗x) = σ (x) for all x ∈ An and so A-free equivalence is ω τ τ symmetric. Lastly,ifσ (x)=σ (Ux)andσ (y)=σ (Wy)forsomeunitariesU,W ∈L(An) ω τ τ κ and all x,y ∈ An then σ (x) = σ (WUx) for all x ∈ An and WU is obviously a ω κ unitaryinM (A). HenceωandκareA-freeequivalentandwehavetransitivity. (cid:3) n Proposition 2.3. Two representations ω and τ are A-free equivalent if and only if there is a unitary U =[u ]∈M (A) for which jk n n ω(v )= τ(v )u i j ji j=1 X for all i=1,...,n. Proof. Suppose that there is a unitary U ∈ L(An) for which σ (x) = σ (Ux) ω τ for all x ∈ An. Since we may identify L(An) with M (A) and thus there is a n matrix [u ] ∈ M (A) such that Ux = n n e u he ,xi for all x ∈ An. In jk n j=1 k=1 j jk k n particular, we have that Ue = e u for all i=1,...,n. Hence i j=1 jPji P P n n n ω(v )=σ (e )=σ (Ue )=σ e u = σ (e )u = τ(v )u i ω i τ i τ j ji τ j ji j ji   j=1 j=1 j=1 X X X   as desired. n Conversely,ifthereisaunitaryU =[u ]∈M (A)forwhichω(v )= τ(v )u jk n i j=1 j ji for all i=1,...,n then P n σ (x) = ω(v )he ,xi ω i i i=1 X n n = τ(v )u he ,xi j ji i i=1j=1 XX n n = σ e u he ,xi τ j ji i   i=1j=1 XX = σ (Ux)  τ for all x∈An. (cid:3) 4 PHILIPM.GIPSON 2.2. Quasifree Equivalence. It is a standard fact that when W ∈ B(H) is a unitary the mapping Ad : T 7→ WTW∗ is a ∗-automorphism of B(H). Of more W interest to us is that when A⊂B(H) is a unital C∗-algebra and Ad restricts to W an automorphism of A we will say that W is A-fixing. Since AdW∗ =(AdW)−1 we find that W is A-fixing if and only if W∗ is A-fixing. It is also obvious that if W 1 and W are both A-fixing then W W is A-fixing as well. 2 1 2 If ω is a representation of E and W ∈ B(H) is a unitary, denote by Ad ◦ω n W the representationgenerated by v 7→Wω(v )W∗. i i Definition2.4. Twonondegeneraterepresentationsωandτ (onH)areA-quasifree A-q.f. equivalent, denoted ω ∼ τ, if there is an A-fixing unitary W ∈ B(H) such that ω and Ad ◦τ are A-free equivalent. W A-q.f. Proposition 2.5. ω ∼ τ if and only if there is an A-fixing unitary W ∈B(H) and a unitary U =[u ]∈M (A) for which jk n n ω(v )= Wτ(v )W∗u i j ji j=1 X for all i=1,...,n. The proof is simply an application of Theorem 4.3 to the representations ω and Adj ◦τ. W Proposition 2.6. For any given unital C∗-algebra A ⊆ B(H), A-quasifree equivalence is an equivalence relation. Proof. Reflexivity is obvious. A-q.f. Suppose that ω ∼ τ,which by Proposition 2.5 allows us to conclude that ω(v )= n Wτ(v )W∗u forsomeunitariesU =[u ]∈M (A)andW ∈B(H) i j=1 j ji jk n (which is A-fixing) and all i = 1,...,n. Set v = W∗u∗ W ∈ A and note that P jk kj V =[v ]∈M (A) is unitary. It remains to check that for each i=1,...,n jk n n n n W∗ω(v )Wv = W∗ Wτ(v )W∗u Wv j ji k kj ji ! j=1 j=1 k=1 X X X n n = W∗ Wτ(v )W∗u WW∗u∗W k kj ij ! j=1 k=1 X X n n = τ(v )W∗u u∗W k kj ij j=1k=1 XX n n = τ(v )W∗ u u∗ W k  kj ij k=1 j=1 X X n   = τ(v )W∗δ W k ki i=1 X = τ(v ) i A-q.f. and so, by the previous proposition, τ ∼ ω. EQUIVALENT REPRESENTATIONS OF TOEPLITZ ALGEBRAS 5 A-q.f. A-q.f. To provetransitivity,suppose that ω ∼ τ andτ ∼ κ, so by Proposition2.5 n ω(v )= W τ(v )W∗u i 1 j 1 ji j=1 X n τ(v )= W κ(v )W∗v j 2 k 2 kj k=1 X for some A-fixing unitaries W ,W ∈ B(H) and unitaries U = [u ],V = [v ] ∈ 1 2 jk jk M (A) and all i,j =1,...,n. Thus n n ω(v ) = W τ(v )W∗u i 1 j 1 ji j=1 X n n = W W κ(v )W∗v W∗u 1 2 k 2 kj 1 ji ! j=1 k=1 X X n n = W W κ(v )(W W )∗ W v W∗u . 1 2 k 1 2  1 kj 1 ji k=1 j=1 X X If we let t := n W v W∗u ∈A then T =[t ]∈M (A) is unitary since ki j=1 1 kj 1 ji ki n ∗ P n n n (T∗T) = t∗ t = W v W∗u W v W∗u ij ki kj 1 kl 1 li 1 km 1 mj ! ! k=1 k=1 l=1 m=1 X X X X n n n = u∗W v∗ v W∗u li 1 kl km 1 mj k=1l=1m=1 XX X n n n = u∗W v∗ v W∗u li 1 kl km 1 mj ! l=1m=1 k=1 X X X n n = u∗W δ W∗u li 1 lm 1 mj l=1m=1 X X n = u∗ u mi mj m=1 X = δ I ij and similarly tedious calculations show (TT∗) =δ . ij ij Denoting byW′ the unitaryW W ∈B(H)(whichis alsoA-fixing),we havefor 1 2 each i=1,...,n n n n ω(v )= W W κ(v )(W W )∗ W v W∗u = W′κ(v )W′∗t i 1 2 k 1 2  1 kj 1 ji j ji k=1 j=1 j=1 X X X   and so ω A-∼q.f.κ as desired. (cid:3) 3. Equivalence and Multiplicity The complex subspace of E spanned by the generating isometries is a Hilbert n space En ∼=Cn. 6 PHILIPM.GIPSON Definition 3.1. The multiplicity of a nondegenerate ∗-representation π : E → n B(H) is the dimension of (π(E )H)⊥. n Our goal for this section is the proof of the following theorem: Theorem 3.2. Two representations of E on H are B(H)-quasifree equivalent if n and only if they have the same multiplicity. We will accomplish this through severalintermediate lemmas. Let p := I − n v v∗ and define J ⊂ E to be the (closed, two-sided) ideal n i=1 i i n n generated by p . Note that the multiplicity of π is also the rank of the projection n P π(p ). Representationsofmultiplicity0(alsocalledessentialrepresentations inthe n literature)factor through the quotient En/Jn ∼=On and thus may be thought of as representations of the Cuntz algebra. Lemma 3.3. Any two essential representations of E on H are B(H)-free equiva- n lent. Proof. If ω and τ are essential representations of E on a Hilbert space H then n necessarily n n ω(v )ω(v )∗ =I = τ(v )τ(v )∗. i i j j i=1 j=1 X X Henceω(v )= n τ(v )τ(v )∗ω(v )foreachi=1,...,n. Definingu :=τ(v )∗ω(v ) i j=1 j j i jk j k for each j,k =1,...,n it is straightforwardto check that P n n u u∗ = τ(v )∗ω(v )∗ω(v )∗τ(v )=τ(v )∗Iτ(v )=δ I ji ki j i i k j k jk i=1 i=1 X X and similarly n u∗u =δ I, hence U =[u ]∈M (B(H)) is unitary. (cid:3) i=1 ij ik jk jk n P The Fock representation of E is a nondegenerate ∗-representation of multiplic- n ity 1 which reduces to the unilateral shift when n = 1. We’ll briefly outline the constructionwhich is originally due to Evans [4]. Let E⊗j denote the j-fold tensor n product Hilbert space (where E⊗0 :=C) and define the full Fock space n ∞ F := E⊗j. E n j=0 M For each i = 1,...,n and all j ≥ 0 we’ll form maps ϕ(v ) : E⊗j → E⊗j+1 by i n n ϕ(v )x = e ⊗x. Taken all together these define a family of linear maps {ϕ(v ) : i i i F → F } which are easily seen to be isometries which have pairwise orthogonal E E ranges. Hence v 7→ ϕ(v ) extends to a representation ϕ : E → B(F ). This is i i n E the canonicalFock representationandit is routine to see that ϕ has multiplicity 1. The Fock representation of multiplicity k is analogously defined but with ϕk(v ) : i E⊗j →E⊗j+k given by ϕk(v )x=(ϕ(v ))kx for each i=1,...,n. n n i i Lemma 3.4. For any Toeplitz algebra E and integers a,b ≥ 1, ϕa is B(F )-free n E equivalent to ϕb if and only if a=b. Proof. The “if” statement is trivial. EQUIVALENT REPRESENTATIONS OF TOEPLITZ ALGEBRAS 7 Suppose that ϕa and ϕb are B(F )-free equivalent. Then there is a unitary E U =[u ]∈M (B(F )) such that jk n E n ϕa(v )= ϕb(v )u i j ji j=1 X for all i=1,...,n. Hence ∗ n n n n ϕa(v )ϕa(v )∗ = ϕb(v )u ϕb(v )u i i j ji k ki   ! i=1 i=1 j=1 k=1 X X X X n n n  = ϕb(v )u u∗ ϕb(v )∗ j ji ki k i=1j=1k=1 XXX n n n = ϕb(v ) u u∗ ϕb(v )∗ j ji ki k ! j=1k=1 i=1 XX X n n = ϕb(v )δ ϕb(v )∗ j jk k j=1k=1 XX n = ϕb(v )ϕb(v )∗. j j j=1 X As a consequence we find that n n ϕa(p )=I − ϕa(v )ϕa(v )∗ =I − ϕb(v )ϕb(v )∗ =ϕb(p ) n i i j j n i=1 j=1 X X and thus (ϕa(p )B(F ))⊥ = (ϕb(p )B(F ))⊥. But that means ϕa and ϕb share n E n E the same multiplicity. (cid:3) We willmakeuseofa(rephrased)resultofPopescu[7,Theorem1.3]whichgen- eralizedtheWolddecomposition: everyrepresentationπofE hasadecomposition n π = π ⊕π where π is an essential representation and π is unitarily equivalent e s e s to a multiple of the Fock representation. Lemma 3.5. For i=1,2 suppose that ω andτ arenondegenerate representations i i of E on H ; A ⊂B(H ) are unital C∗-algebras; and ω Ai∼-q.f.τ . Then ω ⊕ω is n i i i i i 1 2 A ⊕A -quasifree equivalent to τ ⊕τ . 1 2 1 2 Naturally the statement may be generalizedto morethan two pairs of representations, but, because of the Wold decomposition analogue, this is enough for our purposes. Proof. Denote ω = ω ⊕ω and τ = τ ⊕τ . Since ω Ai∼-q.f. τ for i = 1,2 we 1 2 1 2 i i have unitaries W ∈ B(H ) which are, respectively, A -fixing as well as unitaries i i i U =[u(i)]∈M (A ) for which i jk n i n ω (v )= W τ (v )W∗u(i) i j i i k i kj k=1 X for all j =1,...,n and i=1,2. 8 PHILIPM.GIPSON It is clear that W := W ⊕ W is a unitary in B(H ⊕ H ) which fixes the 1 2 1 2 C∗-algebra A ⊕ A . Tedious but routine calculations demonstrate that U := 1 2 [(u(1),u(2))]∈M (A ⊕A ) is unitary. All that remains is to show that jk jk n 1 2 n ω(v )= Wτ(v )(U) k j jk j=1 X for all k =1,...,n. Now for each j,k =1,...,n we have Wπ (v )W∗(U) = (W ⊕W )(τ (v )⊕τ (v ))(W∗⊕W∗)(u(1)⊕u(2)) 2 j jk 1 2 1 j 2 j 1 2 jk jk = W τ (v )W∗u(1) ⊕ W τ (v )W∗u(2) 1 1 j 1 jk 2 2 j 2 jk (cid:16) (cid:17) (cid:16) (cid:17) hence n n Wτ(v )W∗(U) = W τ (v )W∗u(1) ⊕ W τ (v )W∗u(1) j ji 1 1 j 1 jk 2 2 j 2 jk Xj=1 Xj=1(cid:16) (cid:17) (cid:16) (cid:17) n n = W τ (v )W∗u(1) ⊕ W τ (v )W∗u(1)  1 1 j 1 jk  2 2 j 2 jk j=1 j=1 X X = ω(v )⊕ω (v )    1 k 2 k = ω(v ) k as desired. (cid:3) We now have the tools to prove our theorem. Proof of Theorem 3.2. Weshallfirstprovenecessity. Ifωandτ areB(H)-quasifree equivalent then n ω(v )= Wτ(v )W∗u i j ji j=1 X for unitaries W ∈B(H) and U =[u ]∈M (B(H)). Thus for we have jk n n n n n ω(v )ω(v )∗ = Wτ(v )W∗u Wτ(v )WW∗u∗ i i  j ji k ki! i=1 i=1 j=1 k=1 X X X X n n n  = Wτ(v )W∗u u∗ Wτ(v )∗W∗ j ji ki k i=1j=1k=1 XXX n n n = Wτ(v )W∗ u u∗ Wτ(v )∗W∗ j ji ki k ! j=1k=1 i=1 XX X n n = Wτ(v )W∗δ Wτ(v )∗W∗ j jk k j=1k=1 XX n = Wτ(v )τ(v )∗W∗ j j j=1 X Thus ω(p ) = Wτ(p )W∗, where we recall that p = I − n v v∗. As a conse- n n n i=1 i i quence(ω(E )H)⊥ =ω(p )H andτ(p )H =(τ(E )H)⊥ sharethesamedimension n n n n P and we have our result. EQUIVALENT REPRESENTATIONS OF TOEPLITZ ALGEBRAS 9 Now suppose that ω and τ are nondegenerate ∗-representations of E on H n with the same multiplicity. By the analogue of the Wold decomposition we have ω = ω ⊕ω represented on H = K ⊕K and τ = τ ⊕τ represented on H = e s e s e s J ⊕J . Ifthemultiplicity isnonzerothenK andJ arenontrivial,separable,and e s e e consequently unitarily equivalent. Let W ∈B(J ,K ) be a unitary, then W τ W∗ e e e e e e is an essentialrepresentationof E on K and so is B(K )-free equivalent to ω by n e e e Lemma 3.3. Hence there is a unitary V =[v ]∈M (B(K )) jk n e n ω (v )= W τ (v )W∗v e i e e j e ji j=1 X for all i=1,...,n. Sinceω andτ havethesamemultiplicity,ω andτ arebothunitarilyequivalent s s to the same multiple of the Fock representation, hence are unitarily equivalent to each other. Let W ∈B(J ,K ) be a unitary for which ω =W τ W∗. s s s s s s s Notefirstthat,asK ⊕K =H =J ⊕J ,wehavethatW :=W ⊕W isaunitary e s e s e s in B(H). Set u := v I ⊕δ I ∈B(K )⊕B(K ) and U =[u ] ∈ M (B(K )⊕ jk jk jk e s jk n e B(K )) ⊂ M (B(H)). Tedious but straightforward calculations demonstrate that s n U is a unitary matrix. All that remains is to see that for each i=1,...,n n n Wτ(v )W∗u = (W ⊕W )(τ (v )⊕τ (v ))(W ⊕W )∗(v ⊕δ I) j ji e s e j s j e s ji ji j=1 j=1 X X n n = W τ (v )W∗v ⊕ W τ (v )W∗δ I  e e j e ji  s s j s ji  j=1 j=1 X X = ω(v )⊕W τ (v )W∗   e i s s i s = ω (v )⊕ω (v ) e i s i = ω(v ) i and so ω is B(H)-quasifree equivalent to τ. (cid:3) A-q.f. Corollary 3.6. If ω ∼ τ for any A⊂B(H) then they share the same multiplicity. Proof. Any unitary U ∈ M (A) is also a unitary in M (B(H)) and any A-fixing n n unitary W ∈ B(H) is clearly B(H)-fixing. Hence A-quasifree equivalence implies B(H)-quasifree equivalence and so Theorem 3.2 applies. (cid:3) Thattheconverseofthecorollaryisfalseisseenintheresultsofthenextsection, where it is shown that quasifree equivalence is also related to the conjugacy of dynamical systems. 4. Application to Dynamics Ourgoalistorelatefreeandquasifreeequivalenceofrepresentationstoconjugacy of certain ∗-endomorphisms of concretely represented C∗-algebras. To do so we will need the property of Invariant Basis Number (IBN) for C∗-algebrasand some related results. Definition 4.1. A unital C∗-algebra A has IBN if An ≃Am if and only if n=m. 10 PHILIPM.GIPSON Our main tool for detecting IBN is K-theoretical and is a result of the author’s Ph.D. dissertation. Theorem 4.2 (Theorem 3.10 in [5]). A has IBN if and only if [1 ] ∈ K (A) has A 0 finite additive order. While nontrivial to check for arbitrary C∗-algebras, it is an immediate consequence of the theorem that a factor has IBN if and only it is finite. Ourmainresultsforthissection,Theorems4.3and4.4,relatetheequivalenceof representations of Toeplitz algebras to the conjugacy of certain ∗-endomorphisms they induce, and vice versa. Theorem 4.3 is technically a special case of Theorem 4.4, but the proof of the later is markedly simpler if we have already established the former. Theorem4.3. LetA⊆B(H)beaC∗-algebra,ωandτ nondegenerate∗-representations of E and E , respectively, (n,m<∞) on H such that n m n α(a):= ω(v )aω(v )∗ i i i=1 X m β(a):= τ(v )aτ(v )∗ j j j=1 X are ∗-endomorphisms of A. If the relative commutant A′ has Invariant Basis Num- ber then the following are equivalent: (1) α=β, (2) n=m and ω and τ are A′-free equivalent. Proof. 1)⇒2). Consider the subspaces E :={X ∈B(H):Xa=α(a)X for all a∈A} α E :={X ∈B(H):Xa=β(a)X for all a∈A} β which are right C∗-modules over A′ when given the inner product hx,yi = x∗y. Of course when α = β we have E = E and so, to avoid confusion, we’ll call α β this A′-module E. Note that {ω(v ) : i = 1,...,n} and {τ(v ) : j = 1,...,m} are i j orthonormal sets in E and denote by E and E the submodules of E spanned ω τ by {ω(v )} and {τ(v )}, respectively. Since E has a basis of size n it is unitarily i j ω equivalent to (A′)n, and similarly E ≃(A′)m. As τ n m ω(v )ω(v )∗ =α(I)=β(I)= τ(v )τ(v )∗ i i j j i=1 j=1 X X we thus have that ω(v ) = m τ(v )τ(v )∗ω(v ) for all i = 1,...,n and τ(v ) = i j=1 j j i j n ω(v )ω(v )∗τ(v )forallj =1,...,m. Henceω(v )∈E (becauseτ(v )∗ω(v )∈ i=1 i i j P i τ j k A′ always) for all i = 1,...,n and τ(v ) ∈ E for all j = 1,...,m, thus E = E . j ω ω τ PBut then (A′)n ≃E =E ≃(A′)m and so,because A′ has IBN, we conclude that ω τ n=m. Since E =E and, by definition, E =σ ((A′)n) there are vectors f ,...,f ∈ τ ω ω ω 1 n (A′)n such that σ (f ) = τ(v ) for each i = 1,...,n. As {τ(v )} is an orthonormal ω i i i basisforE , andσ is Toeplitz byconstruction,weconclude that{f ,...,f }is an ω ω 1 n orthonormalbasisfor(A′)n,hence thereisaunitaryU =[u ]∈M (A′)forwhich jk n