The isomorphism problem for tensor algebras of multivariable dynamical systems
aa r X i v : . [ m a t h . OA ] J a n THE ISOMORPHISM PROBLEM FOR TENSOR ALGEBRAS OFMULTIVARIABLE DYNAMICAL SYSTEMS
ELIAS G. KATSOULIS AND CHRISTOPHER RAMSEY
Abstract.
We resolve the isomorphism problem for tensor algebras of unital multivariabledynamical systems. Specifically we show that unitary equivalence after a conjugation for multi-variable dynamical systems is a complete invariant for complete isometric isomorphisms betweentheir tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis relatingto work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis. Introduction
Semicrossed products and their variants have appeared in the theory of operator algebrassince the beginning of the subject [1] and continue to be at the forefront of the theory as theylend insight for considerable abstraction [2, 3, 4, 8, 12, 18, 19] .A C ∗ -dynamical system ( A , α ) consists of a unital C ∗ -algebra A and a unital ∗ -endomorphism α : A → A . An isometric covariant representation ( π, V ) of ( A , α ) consists of a non-degenerate ∗ -representation π of A on a Hilbert space H and an isometry V ∈ B ( H ) so that π ( a ) V = V π ( α ( a )), for all a ∈ A . The semicrossed product A ⋊ α Z + is the universal operator algebraassociated with “all” covariant representations of ( A , α ), i.e., the universal algebra generatedby a copy of A and an isometry v satisfying the covariance relations. In the case where α isan automorphism of A , then A ⋊ α Z + is isomorphic to the subalgebra of the crossed productC ∗ -algebra A ⋊ α Z generated by A and the “universal” unitary u implementing the covariancerelations.One of the central problems in the study of semicrossed products is the classification problem,whose study spans more than 50 years. This problem asks if two semicrossed products are iso-morphic as algebras exactly when the corresponding C ∗ -dynamical systems are outer conjugate,that is, unitarily equivalent after a conjugation. The classification problem first appeared in thework of Arveson [1] and it was subsequently investigated by Peters [16] , Hadwin and Hoover [7] ,Power [17] and Davidson and Katsoulis [4] , who finally settled the case where A is abelian. Inthe general case where A may not be abelian, initial consideration for the isomorphism problemwas given in [5, 14] and considerable progress was made by Davidson and Kakariadis [3] whoresolved the problem for isometric isomorphisms and dynamical systems consisting of injectiveendomorphisms. Actually the work of Davidson and Kakariadis went well beyond systems con-sisting of injective endomorphisms. In [3 , Theorem 1.1 ] these authors also worked the case ofepimorphic systems and in [3 , Theorem 1.2 ] they offered 6 additional conditions, with each oneof them guaranteeing a positive resolution for the isomorphism problem. All these partial re-sults offered enough evidence for Davidson and Kakariadis to conjecture that the isomorphismproblem for isometric isomorphisms must have a positive solution for arbitrary systems. Thisconjecture is now being verified here in Corollary 2.7, thus resolving the isomorphism problem forunital dynamical systems and their semicrossed products at the level of isometric isomorphisms.Our initial approach is different from that of [3] and actually allows us to achieve more.A multivariable C ∗ -dynamical system is a pair ( A , α ) consisting of a unital C ∗ -algebra A alongwith unital ∗ -endomorphisms α = ( α , . . . , α n ) of A into itself. A row isometric representationof ( A , α ) consists of a non-degenerate ∗ -representation π of A on a Hilbert space H and a row Mathematics Subject Classification.
Key words and phrases: semicrossed product, tensor algebra, dynamical system, operator algebra. See just below Theorem 1.1 in [3] . isometry V = ( V , V , . . . , V n ) acting on H ( n ) so that π ( a ) V i = V i π ( α ( a )), for all a ∈ A and i = 1 , , . . . , n . The tensor algebra T + ( A , α ) is the universal algebra generated by a copy of A and a row isometry v satisfying the covariance relations. The tensor algebras form a tractablemultivariable generalization of the semicrossed products. One should be careful to note thatthere are semicrossed products of multivariable C ∗ -dynamical systems as well, but these aredifferent from the tensor algebras being discussed here. (See [6, 9, 18] for more information.)In the case where A is abelian, tensor algebras of multivariable systems were first studied indetail by Davidson and Katsoulis [6] . These authors developed a satisfactory dilation theory andprovided invariants of a topological nature for algebraic isomorphisms, which in certain casesturned out to be complete. Specifically, in [6] it was established that if the tensor algebras arealgebraically isomorphic then the two dynamical systems must be piecewise conjugate. However,the converse could only be established for tensor algebras with n = 2 or 3 with a gap in thetopological theory preventing a complete result.The tensor algebras of perhaps non-abelian multivariable systems were studied in [9, 10] . In [9 , Theorem 4.5(ii) ] it was established that unitary equivalence after a conjugation for auto-morphic multivariable systems is a complete invariant for isometric isomorphisms between theirtensor algebras. Under various assumptions Kakariadis and Katsoulis were able to show that theclassification scheme [9 , Theorem 4.5(ii) ] could be extended to more general dynamical systemsbut a complete solution was not obtained in [9] . (This was pointed out as an open problem in [11] just after Theorem 2.6.15). All these assumptions are now being removed in this paper andin Theorem 2.6 we obtain a complete classification of all tensor algebras of (unital) multivariablesystems up to completely isometric isomorphism.The semicrossed products of single variable systems and more generally the tensor algebrasof multivariable systems are prototypical examples of tensor algebras of C ∗ -correspondences.These algebras were pioneered by Muhly and Solel [13] and generalize many concrete classesof operator algebras, including graph algebras and more. The isomorphism problem generalizesin this setting and asks if two tensor algebras of C ∗ -correspondences are isomorphic exactlywhen the C ∗ -correspondences are unitarily equivalent. Muhly and Solel studied this problem in [14] and resolved it affirmatively for aperiodic C ∗ -correspondences and complete isomorphisms.However many natural examples of C ∗ -correspondences, including those associated with tensoralgebras of multivariable systems or graph algebras, may fail to be aperiodic. As it turns out,the unitary equivalence of the correspondences associated with tensor algebras of multivariablesystems coincides with unitary equivalence after a conjugation for the multivariable systemsthemselves. Therefore our Theorem 2.6 resolves Muhly and Solel’s classification problem for animportant class of C ∗ -correspondences and lends support for an overall affirmative answer ofthis problem. We plan to pursue this in a subsequent work.Finally a word about our notation. If ρ : X → Y is any map between linear spaces, then its( m, n )-th ampliation is the matricial map ρ ( m,n ) : M m,n ( X ) −→ M m,n ( Y ); [ x ij ] mi =1 nj =1 [ ρ ( x i,j )] mi =1 nj =1 . In what follows, in order to avoid the use of heavy notation we will be dropping the superscript( m, n ) from ρ ( m,n ) and the symbol ρ will be used not only for the map ρ itself but for all ofits ampliations as well. It goes without saying that the order of an ampliation will be easilyunderstood form the context. 2. The main result
Suppose A is a unital C ∗ -algebra and α i : A → A , ≤ i ≤ n , are unital ∗ -endomorphisms.Recall that the tensor algebra T + ( A , α ) of the C ∗ -dynamical system ( A , α ) is generated by v , . . . , v n , which form a row isometry v = [ v v . . . v n ] and a faithful copy of A . The rowisometry v encodes the dynamics of the dynamical system ( A , α ) in the sense that av i = v i α i ( a ),for all a ∈ A and 1 ≤ i ≤ n . By definition, the tensor algebra T + ( A , α ) is universal over allrepresentations encoding the dynamics of ( A , α ) so that the generators are mapped to a rowisometry. SOMORPHISM OF TENSOR ALGEBRAS 3
Due to its universality, the tensor algebra T + ( A , α ) admits a gauge action ζ : T −→ Aut( T + ( A , α )); T ∋ λ ζ λ so that ζ λ ( a ) = a , for all a ∈ A , and ζ λ ( v p ) = λ | p | v p , where p ∈ F + n and | p | denotes the length of p . (We write p = 0 is the empty word, with the understanding that | | = 0 and v := I .) Fromthis gauge action we deduce that every element x ∈ T + ( A , α ) admits a formal Fourier seriesdevelopment x ∼ P ∞ k =0 E k ( x ). Each E k : T + ( A , α ) → T + ( A , α ) is a completely contractive, A -module projection on the subspace of T + ( A , α ) generated by elements of the form v p a p , with p ∈ F + n , | p | = k and a p ∈ A . Furthermore E is a multiplicative expectation onto A ⊆ T + ( A , α ).Finally the formal series x ∼ P ∞ k =0 E k ( x ) is Cesaro-convergent to x , i.e., x = lim n →∞ n X k =0 (cid:18) − kn + 1 (cid:19) E k ( x ) . See [6] for a comprehensive development of this theory.The following is a key result in our investigation.
Theorem 2.1. If b = [ b . . . b n ] is a strict row contraction in A such that ab i = b i α i ( a ) forall a ∈ A and ≤ i ≤ n then there is a completely isometric automorphism ρ of T + ( A , α ) suchthat ρ ( a ) = a for a ∈ A , ρ ( v ) = ( I − bb ∗ ) / ( I − vb ∗ ) − ( b − v )( I n − b ∗ b ) − / and ρ ◦ ρ = id . Furthermore, E ( ρ ( v i )) = b i , ≤ i ≤ n . Proof.
By hypothesis b is a strict contraction and so k vb ∗ k < I − vb ∗ and D b ∗ := ( I − bb ∗ ) / are indeed invertible in A . Similarly, D b := ( I n − b ∗ b ) / is invertible in M n ( A ). One calculates( I − bv ∗ ) − ( I − bb ∗ )( I − vb ∗ ) − = ( I − bv ∗ ) − ( I − bv ∗ vb ∗ )( I − vb ∗ ) − = ( I − bv ∗ ) − ( I − vb ∗ + ( I − bv ∗ ) vb ∗ )( I − vb ∗ ) − = ( I − bv ∗ ) − + vb ∗ ( I − vb ∗ ) − = ( I − bv ∗ ) − + ( I − vb ∗ ) − − I which gives for w ( v ) := ( I − bb ∗ ) / ( I − vb ∗ ) − ( b − v )( I − b ∗ b ) − / that w ( v ) ∗ w ( v ) = D − b ( b ∗ − v ∗ )( I − bv ∗ ) − ( I − bb ∗ )( I − vb ∗ ) − ( b − v ) D − b = D − b v ∗ ( vb ∗ − I ) (cid:16) ( I − bv ∗ ) − + ( I − vb ∗ ) − − I (cid:17) ( bv ∗ − I ) vD − b = D − b v ∗ (cid:16) ( I − vb ∗ ) + ( I − bv ∗ ) − ( I − vb ∗ )( I − bv ∗ ) (cid:17) vD − b = D − b v ∗ ( I − vb ∗ bv ∗ ) vD − b = D − b ( I n − b ∗ b ) D − b = I n . Hence, w ( v ) is a row isometry. Now by the conjugation relation in the hypothesis we have bb ∗ , bv ∗ and vb ∗ commute with A and b ∗ b commutes with diag( α ( a ) , . . . , α n ( a )) , a ∈ A . Using thisone gets for a ∈ A that aw = a ( I − b ∗ b ) − ( I − vb ∗ ) − ( b − v )( I n − b ∗ b ) − / = ( I − bb ∗ ) / ( I − vb ∗ ) − a ( b − v )( I n − b ∗ b ) − / = ( I − bb ∗ ) / ( I − vb ∗ ) − ( b − v ) diag( α ( a ) , . . . , α n ( a ))( I − b ∗ b ) − / = ( I − bb ∗ ) / ( I − vb ∗ ) − ( b − v )( I − b ∗ b ) − / diag( α ( a ) , . . . , α n ( a ))= w diag( α ( a ) , . . . , α n ( a )) . E.G. KATSOULIS AND C. RAMSEY
Thus, by the universal property there exists a unique completely contractive homomorphism ρ of T + ( A , α ) to itself such that ρ ( a ) = a, ∀ a ∈ A and [ ρ ( v ) . . . ρ ( v n )] = w ( v ) . Lastly, recall the classical Halmos functional calculus trick: since b ( I n − b ∗ b ) = ( I − bb ∗ ) b then bD − b = b ( I n − b ∗ b ) − / = ( I − bb ∗ ) − / b = D − b ∗ b. This allows us to compute[ ρ ◦ ρ ( v ) , . . . , ρ ◦ ρ ( v n )] = w ( w ( v ))= D b ∗ (cid:16) I − D b ∗ ( I − vb ∗ ) − ( b − v ) D − b b ∗ (cid:17) − (cid:16) b − D b ∗ ( I − vb ∗ ) − ( b − v ) D − b (cid:17) D − b = D b ∗ (cid:16) I − D b ∗ ( I − vb ∗ ) − ( b − v ) b ∗ D − b ∗ (cid:17) − (cid:16) b − D b ∗ ( I − vb ∗ ) − ( b − v ) D − b (cid:17) D − b = D b ∗ (cid:16) I − ( I − vb ∗ ) − ( b − v ) b ∗ (cid:17) − D − b ∗ (cid:16) b − D b ∗ ( I − vb ∗ ) − ( b − v ) D − b (cid:17) D − b = D b ∗ (cid:16) I − ( I − vb ∗ ) − ( b − v ) b ∗ (cid:17) − (cid:16) D − b ∗ b − ( I − vb ∗ ) − ( b − v ) D − b (cid:17) D − b = D b ∗ (cid:16) I − ( I − vb ∗ ) − ( b − v ) b ∗ (cid:17) − (cid:16) b − ( I − vb ∗ ) − ( b − v ) (cid:17) D − b = D b ∗ (cid:16) ( I − vb ∗ ) − (( b − v ) b ∗ (cid:17) − ( I − vb ∗ ) (cid:16) b − ( I − vb ∗ ) − ( b − v ) (cid:17) D − b = D b ∗ ( I − bb ∗ ) (cid:16) ( I − vb ∗ ) b − ( b − v ) (cid:17) D − b = ( v − vb ∗ b ) D − b = v. Therefore, by the universal property ρ ◦ ρ = id and ρ is a completely isometric automorphismof T + ( A , α ).To find the first Fourier coefficients of the ϕ ( v i ) one needs to recall that E : T + ( A , α ) → A is a completely contractive homomorphism given by sending A to itself and sending v i to 0,1 ≤ i ≤ n . Thus, E ([ ϕ ( v ) , . . . , ϕ ( v n ]) = E ( w ( v ))= E (cid:0) ( I − bb ∗ ) / ( I − vb ∗ ) − ( b − v )( I n − b ∗ b ) − / (cid:1) = ( I − bb ∗ ) / (( I − b ∗ )) − ( b − I n − b ∗ b ) − / = ( I − bb ∗ ) / b ( I n − b ∗ b ) − / = b with the last equality arising from a familiar functional calculus argument. Proposition 2.2.
Let ψ : T + ( A , α ) → T + ( A , β ) be a completely isometric isomorphism andassume that ψ | A = id . If v = [ v , v , . . . , v n a ] is the generating row isometry for T + ( A , α ) , then k E ( ψ ( v )) k < . Proof.
Let w = [ w , w , . . . , w n b ] be the generating row isometry for T + ( A , β ). Concretelyembed both tensor algebras into B ( H ). Note that we are not assuming n a = n b SOMORPHISM OF TENSOR ALGEBRAS 5
By the Fourier analysis discussed earlier ψ ( v ) = lim n →∞ n X k =0 (cid:18) − kn + 1 (cid:19) E k ( ψ ( v ))= lim n →∞ n X k =0 X | p | = k (cid:18) − kn + 1 (cid:19) w p b p (2.1)with each b p = [ b p, , b p, , . . . , b p,n a ] being a row contraction.Assume by contradiction that k b k = k E ( ψ ( v )) k = 1. Then there exists a sequence ofunit vectors ξ j ∈ H ( n b ) such that lim j →∞ k b ξ j k = 1. Using these contractive Cesaro sums inconjunction with the orbit representation applied to the vectors ~ξ j = ( ξ j , , , . . . ) we get1 ≥ lim j →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) π n X k =0 X | p | = k (cid:18) − kn + 1 (cid:19) w p b p ~ξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim j →∞ n X k =0 X | p | = k (cid:18) − kn + 1 (cid:19) k b p ξ j k ≥ lim j →∞ k b ξ j k = 1 . This implies that lim j →∞ k b p ξ j k = 0, for all non-empty words p ∈ F + n a .Pick an n ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =0 (cid:18) − kn + 1 (cid:19) E k ( ψ ( v )) − ψ ( v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ . Then for every j ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ◦ ψ − n X k =0 (cid:18) − kn + 1 (cid:19) E k ( ψ ( v )) ! ξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ◦ ψ − n X k =0 (cid:18) − kn + 1 (cid:19) E k ( ψ ( v )) − ψ ( v ) ! ξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + k E ◦ ψ − ◦ ψ ( v ) ξ j k≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =0 (cid:18) − kn + 1 (cid:19) E k ( ψ ( v )) − ψ ( v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k ξ j k + k E ( v ) ξ j k≤
12 + 0 = 12 . Using the fact that lim j →∞ b p ξ j = 0, for all non-empty words p ∈ F + n a , we obtain thatlim j →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E ◦ ψ − n X k =0 (cid:18) − kn + 1 (cid:19) E k ( ψ ( v )) ! ξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim j →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) E n X k =0 X | p | = k (cid:18) − kn + 1 (cid:19) ψ − ( w p ) b p ξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim j →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =0 X | p | = k (cid:18) − kn + 1 (cid:19) E (cid:0) ψ − ( w p ) (cid:1) b p ξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim j →∞ (cid:13)(cid:13) E (cid:0) ψ − ( w ) (cid:1) b ξ j (cid:13)(cid:13) = 1 , E.G. KATSOULIS AND C. RAMSEY which is a contradiction. Therefore, b = E ( ψ ( v )) must be a strict contraction.Continuing with the assumptions and notation of the previous proposition and its proof, thematrix [ b ij ] ∈ M n b ,n a ( A ), with entries b ij as appearing in (2.1), is called the matrix associatedwith ψ . It has the property of intertwining the endomorphisms α , α , . . . , α n a and β , β , . . . , β n b in the sense,(2.2) β i ( a ) b ij = b ij α j ( a )for all a ∈ A , 1 ≤ i ≤ n b and 1 ≤ j ≤ n a . A matrix [ a ij ] ∈ M n a ,n b ( A ) with similar properties isassociated with ψ − . Let us observe this more closely.Indeed, assume that ψ ( v ) admits a Fourier development ψ ( v ) = lim n →∞ n X k =0 X | p | = k (cid:18) − kn + 1 (cid:19) w p b p = b + w [ b ij ] + lim n →∞ n X k =2 X | p | = k (cid:18) − kn + 1 (cid:19) w p b p (2.3)as in (2.1). For any a ∈ A and 1 ≤ j ≤ n a , we have aψ ( v j ) = ψ ( av j ) = ψ ( v j α j ( a )) = ψ ( v j ) α j ( a )and so a (cid:16) b j + n b X i =1 w i b ij + lim n →∞ n X k =2 X | p | = k (cid:18) − kn + 1 (cid:19) w p b pj (cid:17) == (cid:16) b j + n b X i =1 w i b ij + n X k =2 X | p | = k (cid:18) − kn + 1 (cid:19) w p b pj (cid:17) α j ( a )(2.4)Apply E to (2.4) to obtain(2.5) ab j = b j α j ( a ) , for all a ∈ A and 1 ≤ j ≤ n a . The relations (2.2) are obtained by applying E to (2.4). Lemma 2.3.
Let ψ : T + ( A , α ) → T + ( A , β ) be a completely isometric isomorphism and assumethat ψ | A = id . Let v and w be the generating row isometries for T + ( A , α ) and T + ( A , β ) respectively. Then E ( ψ ( v )) = 0 if and only if E ( ψ − ( w )) = 0 . In such a case, the matricesassociated with ψ and ψ − are inverses of each other. Proof.
Assume that ψ ( v ) admits a Fourier development as in (2.3) and similarly ψ − ( w ) = a + v [ a ij ] + lim n →∞ n X k =2 X | q | = k (cid:18) − kn + 1 (cid:19) v q a q Assume that E ( ψ − ( w )) = a = 0. Claim. E ( ψ − ( w p )) = E ( ψ − ( w p )) = 0 for all p ∈ F + n b with | p | ≥ . Indeed, if p = p p . . . p l , l ≥
2, then ψ − ( w p ) = l Y i =1 ψ − ( w p i )= lim n →∞ l Y i =1 n X k =1 X | q | = k (cid:18) − kn + 1 (cid:19) v q a q,p i . (2.6)Since in the limit above we have l ≥ k ≥
1, a development of the product involved willreveal only terms of the form v u a u , with u ∈ F + n a , | u | ≥ a u ∈ A . Since both E and E are continuous, this suffices to prove the claim. SOMORPHISM OF TENSOR ALGEBRAS 7
For the proof consider i = 1 ,
2. Apply ψ − to (2.3) and use the Claim to obtain E i ( v ) = E i ( ψ − ( ψ ( v )))= E i (( b ) + E i ( ψ − ( w ))[ b ij ] + lim n →∞ n X k =2 X | p | = k (cid:18) − kn + 1 (cid:19) E i ( ψ − ( w p )) b p = E i ( b ) + E i ( ψ − ( w ))[ b ij ]= E i ( b ) + E i (cid:16) v [ a ij ] + lim n →∞ n X k =2 X | q | = k (cid:18) − kn + 1 (cid:19) v q a q (cid:17) [ b ij ]= E i ( b ) + E i ( v )[ a ij ][ b ij ] , For i = 0 we obtain 0 = b = E ( ψ ( v )). For i = 1 we obtain v = v [ a ij ][ b ij ] and so I n a = v ∗ v = v ∗ v [ a ij ][ b ij ] = [ a ij ][ b ij ] . By reversing the roles of ψ and ψ − and using what has been proven so far, we obtain [ b ij ] . [ a ij ] = I n b . This completes the proof. Corollary 2.4.
Let ψ : T + ( A , α ) → T + ( A , β ) be a completely isometric isomorphism andassume that ψ | A = id . Let v be the generating row isometry for T + ( A , α ) and assume that E ( ψ ( v )) = 0 . Then there exists a unitary matrix u ∈ M n b ,n a ( A ) intertwining the endomor-phisms α , α , . . . , α n a and β , β , . . . , β n b . Proof.
According to (2.2), the matrix b := [ b ij ] ∈ M n b ,n a ( A ) associated with ψ intertwines theendomorphisms α , α , . . . , α n a and β , β , . . . , β n b . Hence(2.7) diag( β ( a ) , . . . , β n b ( a )) b = b diag( α ( a ) , . . . , α n a ( a )) , for all a ∈ A . Hence b ∗ b (and therefore | b | ) commutes with diag( α ( a ) , . . . , α n a ( a )), for all a ∈ A . By Lemma 2.3, b is invertible and so it admits a polar decomposition b = u | b | , with u = M n b ,n a ( A ) a unitary matrix. Furthermore, (2.7) implies thatdiag( β ( a ) , . . . , β n b ( a )) u | b | = u | b | diag( α ( a ) , . . . , α n a ( a ))= u diag( α ( a ) , . . . , α n a ( a )) | b | , for all a ∈ A . Since | b | is invertible, diag( β ( a ) , . . . , β n b ( a )) u = u diag( α ( a ) , . . . , α n a ( a )) andthe conclusion follows.Motivated by the statement of the previous result, we introduce the following Definition 2.5.
Two multivariable dynamical systems ( A , α ) and ( A , β ) are said to be unitarilyequivalent if there exists a unitary matrix with entries in A intertwining the two systems. Twomultivariable dynamical systems ( A , α ) and ( B , β ) are said to be are unitarily equivalent after aconjugation if there exists a ∗ -isomorphism γ : A → B so that the systems ( A , α ) and ( A , γ − ◦ β ◦ γ ) are unitarily equivalent.Note that the above definition does not require that the multivariable systems ( A , α ) and( A , β ) should have the same number of maps, i.e., n α = n β , where α = ( α , α , . . . , α n α ) and β = ( β , β , . . . , β n β ). On the contrary, it is possible for two dynamical systems with a differentnumber of maps to be unitarily equivalent; see [9 , Example 5.1 ] . Nevertheless in that case atleast one of the systems will fail to be automorphic, as [9 , Theorem 4.4 ] clearly indicates.Recall that two (single variable) dynamical systems are said to be outer conjugate if there isa ∗ -isomorphism γ : A → B and a unitary u ∈ A such that α ( c ) = u ( γ − ◦ β ◦ γ ( c )) u ∗ Therefore in that case the concept of unitary equivalence after a conjugation coincides with thatof outer conjugacy. Davidson and Kakariadis [3] established that outer conjugacy of the systemsimplies that the associated tensor algebras are completely isometrically isomorphic. They showedthat the converse is true in several broad cases (injective, surjective, etc.) and specifically when
E.G. KATSOULIS AND C. RAMSEY k E ( ψ ( v )) k < √ − ≈ . [3 , Remark 3.6 ] . For multivariable dynamical systems consistingof automorphisms Kakariadis and Katsoulis have shown [9 , Theorem 4.5 ] that the isomorphismof the tensor algebras is equivalent to unitary equivalence after a conjugation for the associateddynamical systems. A similar result was shown for arbitrary dynamical systems provided thatthe pertinent C ∗ -algebras are stably finite [9 , Theorem 5.2 ] . Our next result removes all theseconditions and establishes a complete result for the unital case. Theorem 2.6. If ψ : T + ( A , α ) → T + ( B , β ) is a completely isometric isomorphism then ( A , α ) and ( B , β ) are unitarily equivalent after a conjugation. Proof.
Without loss of generality we may assume that A = B and ψ | A = id. Indeed, it is well-known that the restriction of the isomorphism ψ on A ⊆ T + ( A , α ) induces a ∗ -isomorphism γ : A → B . The dynamical systems ( B , β ) and ( A , γ − ◦ β ◦ γ ) are conjugate via γ and sothere exists a completely isometric isomorphism ϕ : T + ( B , β ) → T + ( A , γ − ◦ β ◦ γ ) so that ϕ | B = γ − . Hence ϕ ◦ ψ establishes a completely isometric isomorphism between T + ( A , α ) and T + ( A , γ − ◦ β ◦ γ ), whose restriction on A is the identity map. Therefore if γ is not the identitymap to begin with, then replace ψ with ϕ ◦ ψ and establish the conclusion for the dynamicalsystems ( A , α ) and ( A , γ − ◦ β ◦ γ ).For the proof, Proposition 2.2 gives that b = E ( ψ ( v )) is a strict row contraction. Com-bined with (2.5), this implies that b satisfies the conditions of Theorem 2.1 and so there existsa completely isometric automorphism ρ of T + ( A , α ) such that E ( ρ ( v )) = b . Since E ismultiplicative, we obtain that E ( ψ ◦ ρ ( v )) = E ◦ ψ (cid:0) D b ∗ ( I − vb ∗ ) − ( b − v ) D b (cid:1) = D b ∗ E (( I − ψ ( vb ∗ )) − ( b − E ( ψ ( v ))) D b = D b ∗ E (( I − ψ ( vb ∗ )) − ( b − b ) D b = 0 . Therefore, ψ ◦ ρ satisfies the requirements of Corollary 2.4 and the conclusion follows.Of course, the converse of Theorem 2.6 is also true. If two multivariable systems ( A , α ) and( B , β ) are unitarily equivalent after a conjugation, then the associated C ∗ -correspondences areunitarily equivalent and so the tensor algebras T + ( A , α ) and T + ( B , β ) are completely isometri-cally isomorphic. (See [10 , Theorem 4.5 ] and the discussion preceding it.) Hence Theorem 2.6provides a complete classification of tensor algebras up to complete isomorphism. For semi-crossed products we can say something more. Corollary 2.7.
Let ( A , α ) and ( B , β ) be two unital C ∗ -dynamical systems. Then A ⋊ α Z + and B ⋊ β Z + are isometrically isomorphic if and only ( A , α ) and ( B , β ) are outer conjugate. Proof.
The result follows from Theorem 2.6 and the fact that contractive representations ofsemicrossed products associated with unital endomorphisms are always completely contractive [13 , Corollary 3.14 ] . Acknowledgement.
The second author was supported by NSERC Discovery Grant 2019-05430.
Conflict of interest statement.
On behalf of all authors, the corresponding author states thatthere is no conflict of interest.
Data availability statement.
Data sharing not applicable to this article as no datasets weregenerated or analyzed during the current study.
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