Quantum Cuntz-Krieger algebras
aa r X i v : . [ m a t h . OA ] S e p QUANTUM CUNTZ-KRIEGER ALGEBRAS
MICHAEL BRANNAN, KARI EIFLER, CHRISTIAN VOIGT, AND MORITZ WEBER
Abstract.
Motivated by the theory of Cuntz-Krieger algebras we define andstudy C ∗ -algebras associated to directed quantum graphs. For classical graphsthe C ∗ -algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear.We study two particular classes of quantum graphs in detail, namely thetrivial and the complete quantum graphs. For the trivial quantum graph on asingle matrix block, we show that the associated quantum Cuntz-Krieger al-gebra is neither unital, nuclear nor simple, and does not depend on the size ofthe matrix block up to KK -equivalence. In the case of the complete quantumgraphs we use quantum symmetries to show that, in certain cases, the corre-sponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras.These isomorphisms, which seem far from obvious from the definitions, implyin particular that these C ∗ -algebras are all pairwise non-isomorphic for com-plete quantum graphs of different dimensions, even on the level of KK -theory.We explain how the notion of unitary error basis from quantum informationtheory can help to elucidate the situation.We also discuss quantum symmetries of quantum Cuntz-Krieger algebrasin general. Introduction
Cuntz-Krieger algebras were introduced in [12], generalizing the Cuntz algebrasin [9]. These algebras have intimate connections with symbolic dynamics, andhave been studied intensively in the framework of graph C ∗ -algebras over the pastdecades, thus providing a rich supply of interesting examples [29]. The structure ofgraph C ∗ -algebras is understood to an impressive level of detail, and many proper-ties can be interpreted geometrically in terms of the underlying graphs. Motivatedby this success, the original constructions and results have been generalized inseveral directions, including higher rank graphs [21], Exel-Laca algebras [17] andultragraph algebras [32], among others.The aim of the present paper is to study a generalization of Cuntz-Krieger al-gebras of a quite different flavor, based on the concept of a quantum graph. Thelatter notion goes back to work of Erdos-Katavolos-Shulman [16] and Weaver [34],and was subsequently developed further by Duan-Severini-Winter [14] and Musto-Reutter-Verdon [27]. Quantum graphs play an intriguing role in the study of thegraph isomorphism game in quantum information via their connections with quan-tum symmetries of graphs [7]. Moreover, based on the use of quantum symmetries,fascinating results on the graph theoretic interpretation of quantum isomorphismsbetween finite graphs were recently obtained by Manˇcinska-Roberson [24].Our main idea is to replace the matrix A in the definition of the Cuntz-Krieger al-gebra O A by the quantum adjacency matrix of a directed quantum graph. Roughlyspeaking, this means that the standard generators in a Cuntz-Krieger algebra arereplaced by matrix-valued valued partial isometries, with matrix sizes determinedby the quantum graph, and the Cuntz-Krieger relations are expressed using thequantum adjacency matrix of the quantum graph, in analogy to the scalar case. Mathematics Subject Classification.
An important difference to the classical situation is that the matrix partial isome-tries are not required to have mutually orthogonal ranges, as this condition doesnot generalize to matrices in a natural way. Therefore, the quantum Cuntz-Kriegeralgebra of a classical graph is typically not isomorphic to an ordinary Cuntz-Kriegeralgebra, and will often neither be unital nor nuclear. However, we show that freeCuntz-Krieger algebras, or equivalently, quantum Cuntz-Krieger algebras associ-ated with classical graphs, are always KK -equivalent to Cuntz-Krieger algebras.Our main results concern the quantum Cuntz-Krieger algebras associated withtwo basic examples of quantum graphs, namely the trivial and complete quantumgraphs associated to an arbitrary measured finite-dimensional C ∗ -algebra ( B, ψ ).The first example we consider in detail, namely the quantum Cuntz-Krieger algebraof the trivial quantum graph
T M N on the full matrix algebra M N ( C ), can be viewedas a non-unital version of Brown’s universal algebra generated by the entries of aunitary N × N -matrix [8]. For N >
1, the quantum Cuntz-Krieger algebra of
T M N is neither unital, nuclear nor simple, but it is always KK -equivalent to C ( S ).We exhibit a description of matrices over this algebra in terms of a free product.The second example, namely the quantum Cuntz-Krieger algebra associated to thecomplete quantum graph K ( B, ψ ), is even more intriguing. We show that this C ∗ -algebra always admits a canonical quotient map onto the Cuntz algebra O n ,where n = dim( B ). Moreover, for certain quantum complete graphs we are ableto show that this map is an isomorphism. This fact, which seems far from obviousfrom the defining relations, is proved using monoidal equivalence of the quantumautomorphism groups of the underlying quantum graphs. In particular, our resultsshow that for N > K ( M N ( C ) , tr) are unital, nuclear, simple, and pairwise non-isomorphic, evenon the level of KK -theory.We also discuss how quantum symmetries of directed quantum graphs inducequantum symmetries of their associated quantum Cuntz-Krieger algebras in general.This is particularly interesting when one tries to relate quantum Cuntz-Kriegeralgebras associated to graphs which are quantum isomorphic, as in our analysisof the examples mentioned above. In particular, we indicate how the notion of aunitary error basis [35], which is well-known in quantum information theory, canbe used to find finite-dimensional quantum isomorphisms, which in turn inducecrossed product relations between quantum Cuntz-Krieger algebras. In a sense, theexistence of quantum symmetries can be viewed as a substitute for the gauge actionwhich features prominently in the study of ordinary Cuntz-Krieger algebras. Whilethere exists a gauge action in the quantum case as well, it seems to be of limiteduse for understanding the structure of quantum Cuntz-Krieger algebras in general.Let us briefly explain how the paper is organized. In section 2 we collect somebackground material on graphs and their associated C ∗ -algebras, and introducefree graph C ∗ algebras and free Cuntz-Krieger algebras. We show that these alge-bras are KK -equivalent to ordinary graph C ∗ -algebras and Cuntz-Krieger algebras,respectively. After reviewing some facts about finite quantum spaces, that is, mea-sured finite-dimensional C ∗ -algebras, we define directed quantum graphs in section3. We then introduce our main object of study, namely quantum Cuntz-Kriegeralgebras. In section 4 we discuss some examples of quantum graphs and their asso-ciated C ∗ -algebras. We show that the quantum Cuntz-Krieger algebras associatedwith classical graphs lead precisely to free Cuntz-Krieger algebras, and look at sev-eral concrete examples of quantum graphs. We also discuss two natural operationson directed quantum graphs, obtained by taking direct sums and tensor productsof their underlying C ∗ -algebras, respectively. Section 5 is concerned with a generalprocedure to assign quantum graphs to classical graphs, essentially by replacing all UANTUM CUNTZ-KRIEGER ALGEBRAS 3 vertices with matrix blocks of a fixed size. We analyze the structure of the resultingquantum Cuntz-Krieger algebras, and show that they are always KK -equivalent totheir classical counterparts. This allows one in particular to determine the K -theory of the quantum Cuntz-Krieger algebra of the trivial quantum graph on amatrix algebra mentioned above. In section 6 we explain how quantum symme-tries of quantum graphs induce actions on quantum Cuntz-Krieger algebras. Wealso discuss the canonical gauge action, in analogy to the classical situation. Theconstruction of quantum symmetries works in fact at the level of linking algebrasassociated with arbitrary quantum isomorphisms of quantum graphs. This is usedtogether with the some unitary error basis constructions in section 7 to study thestructure of the quantum Cuntz-Krieger algebras of the trivial and complete quan-tum graphs associated to a full matrix algebra equipped with its standard trace. Inthe final section 8 we gather the required results from the preceding sections to fur-nish a proof of our main theorem for quantum Cuntz-Krieger algebras of completequantum graphs.Let us conclude with some remarks on notation. The closed linear span of asubset X of a Banach space is denoted by [ X ]. If F is a finite set and A a C ∗ -algebra we shall write M F ( A ) for the C ∗ -algebra of matrices indexed by elementsfrom F with entries in A . We write ⊗ both for algebraic tensor products and for theminimal tensor product of C ∗ -algebras. For operators on multiple tensor productswe use the leg numbering notation. Acknowledgements.
The authors are indebted to Li Gao for fruitful discussionson unitary error bases and their connections with quantum isomorphisms. MBand KE were partially supported by NSF Grant DMS-2000331. CV and MW werepartially supported by SFB-TRR 195 “Symbolic Tools in Mathematics and theirApplication” at Saarland University. Parts of this project were completed while theauthors participated in the March 2019 Thematic Program “New Developmentsin Free Probability and its Applications” at CRM (Montreal) and the October2019 Mini-Workshop “Operator Algebraic Quantum Groups” at MathematischesForschungsinstitut Oberwolfach. The authors gratefully acknowledge the supportand productive research environments provided by these institutes.2.
Cuntz-Krieger algebras
In this section we review the definition of Cuntz-Krieger algebras and graph C ∗ -algebras [12], [15], [22], [29], and introduce a free variant of these algebras. Ourconventions for graphs and graph C ∗ -algebras will follow [22].2.1. Graphs.
A directed graph E = ( E , E ) consists of a set E of vertices anda set E of edges, together with source and range maps s, r : E → E . A vertex v ∈ E is called a sink iff s − ( v ) is empty, and a source iff r − ( v ) is empty. Thatis, a sink is a vertex which emits no edges, and a source is a vertex which receivesno edges. A self-loop is an edge with s ( e ) = r ( e ). The graph E is called simple ifthe map E → E × E , e ( s ( e ) , r ( e )) is injective.The line graph LE of E is the directed graph with vertex set EL = E , edge set EL = { ( e, f ) | r ( e ) = s ( f ) } ⊂ E × E, and the source and range maps s, r : LE → LE given by projection to the firstand second coordinates, respectively. By construction, the line graph LE is simple.The adjacency matrix of E = ( E , E ) is the E × E -matrix B E ( v, w ) = |{ e ∈ E | s ( e ) = v, r ( e ) = w }| , MICHAEL BRANNAN, KARI EIFLER, CHRISTIAN VOIGT, AND MORITZ WEBER and the edge matrix of E is the E × E -matrix with entries A E ( e, f ) = ( r ( e ) = s ( f )0 else . Note that the edge matrix A E of E equals the adjacency matrix B LE of LE .We will only be interested in finite directed graphs in the sequel, that is, directedgraphs E = ( E , E ) such that both E and E are finite sets. This requirementcan be substantially relaxed [22].2.2. Graph C ∗ -algebras and Cuntz-Krieger algebras. We recall the definitionof the graph C ∗ -algebra of a finite directed graph E = ( E , E ).A Cuntz-Krieger E -family in a C ∗ -algebra D consists of mutually orthogonalprojections p v ∈ D for all v ∈ E together with partial isometries s e ∈ D for all e ∈ E such thata) s ∗ e s e = p r ( e ) for all edges e ∈ E b) p v = P s ( e )= v s e s ∗ e whenever v ∈ E is not a sink.The graph C ∗ -algebra of E can then be defined as follows. Definition 2.1.
Let E = ( E , E ) be a finite directed graph. The graph C ∗ -algebra C ∗ ( E ) is the universal C ∗ -algebra generated by a Cuntz-Krieger E -family.We write P v and S e for the corresponding projections and partial isometries in C ∗ ( E ), associated with v ∈ E and e ∈ E , respectively.That is, given any Cuntz-Krieger E -family in a C ∗ -algebra D , with projec-tions p v for v ∈ E and partial isometries s e for e ∈ E , there exists a unique ∗ -homomorphism φ : C ∗ ( E ) → D such that φ ( P v ) = p v and φ ( S e ) = s e .Next we recall the definition of Cuntz-Krieger algebras [12]. If A ∈ M N ( Z ) is amatrix with entries A ( i, j ) ∈ { , } then a Cuntz-Krieger A -family in a C ∗ -algebra D consists of partial isometries s , . . . , s N ∈ D with mutually orthogonal rangessuch that the Cuntz-Krieger relations s ∗ i s i = N X j =1 A ( i, j ) s j s ∗ j hold for all 1 ≤ i ≤ N . Definition 2.2.
Let A ∈ M N ( Z ) be a matrix with entries A ( i, j ) ∈ { , } . TheCuntz-Krieger algebra O A is the C ∗ -algebra generated by a universal Cuntz-Krieger A -family, that is, it is the universal C ∗ -algebra generated by partial isometries S , . . . , S N with mutually orthogonal ranges, satisfying S ∗ i S i = N X j =1 A ( i, j ) S j S ∗ j for all 1 ≤ i ≤ N .In contrast to [12], we do not make any further assumptions on the matrix A inDefinition 2.2 in the sequel, except that it should have entries in { , } . Accordingly,the algebras O A may sometimes be rather degenerate or even trivial, as for instanceif A = 0. However, we have adopted this setting for the sake of consistency withour definitions in the quantum case further below.If E is a graph with no sinks and no sources then the graph C ∗ -algebra C ∗ ( E )can be canonically identified with the Cuntz-Krieger algebra associated with theedge matrix A E of E . In particular, the projections in C ∗ ( E ) associated to verticesof E need not be mentioned explicitly in this case. UANTUM CUNTZ-KRIEGER ALGEBRAS 5
We note that the graph C ∗ -algebra of a graph E with no sinks and no sourcesis completely determined by the line graph LE of E , keeping in mind that theedge matrix A E equals the adjacency matrix B LE , see also [23]. Viewing C ∗ ( E )as being associated with the line graph of E motivates our generalizations furtherbelow, where we will replace the matrix A in Definition 2.2 with the quantumadjacency matrix of a directed quantum graph. Remark 2.3.
It is known that all graph C ∗ -algebras of finite directed graphswithout sinks are isomorphic to Cuntz-Krieger algebras [1].2.3. Free graph C ∗ -algebras and free Cuntz-Krieger algebras. Borrowingterminology from [5], we shall now consider “liberated” analogues of graph C ∗ -algebras and Cuntz-Krieger algebras.In the case of graphs, the input for this construction is a finite directed graph E = ( E , E ) as above. By a free Cuntz-Krieger E -family in a C ∗ -algebra D weshall mean a collection of projections p v ∈ D for all v ∈ E together with partialisometries s e ∈ D for all e ∈ E such thata) s ∗ e s e = p r ( e ) for all edges e ∈ E b) p v = P s ( e )= v s e s ∗ e whenever v ∈ E is not a sink.That is, the only difference to an ordinary Cuntz-Krieger E -family is that theprojections p v are no longer required to be mutually orthogonal.Stipulating that the p v are mutually orthogonal is equivalent to saying that the C ∗ -algebra generated by the projections p v is commutative. In the same way asin the liberation of matrix groups [5], removing commutation relations of this typeleads to the following free version of the notion of a graph C ∗ -algebra. Definition 2.4.
Let E = ( E , E ) be a finite directed graph. The free graph C ∗ -algebra F C ∗ ( E ) is the universal C ∗ -algebra generated by a free Cuntz-Krieger E -family. We write P v and S e for the corresponding projections and partial isometriesin F C ∗ ( E ), associated with v ∈ E and e ∈ E , respectively.Of course, a similar definition can be made in the Cuntz-Krieger case as well.For the sake of definiteness, let us say that a free Cuntz-Krieger A -family in a C ∗ -algebra D , associated with a matrix A ∈ M N ( Z ) with entries A ( i, j ) ∈ { , } ,consists of partial isometries s , . . . , s N ∈ D such that the Cuntz-Krieger relations s ∗ i s i = N X j =1 A ( i, j ) s j s ∗ j hold for all 1 ≤ i ≤ N . Definition 2.5.
Let A ∈ M N ( Z ) be a matrix with entries A ( i, j ) ∈ { , } . Thefree Cuntz-Krieger algebra F O A is the universal C ∗ -algebra generated by partialisometries S , . . . , S N , satisfying the relations S ∗ i S i = N X j =1 A ( i, j ) S j S ∗ j for all i .We note that free graph C ∗ -algebras and free Cuntz-Krieger algebras alwaysexist, keeping in mind that the norms of all generators are uniformly bounded inany representation of the universal ∗ -algebra generated by a free Cuntz-Kriegerfamily. Let us also remark that the free graph C ∗ -algebra of a finite directedgraph E with no sinks and no sources agrees with the free Cuntz-Krieger algebraassociated with the edge matrix A E . MICHAEL BRANNAN, KARI EIFLER, CHRISTIAN VOIGT, AND MORITZ WEBER
For any finite directed graph E and any matrix A as above there are canon-ical surjective ∗ -homomorphisms π : F C ∗ ( E ) → C ∗ ( E ) and π : F O A → O A ,respectively, obtained directly from the universal property. These maps are notisomorphisms in general.For instance, if E is the graph with two vertices and no edges then C ∗ ( E ) = C ⊕ C ,whereas F C ∗ ( E ) = C ∗ C is the non-unital free product of two copies of C . However,we note that if E is the graph with a single vertex and N self-loops then thecanonical projection induces an isomorphism F C ∗ ( E ) ∼ = C ∗ ( E ), identifying the freegraph C ∗ -algebra with the Cuntz algebra O N .Let us now elaborate on the relation between F C ∗ ( E ) and C ∗ ( E ) for an arbitraryfinite directed graph E , and similarly on the relation between F O A and O A . Theorem 2.6.
Let E be a finite directed graph. Then the canonical projectionmap F C ∗ ( E ) → C ∗ ( E ) is a KK -equivalence. Similarly, if A ∈ M N ( Z ) is a matrixwith entries A ( i, j ) ∈ { , } then the canonical projection F O A → O A is a KK -equivalence.Proof. The proof is analogous for graph algebras and Cuntz-Krieger algebras, there-fore we shall restrict attention to the case of graph algebras.Adapting a well-known argument from [11], we will show more generally that C ∗ ( E ) and F C ∗ ( E ) cannot be distinguished by any homotopy invariant functoron the category of C ∗ -algebras which is stable under tensoring with finite matrixalgebras.Firstly, we claim that there exists a ∗ -homomorphism φ : C ∗ ( E ) → M E ( F C ∗ ( E ))satisfying φ ( P v ) xy = δ x,v δ y,v P v ,φ ( S e ) xy = δ x,s ( e ) δ y,r ( e ) S e for v ∈ E and e ∈ E . For this it suffices to show that the elements φ ( P v ) , φ ( S e )in M E ( F C ∗ ( E )) given by the above formulas define a Cuntz-Krieger E -family.Clearly, the elements P v are mutually orthogonal projections, and the elements φ ( S e ) are partial isometries such that( φ ( S e ) ∗ φ ( S e )) xy = δ x,r ( e ) δ y,r ( e ) S ∗ e S e = δ x,r ( e ) δ y,r ( e ) P r ( e ) = φ ( P r ( e ) ) xy and φ ( P v ) xy = δ x,v δ y,v P v = δ x,v δ y,v X s ( f )= v S f S ∗ f = X s ( f )= v δ x,s ( f ) δ y,s ( f ) S f S ∗ f = X s ( f )= v X z ∈ E φ ( S f ) xz ( φ ( S f ) yz ) ∗ = X s ( f )= v ( φ ( S f ) φ ( S f ) ∗ ) xy if v ∈ E is not a sink, as required.Recall that we write π : F C ∗ ( E ) → C ∗ ( E ) for the canonical projection. Fixinga vertex w ∈ E , we claim that M E ( π ) ◦ φ is homotopic to the embedding ι of C ∗ ( E ) into the corner of M E ( C ∗ ( E )) corresponding to w . For this we considerthe ∗ -homomorphisms µ t : C ∗ ( E ) → M E ( C ∗ ( E )) for t ∈ [0 ,
1] given by µ t ( P v ) = u vt ι ( P v )( u vt ) ∗ , µ t ( S e ) = u s ( e ) t ι ( S e )( u r ( e ) t ) ∗ , UANTUM CUNTZ-KRIEGER ALGEBRAS 7 where u xt for x ∈ E with x = w is the rotation matrix u t = (cid:18) cos(2 πt ) sin(2 πt ) − sin(2 πt ) cos(2 πt ) (cid:19) placed in the block corresponding to the indices w and x , and u xt = id for x = w .In a similar way as above one checks that µ t preserves the relations for C ∗ ( E ).Indeed, the elements µ t ( P v ) are mutually orthogonal projections since P v , P w for v = w are orthogonal in C ∗ ( E ) and the unitaries u xt have scalar entries. Moreover,for t ∈ [0 ,
1] the elements µ t ( S e ) are partial isometries such that µ t ( S e ) ∗ µ t ( S e ) = u r ( e ) t ι ( S ∗ e S e )( u r ( e ) t ) ∗ = u r ( e ) t ι ( P r ( e ) )( u r ( e ) t ) ∗ = µ t ( P r ( e ) ) , and µ t ( P v ) = u vt ι ( P v )( u vt ) ∗ = X s ( f )= v u s ( f ) t ι ( S f S ∗ f )( u s ( f ) t ) ∗ = X s ( f )= v µ t ( S f ) µ t ( S f ) ∗ if v is not a sink. By construction we have µ = ι and µ = M E ( π ) ◦ φ .The composition φ ◦ π looks the same as M E ( π ) ◦ φ on generators, and a similarhomotopy shows that φ ◦ π is homotopic to the embedding F C ∗ ( E ) → M E ( F C ∗ ( E ))associated with a fixed vertex w . This finishes the proof. (cid:3) Quantum Cuntz-Krieger algebras
In this section we define our quantum analogue of Cuntz-Krieger algebras. Sincethe input for this construction is the quantum adjacency matrix of a directed quan-tum graph, we shall first review the concept of a quantum graph.3.1.
Quantum graphs.
The notion of a quantum graph has been considered withsome variations by a number of authors, see [16], [34], [14], [27], [7]. We will followthe approach in [27], [7], and adapt it to the setting of directed graphs.Assume that B is a finite dimensional C ∗ -algebra B and let ψ : B → C bea faithful state. We denote by L ( B ) = L ( B, ψ ) the Hilbert space obtained byequipping B with the inner product h x, y i = ψ ( x ∗ y ). Moreover let us write m : B ⊗ B → B for the multiplication map of B and m ∗ for its adjoint, noting that m can be viewed as a linear operator L ( B ) ⊗ L ( B ) → L ( B ).If B = C ( X ) is the algebra of functions on a finite set X then states on B correspond to probability measures on X . The most natural choice is to take for ψ the state corresponding to the uniform measure in this case. For an arbitrary finitedimensional C ∗ -algebra B we have the following well-known condition, singling outcertain natural choices among all possible states on B in a similar way [3]. Definition 3.1.
Let B be a finite dimensional C ∗ -algebra and δ >
0. A faithfulstate ψ : B → C is called a δ -form if mm ∗ = δ id. By a finite quantum space ( B, ψ )we shall mean a finite dimensional C ∗ -algebra B together with a δ -form ψ : B → C .If B is a finite dimensional C ∗ -algebra then we have B ∼ = L da =1 M N a ( C ) for some N , . . . , N d . A state ψ on B can be written uniquely in the form ψ ( x ) = d X a =1 Tr( Q ( a ) x i )for x = ( x , . . . , x d ), where the Q ( a ) ∈ M N a ( C ) are positive matrices satisfying P da =1 Tr( Q ( a ) ) = 1. Then ψ is a δ -form iff Q ( a ) is invertible and Tr( Q − a ) ) = δ forall a . Here Tr denotes the natural trace, given by summing all diagonal terms of amatrix. MICHAEL BRANNAN, KARI EIFLER, CHRISTIAN VOIGT, AND MORITZ WEBER
Note that we may assume without loss of generality that all matrices Q ( a ) in thedefinition of ψ are diagonal. We shall say that ( B, ψ ) as above is in standard formin this case.Any finite dimensional C ∗ -algebra B admits a unique tracial δ -form for a uniquelydetermined value of δ . Explicitly, this is the tracial state given bytr( x ) = 1dim( B ) d X a =1 N a Tr( x i ) , and we have δ = dim( B ). Note that if B = C ( X ) is commutative then thiscorresponds to the uniform measure on X , and δ is the cardinality of X .For later purposes it will be useful to record an explicit formula for the adjointof the multiplication map in a finite quantum space. Lemma 3.2.
Let ( B, ψ ) be a finite quantum space in standard form as describedabove, and consider the linear basis of B given by the adapted matrix units f ( a ) ij = ( Q − / a ) ) ii e ( a ) ij ( Q − / a ) ) jj , where e ( a ) ij in M N a ( C ) are the standard matrix units. Then we have ( f ( a ) ij ) ∗ = f ( a ) ji and m ∗ ( f ( a ) ij ) = X k f ( a ) ik ⊗ f ( a ) kj for all a, i, j .Proof. Since the matrices Q ( a ) are positive we clearly have ( f ( a ) ij ) ∗ = f ( a ) ji . More-over, observing f ( b ) rs f ( c ) pq = δ bc ( Q − b ) ) sp f ( b ) rq and ψ ( f ( a ) kl ) = δ kl , we compute h f ( a ) ij , m ( f ( b ) rs ⊗ f ( c ) pq ) i = δ bc ψ ( f ( a ) ji ( Q − b ) ) sp f ( b ) rq ) = δ abc ( Q − a ) ) sp ( Q − a ) ) ir δ jq and X k h f ( a ) ik ⊗ f ( a ) kj , f ( b ) rs ⊗ f ( c ) pq i = X k ψ ( f ( a ) ki f ( b ) rs ) ψ ( f ( a ) jk f ( c ) pq )= δ ab δ ac X k ( Q − a ) ) ir ( Q − a ) ) kp ψ ( f ( a ) ks ) ψ ( f ( a ) jq )= δ abc ( Q − a ) ) ir ( Q − a ) ) sp δ jq . This yields the claim. (cid:3)
Let us now discuss the concept of a quantum graph. We shall be interested indirected quantum graphs in the following sense.
Definition 3.3.
Let B be a finite dimensional C ∗ -algebra and ψ : B → C a δ -form.A linear operator A : L ( B ) → L ( B ) is called a quantum adjacency matrix if m ( A ⊗ A ) m ∗ = δ A. A directed quantum graph G = ( B, ψ, A ) is a finite quantum space (
B, ψ ) togetherwith a quantum adjacency matrix.In order to explain Definition 3.3 let us consider the case that B = C ( X ) is thequantum space associated with a finite set X , with ψ being given by the uniformmeasure. A linear operator A : L ( B ) → L ( B ) can be identified canonically witha matrix in M X ( C ). Moreover, a straightforward calculation shows that1 | X | m ( A ⊗ B ) m ∗ UANTUM CUNTZ-KRIEGER ALGEBRAS 9 is the Schur product of
A, B ∈ M X ( C ), given by entrywise multiplication. Hence A is a quantum adjacency matrix iff it is an idempotent with respect to the Schurproduct in this case, which is equivalent to saying that A has entries in { , } .According to the above discussion, every simple finite directed classical graph E = ( E , E ) gives rise to a directed quantum graph in a natural way. More pre-cisely, if A E denotes the adjacency matrix of E then we obtain a directed quantumgraph structure on B = C ( E ) by taking the state ψ which corresponds to count-ing measure, and the operator A : L ( B ) → L ( B ) given by A ( e i ) = P j A ( i, j ) e j .Conversely, every directed quantum graph structure on a finite dimensional com-mutative C ∗ -algebra B = C ( X ) arises from a simple finite directed graph on thevertex set X .For a general finite quantum space ( B, ψ ) it will be convenient for our consid-erations further below to write down the quantum adjacency matrix condition interms of bases.
Lemma 3.4.
Let ( B, ψ ) be a finite quantum space in standard form. Then a linearoperator A : L ( B ) → L ( B ) , given by A ( f ( a ) ij ) = X brs A rsbija f ( b ) rs in terms of the adapted matrix units, is a directed quantum adjacency matrix iff X ks ( Q − b ) ) ss A rsbika A snbkja = δ A rnbija for all a, b, i, j, r, n .Proof. Using Lemma 3.2 we calculate m ( A ⊗ A ) m ∗ ( f ( a ) ij ) = X k A ( f ( a ) ik ) A ( f ( a ) kj )= X k X brs X cmn A rsbika f ( b ) rs A mnckja f ( c ) mn = X k X brsn ( Q − b ) ) ss A rsbika A snbkja f ( b ) rn , so that comparing coefficients yields the claim. (cid:3) We point out that there is a rich supply of directed quantum adjacency matricesand quantum graphs. Let B be a finite dimensional C ∗ -algebra and let tr be theunique tracial δ -form on B . Every element P ∈ B ⊗ B op has a Choi-Jamio lkowskiform, that is, there exists a unique linear map A : B → B such that P = P A = 1dim( B ) (1 ⊗ A ) m ∗ (1) , where m ∗ : B → B ⊗ B is the adjoint of multiplication with respect to tr. Then A is a quantum adjacency matrix with respect to ( B, tr) iff P is idempotent, that is,iff P = P .Moreover, idempotents in B ⊗ B op can be naturally obtained as follows. Assumethat B ֒ → B ( H ) is unitally embedded into the algebra of bounded operators onsome finite dimensional Hilbert space H , and let B ′ ⊂ B ( H ) be the commutantof B . Then B ⊗ B op identifies with the space of all completely bounded B ′ - B ′ -bimodule maps from B ( H ) to itself. In particular, idempotents in B ⊗ B op are thesame thing as direct sum decompositions B ( H ) = S ⊕ R of B ′ - B ′ -bimodules.Taking B = M N ( C ) and the identity embedding into B ( C N ) = M N ( C ) wesee that there is a bijective correspondence between quantum graph structures on( M N ( C ) , tr) and vector space direct sum decompositions M N ( C ) = S ⊕ R . Remark 3.5.
One could work more generally with arbitrary faithful positive linearfunctionals ψ instead of δ -forms, by modifying the defining relation of a quantumadjacency matrix in Definition 3.3 to m ( A ⊗ A ) m ∗ = Amm ∗ . We will however restrict ourselves to δ -forms in the sequel, as this will allow us toremain closer to the classical theory in the commutative case. Remark 3.6.
The definition of a quantum graph in [27], [7] contains further con-ditions on the quantum adjacency matrix. If B = C ( X ) is commutative then theseconditions correspond to requiring that the matrix A ∈ M X ( C ) is symmetric andhas entries 1 on the diagonal, respectively. That is, the quantum graphs consideredin these papers are undirected and have all self-loops. Neither of these conditionsis appropriate in connection with Cuntz-Krieger algebras.3.2. Quantum Cuntz-Krieger algebras.
Let us now define the quantum Cuntz-Krieger algebra associated to a directed quantum graph. Comparing with the def-inition of graph C ∗ -algebras, we note that the quantum graph used as an inputin our definition may be thought of as an analogue of the line graph of a classicalgraph.If G = ( B, ψ, A ) is a directed quantum graph then we shall say that a quantumCuntz-Krieger G -family in a C ∗ -algebra D is a linear map s : B → D such thata) µ D (id ⊗ µ D )( s ⊗ s ∗ ⊗ s )(id ⊗ m ∗ ) m ∗ = s b) µ D ( s ∗ ⊗ s ) m ∗ = µ D ( s ⊗ s ∗ ) m ∗ A ,where µ D : D ⊗ D → D is the multiplication map for D and s ∗ ( b ) = s ( b ∗ ) ∗ for b ∈ B . We also recall that m ∗ denotes the adjoint of the multiplication map for B with respect to the inner product given by ψ . Definition 3.7.
Let G = ( B, ψ, A ) be a directed quantum graph. The quantumCuntz-Krieger algebra F O ( G ) is the universal C ∗ -algebra generated by a quantumCuntz-Krieger G -family S : B → F O ( G ).In other words, the quantum Cuntz-Krieger algebra F O ( G ) satisfies the followinguniversal property. If D is a C ∗ -algebra and s : B → D a quantum Cuntz-Krieger G -family, then there exists a unique ∗ -homomorphism ϕ : F O ( G ) → D such that ϕ ( S ( b )) = s ( b ) for all b ∈ B . Remark 3.8.
We note that Definition 3.7 makes sense for a finite dimensional C ∗ -algebra B together with a faithful positive linear functional ψ and an arbitrarylinear map A : L ( B ) → L ( B ). At this level of generality one can shift informationfrom ψ into the matrix A and vice versa, without changing the resulting C ∗ -algebra.Our definition will allow us to remain closer to the standard setup for Cuntz-Kriegeralgebras.It is not difficult to check that the quantum Cuntz-Krieger algebra F O ( G ) alwaysexists. This is done most easily by rewriting Definition 3.7 in terms of a linearbasis for the algebra B . In the sequel we shall say that a directed quantum graph G = ( B, ψ, A ) is in standard form if its underlying finite quantum space is, compareparagraph 3.1.
Proposition 3.9.
Let G = ( B, ψ, A ) be a directed quantum graph in standard form,and let A ( f ( a ) ij ) = X brs A rsbija f ( b ) rs be the quantum adjacency matrix written in terms of the adapted matrix units asdiscussed further above. Then the quantum Cuntz-Krieger algebra F O ( G ) identifies UANTUM CUNTZ-KRIEGER ALGEBRAS 11 with the universal C ∗ -algebra F O A with generators S ( a ) ij for ≤ a ≤ d and ≤ i, j ≤ N a , satisfying the relations X rs S ( a ) ir ( S ( a ) sr ) ∗ S ( a ) sj = S ( a ) ij X l ( S ( a ) li ) ∗ S ( a ) lj = X brs A rsbija X l S ( b ) rl ( S ( b ) sl ) ∗ for all a, i, j .Proof. Let us first consider the elements S ( a ) ij = S ( f ( a ) ij ) in F O ( G ). If µ denotes themultiplication map for F O ( G ), then according to Lemma 3.2 we get X rs S ( a ) ir ( S ( a ) sr ) ∗ S ( a ) sj = X rs S ( f ( a ) ir ) S ∗ ( f ( a ) rs ) S ( f ( a ) sj )= X rs µ (id ⊗ µ )( S ⊗ S ∗ ⊗ S )( f ( a ) ir ⊗ f ( a ) rs ⊗ f ( a ) sj )= µ (id ⊗ µ )( S ⊗ S ∗ ⊗ S )(id ⊗ m ∗ ) m ∗ ( f ( a ) ij )= S ( f ( a ) ij ) = S ( a ) ij , and similarly X r ( S ( a ) ri ) ∗ S ( a ) rj = X r µ ( S ∗ ⊗ S )( f ( a ) ir ⊗ f ( a ) rj )= µ ( S ∗ ⊗ S ) m ∗ ( f ( a ) ij )= µ ( S ⊗ S ∗ ) m ∗ A ( f ( a ) ij )= X brs A rsbija µ ( S ⊗ S ∗ ) m ∗ ( f ( b ) rs )= X brsl A rsbija µ ( S ⊗ S ∗ )( f ( b ) rl ⊗ f ( b ) ls )= X brsl A rsbija S ( b ) rl ( S ( b ) sl ) ∗ . Hence, by the definition of F O A , there exists a unique ∗ -homomorphism φ : F O A → F O ( G ) such that φ ( S ( a ) ij ) = S ( f ( a ) ij ) for all a, i, j .Conversely, let us define a linear map s : B → F O A by s ( f ( a ) ij ) = S ( a ) ij . Essentiallythe same computation as above shows that s defines a quantum Cuntz-Krieger G -family in F O A , so that there exists a unique ∗ -homomorphism ψ : F O ( G ) → F O A satisfying ψ ( S ( b )) = s ( b ) for all b ∈ B .It is straightforward to check that the maps φ and ψ are mutually inverse iso-morphisms. (cid:3) Using matrix notation we can rephrase the relations from Proposition 3.9 in avery concise way. More precisely, writing S ( a ) ∈ M N a ( F O ( G )) for the matrix withentries S ( a ) ij = S ( f ( a ) ij ) and ˆ A for the d × d -matrix with coefficients ˆ A ba = A rsbija weobtain S ( a ) ( S ( a ) ) ∗ S ( a ) = S ( a ) ( S ( a ) ) ∗ S ( a ) = X b ˆ A ba S ( b ) ( S ( b ) ) ∗ for all 1 ≤ a ≤ d . The first formula says that the elements S ( a ) ∈ M N a ( F O ( G )) arepartial isometries. This means in particular that their entries are bounded in normby 1, which implies in turn that the universal C ∗ -algebras F O A and F O ( G ) always exist. The second formula can be viewed as a matrix-valued version of the classicalCuntz-Krieger relation. Remark 3.10.
From Proposition 3.9 and the above remarks it may appear atfirst sight that F O ( G ) ∼ = F O A does not depend on the δ -form ψ in G = ( B, ψ, A ).However, recall from Lemma 3.4 that the choice of ψ is reflected in the definingrelations for the coefficients A rsbija of the quantum adjacency matrix. Remark 3.11.
As will be discussed in more detail at the start of the next section,the notation F O A used in Proposition 3.9 is compatible with our notation for freeCuntz-Krieger algebras introduced in Definition 2.5.4. Examples
In this section we take a look at some examples of quantum graphs and theirassociated quantum Cuntz-Krieger algebras in the sense of Definition 3.7.4.1.
Classical graphs.
Assume that E = ( E , E ) is a finite simple directedgraph with N vertices. The directed quantum graph G associated with E has B = C ( E ) = C N as underlying C ∗ -algebra. We work with the canonical ba-sis e , . . . , e N of minimal projections in B and the normalized standard trace tr : B → C . That is, tr( e i ) = 1 /N for all i , and we have m ( e i ⊗ e j ) = δ ij e i and m ∗ ( e i ) = N e i ⊗ e i . If B E denotes the adjacency matrix of E then A ( e i ) = N X j =1 B E ( i, j ) e j determines a quantum adjacency matrix A : L ( B ) → L ( B ). Proposition 4.1.
Let E be a finite simple directed graph and let G = ( B, ψ, A ) be the quantum graph corresponding to E as above. Then the free Cuntz-Kriegeralgebra associated with the adjacency matrix B E of E is canonically isomorphic tothe quantum Cuntz-Krieger algebra F O ( G ) .Proof. This can be viewed as a special case of Proposition 3.9, but let us writedown the key formulas explicitly. Note that tr is a δ -form with δ = N and consider S i = N S ( e i ) ∈ F O ( G ). Then the defining relations for a free Cuntz-Krieger B E -family are obtained from S i S ∗ i S i = N µ (id ⊗ µ )( S ( e i ) ⊗ S ∗ ( e i ) ⊗ S ( e i ))= N µ (id ⊗ µ )( S ⊗ S ∗ ⊗ S )(id ⊗ m ∗ ) m ∗ ( e i )= N S ( e i ) = S i and S ∗ i S i = N µ ( S ∗ ⊗ S )( e i ⊗ e i )= N µ ( S ∗ ⊗ S ) m ∗ ( e i )= N µ ( S ⊗ S ∗ ) m ∗ ( A ( e i ))= N N X j =1 B E ( i, j ) µ ( S ⊗ S ∗ )( e j ⊗ e j )= N X j =1 B E ( i, j ) S j S ∗ j UANTUM CUNTZ-KRIEGER ALGEBRAS 13 for all i . This yields a ∗ -homomorphism F O B E → F O ( G ). Similarly, one checksthat the linear map s : B → F O B E given by s ( e i ) = N S i is a quantum Cuntz-Krieger G -family, which induces a ∗ -homomorphism F O ( G ) → F O B E . These mapsare mutually inverse isomorphisms. (cid:3) It follows from the remarks after Definition 3.3 that every quantum Cuntz-Krieger algebra F O ( G ) over a directed quantum graph G = ( B, ψ, A ) with B abelianis a free Cuntz-Krieger algebra associated to some 0 , Complete quantum graphs and quantum Cuntz algebras.
Let us nextconsider an arbitrary finite quantum space (
B, ψ ) in standard form, using the samenotation as after Definition 3.1. Following [7], we can form the complete quantumgraph on (
B, ψ ), which is the directed quantum graph K ( B, ψ ) = (
B, ψ, A ) withquantum adjacency matrix A : L ( B ) → L ( B ) given by A ( b ) = δ ψ ( b )1. In termsof the adapted matrix units f ( a ) ij ∈ B defined in Lemma 3.2 we get A ( f ( a ) ij ) = δ ij δ X b X k δ ij δ ( Q ( a ) ) kk f ( b ) kk . Therefore, relative to this basis, we have the matrix representation A = ( A klbija ),where A klbija = δ ij δ kl δ ( Q ( b ) ) kk . It follows from Proposition 3.9 and the preceding discussion that the quantumCuntz-Krieger algebra F O ( K ( B, ψ )) is the universal C ∗ -algebra with generators S ( a ) ij for 1 ≤ a ≤ d, ≤ i, j ≤ N a and relations X rs S ( a ) ir ( S ( a ) sr ) ∗ S ( a ) sj = S ( a ) ij , X r ( S ( a ) ri ) ∗ S ( a ) rj = δ ij δ X b X kl ( Q ( b ) ) kk S ( b ) kl ( S ( b ) kl ) ∗ for all a, i, j . Example 4.2.
Let us consider explicitly the special case of the complete quan-tum graph K ( M N ( C ) , tr) on a full matrix algebra B = M N ( C ). The C ∗ -algebra F O ( K ( M N ( C ) , tr)) has generators S ij for 1 ≤ i, j ≤ N satisfying the relations X kl S ik S ∗ lk S lj = S ij X r S ∗ ri S rj = δ ij N X rs S rs S ∗ rs for all i, j .Note that when B = C d is abelian, Proposition 4.1 implies that F O ( K ( C d , tr))is nothing other than the free Cuntz-Krieger algebra associated to the completegraph K d , or equivalently, the free graph C ∗ -algebra associated to the graph with asingle vertex and d self-loops. Thus F O ( K ( C d , tr)) identifies with the Cuntz algebra O d , compare the remarks after Definition 2.5. With this in mind, we may call anyquantum Cuntz-Krieger algebra of the form F O ( K ( B, ψ )) a quantum Cuntz algebra .The algebras obtained in this way are in fact rather closely related to Cuntzalgebras, as we discuss next.
Lemma 4.3.
Let F O ( K ( B, ψ )) be as above and write n = dim( B ) . Then thereexists a surjective ∗ -homomorphism φ : F O ( K ( B, ψ )) → O n such that φ ( S ( a ) ij ) = 1( Q ( a ) ) / ii δ s ( a ) ij for all a, i, j , where s ( a ) ij are standard generators of the Cuntz algebra O n .Proof. We just have to check that the elements φ ( S ( a ) ij ) satisfies the defining rela-tions of F O ( K ( B, ψ )) from above. Indeed, we obtain X rs φ ( S ( a ) ir ) φ ( S ( a ) sr ) ∗ φ ( S ( a ) sj ) = X rs Q ( a ) ) / ii ( Q ( a ) ) ss δ s ( a ) ir ( s ( a ) sr ) ∗ s ( a ) sj = X s Q ( a ) ) / ii ( Q ( a ) ) ss δ s ( a ) ij = 1( Q ( a ) ) / ii δ s ( a ) ij = φ ( S ( a ) ij ) , and similarly X r φ ( S ( a ) ri ) ∗ φ ( S ( a ) rj ) = X r Q ( a ) ) rr δ ( s ( a ) ri ) ∗ s ( a ) rj = δ ij = δ ij X bkl s ( b ) kl ( s ( b ) kl ) ∗ = δ ij δ X bkl ( Q ( b ) ) kk φ ( S ( b ) kl ) φ ( S ( b ) kl ) ∗ as required. (cid:3) Remark 4.4.
Lemma 4.3 implies in particular that the canonical linear map S : B → F O ( K ( B, ψ )) is injective. This is not always the case for general quantumCuntz-Krieger algebras. An explicit example will be given in Example 4.9 furtherbelow.Our main structure result regarding the quantum Cuntz algebras F O ( K ( B, ψ ))can be stated as follows.
Theorem 4.5.
Let B be an n -dimensional C ∗ -algebra and let ψ : B → C be a δ -form satisfying δ ∈ N . Then F O ( K ( B, ψ )) ∼ = O n . We will prove Theorem 4.5 using methods from the theory of quantum groupsin section 8. Under the hypothesis δ ∈ N , Theorem 4.5 implies that the ∗ -homomorphism φ : F O ( K ( B, ψ )) → O n constructed in Lemma 4.3 is an isomor-phism.It seems remarkable that the relations defining F O ( K ( B, ψ )) do indeed charac-terize the Cuntz algebra O n , at least when we restrict to δ -forms satisfying theabove integrality condition. Already in the special case ( B, ψ ) = ( M N ( C ) , tr) fromExample 4.2 it seems not even obvious that F O ( K ( B, ψ )) is unital . In fact, aneasy argument shows that the element e = N P kl S kl ( S kl ) ∗ ∈ F O ( K ( M N ( C ) , tr))satisfies S ij e = S ij for all 1 ≤ i, j ≤ N . In section 8 we will verify in particular theless evident relation eS ij = S ij for all i, j .We note at the same time that F O ( K ( M N ( C ) , tr)) is very different from theuniversal C ∗ -algebra generated by the coefficients of a N × N -matrix S = ( S ij )satisfying S ∗ S = id, as the latter algebra admits many characters. UANTUM CUNTZ-KRIEGER ALGEBRAS 15
Trivial quantum graphs.
If (
B, ψ ) is a finite quantum space as above, thenthe trivial quantum graph T ( B, ψ ) on (
B, ψ ) is given by the quantum adjacencymatrix A = id, so that we have the matrix representation A klbija = δ ab δ ik δ jl . UsingProposition 3.9 we see that the quantum Cuntz-Krieger algebra F O ( T ( B, ψ )) isthe universal C ∗ -algebra with generators S ( a ) ij for 1 ≤ a ≤ d, ≤ i, j ≤ N a , andrelations X kl S ( a ) ik ( S ( a ) lk ) ∗ S ( a ) lj = S ( a ) ij X k ( S ( a ) ki ) ∗ S ( a ) kj = X k S ( a ) ik ( S ( a ) jk ) ∗ for all a, i, j . We note that F O ( T ( B, ψ )) is independent of the δ -form ψ on B , andwe will therefore also write F O ( T B ) instead of F O ( T ( B, ψ )) in the sequel.
Example 4.6.
Let us consider explicitly the special case of the trivial quantumgraph
T M N = T M N ( C ) on a full matrix algebra B = M N ( C ). The C ∗ -algebra F O ( T M N ) has generators S ij for 1 ≤ i, j ≤ N satisfying the relations X kl S ik S ∗ lk S lj = S ij X k S ∗ ki S kj = X k S ik S ∗ jk for all i, j .It is easy to check that F O ( T M N ) maps onto Brown’s algebra [8], that is, theuniversal C ∗ -algebra U ncN generated by the entries of a unitary N × N -matrix u =( u ij ), by sending S ij to u ij . This shows in particular that F O ( T M N ) for N > F O ( T M N ) onto the non-unital free product C ∗· · ·∗ C of N copies of C , by sending S ij to δ ij i , where 1 i denotes the unit element in the i -th copy of C . It follows that F O ( T M N ) is not unital for N > F O ( T M N ) as a special case of Theorem 5.3. Theorem 4.7.
Let
T M N be the trivial quantum graph as above. Then there existsa ∗ -isomorphism M N ( F O ( T M N ) + ) ∼ = M N ( C ) ∗ ( C ( S ) ⊕ C ) , and the quantum Cuntz-Krieger algebra F O ( T M N ) is KK -equivalent to C ( S ) forall N ∈ N . In particular K ( F O ( T M N )) = Z ,K ( F O ( T M N )) = Z . Here ∗ denotes the unital free product and F O ( T M N ) + is the minimal unita-rization of F O ( T M N ).With little extra effort one can also determine generators for the K -groups inTheorem 4.7. More precisely, if we write S = ( S ij ) for the matrix of generators of F O ( T M N ), then these are represented by the projection S ∗ S ∈ M N ( F O ( T M N ))and the unitary S − (1 − S ∗ S ) ∈ M N ( F O ( T M N ) + ), respectively. Remark 4.8.
Combining Theorem 4.7 and Proposition 4.10 below one can deter-mine the K -theory of F O ( T B ) for general B . More precisely, if B ∼ = L da =1 M N a ( C )then we obtain K ( F O ( T B )) = Z d ,K ( F O ( T B )) = Z d , taking into account [11]. Diagonal quantum graphs.
A natural generalization of the trivial quantumgraphs described in the previous paragraph are the diagonal quantum graphs . Here,we take (
B, ψ ) again to be an arbitrary finite quantum space in standard form, butreplace the trivial quantum adjacency matrix A = id with a map of the form A ( f ( a ) ij ) = x ( a ) ij f ( a ) ij for some suitable complex numbers x ( a ) ij ∈ C for 1 ≤ a ≤ d, ≤ i, j ≤ N a . Notethat if B is abelian then the associated adjacency matrix is a diagonal matrix withentries in { , } . That is, the only edges possible are self-loops, and we recoverprecisely the classical notion of a diagonal graph.In the non-commutative setting the notion of a diagonal graph is somewhatricher. Namely, Lemma 3.4 shows that the only requirements on the coefficients x ( a ) ij are X s ( Q − b ) ) ss x ( b ) ks x ( b ) sl = δ x ( b ) kl for all 1 ≤ b ≤ d, ≤ k, l ≤ N b . Example 4.9.
Let B = M N ( C ) be equipped with the δ -form ψ corresponding tothe diagonal matrix Q with entries q , . . . , q N satisfying q + · · · + q N = 1. Moreoverlet A be the diagonal quantum adjacency matrix with coefficients A ijkl = x ij δ ik δ jl for some scalars x ij satisfying P s q − s x ks x sl = δ x kl . The quantum Cuntz-Kriegeralgebra F O ( G ) associated with the diagonal quantum graph G = ( B, ψ, A ) hasgenerators S ij for 1 ≤ i, j ≤ N satisfying the relations X kl S ik S ∗ lk S lj = S ij X k S ∗ ki S kj = X k x ij S ik S ∗ jk for all i, j .Consider the special case x = q δ and x ij = 0 else. From the second relationabove we get P i S ∗ ij S ij = 0 for j >
1, and hence S ij = 0 for all 1 ≤ i ≤ N and j >
1. This shows that the canonical linear map S : B → F O ( G ) in the definitionof a quantum Cuntz-Krieger algebra need not be injective.One may interpret this as a reflection of the fact that we work with rather generalquantum adjacency matrices. It would be interesting to identify a suitable conditionon directed quantum graphs G which ensures that the map S : B → F O ( G ) isinjective.Note also that we have P l S i S ∗ l S l = S i and P k S ∗ k S k = x S S ∗ in theabove special case. Hence for all complex numbers ǫ , . . . , ǫ N satisfying | ǫ | + · · · + | ǫ N | = 1 and x | ǫ | = 1 there exists a ∗ -homomorphism ǫ : F O ( G ) → C satisfying ǫ ( S ij ) = ( ǫ i j = 10 j > . It follows in particular that the C ∗ -algebra F O ( G ) admits a trace.4.5. Direct sums and tensor products of quantum graphs.
Assume that G = ( B , ψ , A ) and G = ( B , ψ , A ) are directed quantum graphs. We obtain afinite quantum space structure on the direct sum B ⊕ B by considering the state ψ = δ δ ψ ⊕ δ δ ψ , with δ = δ + δ . It is easy to check that A = A ⊕ A defines a quantum adjacencymatrix on ( B ⊕ B , ψ ), so that G ⊕ G = ( B ⊕ B , ψ, A ) is a directed quantum UANTUM CUNTZ-KRIEGER ALGEBRAS 17 graph. Classically, this construction corresponds to taking the disjoint union ofgraphs.
Proposition 4.10.
Let G = ( B , ψ , A ) and G = ( B , ψ , A ) be directed quan-tum graphs. Then F O ( G ⊕ G ) ∼ = F O ( G ) ∗ F O ( G ) is the non-unital free product of F O ( G ) and F O ( G ) .Proof. This follows directly from the universal properties of the algebras involved,noting that the quantum adjacency matrix A ⊕ A does not mix generators from B and B . (cid:3) We can also form tensor products in a natural way. If G = ( B , ψ , A ) and G = ( B , ψ , A ) are directed quantum graphs then ψ = ψ ⊗ ψ is a δ -form on thetensor product B ⊗ B with δ = δ δ . Moreover A = A ⊗ A defines a quantumadjacency matrix on ( B ⊗ B , ψ ). We let G ⊗ G be the corresponding directedquantum graph.Compared to the case of direct sums, it seems less obvious how to describe thestructure of F O ( G ⊗ G ) in terms of F O ( G ) and F O ( G ) in general. We shalldiscuss a special case in the next section.5. Amplification
In this section we study examples of quantum Cuntz-Krieger algebras obtainedfrom classical graphs by replacing the vertices with matrix blocks. This amplifi-cation procedure is a special case of the tensor product construction for quantumgraphs described in paragraph 4.5.Given a directed quantum graph G = ( B, ψ, A ) and N ∈ N we define the ampli-fication M N ( G ) of G to be the tensor product M N ( G ) = G ⊗
T M N , where T M N is the trivial quantum graph on M N ( C ) as defined in paragraph 4.3. Explicitly, M N ( G ) is the directed quantum graph with underlying C ∗ -algebra B ⊗ M N ( C ),state φ = ψ ⊗ tr, and quantum adjacency matrix A ( N ) = A ⊗ id.In the sequel we shall restrict ourselves to the case that G is associated with aclassical graph. Recall from paragraph 4.1 that if E = ( E , E ) is a simple finitedirected classical graph then the adjacency matrix B E of E induces canonically adirected quantum graph structure on C ( E ) with its unique δ -form. Lemma 5.1.
Let E = ( E , E ) be a simple finite directed classical graph and denoteby G = ( C ( E ) , tr , B E ) the directed quantum graph corresponding to E . Thenthe quantum Cuntz-Krieger algebra F O ( M N ( G )) associated with the amplification M N ( G ) is the universal C ∗ -algebra with generators S eij for e ∈ E and ≤ i, j ≤ N ,satisfying the relations X rs S eir S ∗ esr S esj = S eij X k S ∗ eki S ekj = X k X f ∈ E B E ( e, f ) S fik S ∗ fjk . Proof.
Consider the generators S eij = S ( f ( e ) ij ) in F O ( M N ( G )) associated with theadapted matrix units f ( e ) ij = nN δ e ⊗ e ij , where e ∈ E and n is the number ofvertices of E . Noting that the quantum adjacency matrix of M N ( G ) is given by A ( N ) ( f ( e ) ij ) = X f ∈ E B E ( e, f ) f ( f ) ij , the assertion is a direct consequence of Proposition 3.9. (cid:3) We will follow arguments of McClanahan [25] to study the structure of the quan-tum Cuntz-Krieger algebras in Lemma 5.1. As a first step we discuss a slightstrengthening of Theorem 2.3 in [25]. If A is a C ∗ -algebra we write A + for theunital C ∗ -algebra obtained by adjoining an identity element to A , and if A, B areunital C ∗ -algebras we denote by A ∗ B their unital free product. Proposition 5.2.
Let A be a separable C ∗ -algebra. Then M N ( C ) ∗ A + is KK -equivalent to A + .Proof. This fact is certainly known to experts, but we shall give the details for theconvenience of the reader.Note first that A + is KK -equivalent to the direct sum A ⊕ C . This equivalence isimplemented by taking the direct sum of the canonical ∗ -homomorphisms A → A + and C → A + at the level of KK -theory.We consider the unital ∗ -homomorphism φ : M N ( C ) ∗ A + → M N ( C ) ⊗ A + givenby φ ( e ij ) = e ij ⊗ , φ ( a ) = e ⊗ a, for 1 ≤ i, j ≤ N and a ∈ A , and view this as a class [ φ ] ∈ KK ( M N ( C ) ∗ A + , A + ).In the opposite direction we define a map ψ A : A → M N ( C ) ⊗ ( M N ( C ) ∗ A + )by ψ A ( a ) = X kl e kl ⊗ e k ae l . Then ψ A ( a ) ψ A ( b ) = X klrs e kl e rs ⊗ e k ae l e r be s = X kls e ks ⊗ e k ae l e l be s = X ks e ks ⊗ e k abe s = ψ A ( ab )and ψ A ( a ∗ ) = ψ A ( a ) ∗ , so that the map ψ A is a ∗ -homomorphism.Consider also the ∗ -homomorphism ψ C : C → M N ( C ) ⊗ ( M N ( C ) ∗ A + ) givenby ψ C (1) = e ⊗ e . Combining the maps ψ A and ψ C , and using that A + is KK -equivalent to A ⊕ C , we obtain a class in KK ( A + , M N ( C ) ∗ A + ), which weshall denote by [ ψ ].We claim that the classes [ φ ] and [ ψ ] are mutually inverse. In order to determinethe composition [ φ ] ◦ [ ψ ] ∈ KK ( A + , A + ) it suffices to compute M N ( φ ) ◦ ψ A and M N ( φ ) ◦ ψ C , respectively.We calculate( M N ( φ ) ◦ ψ A )( a ) = X kl e kl ⊗ φ ( e k ae l ) = X kl e kl ⊗ e k e e l ⊗ a = e ⊗ e ⊗ a for a ∈ A and ( M N ( φ ) ◦ ψ C )(1) = M N ( φ )( e ⊗ e ) = e ⊗ e ⊗
1. This immediatelyyields [ φ ] ◦ [ ψ ] = id.Next consider [ ψ ] ◦ [ φ ] ∈ KK ( M N ( C ) ∗ A + , M N ( C ) ∗ A + ). Let us write j A + : A + → M N ( C ) ∗ A + and j M N ( C ) : M N ( C ) → M N ( C ) ∗ A + for the canonicalinclusion homomorphisms. Moreover write u : C → M N ( C ) ⊕ A + for the unitmap. According to [20], [18], the suspension of M N ( C ) ∗ A + is KK -equivalentto the cone of u . In order to show [ ψ ] ◦ [ φ ] = id it therefore suffices to verify[ ψ ] ◦ [ φ ] ◦ [ j A + ] = [ j A + ] and [ ψ ] ◦ [ φ ] ◦ [ j M N ( C ) ] = [ j M N ( C ) ].We calculate( M N ( ψ A ) ◦ φ )( a ) = M N ( ψ )( e ⊗ a ) = X kl e ⊗ e kl ⊗ e k ae l for a ∈ A . Pick a continuous path of unitaries U t in M N ( C ) ⊗ M N ( C ) such that U = id and U ( e k ⊗ e ) = e ⊗ e k for all k , and push this into the last two tensor factors of M N ( C ) ⊗ M N ( C ) ⊗ ( M N ( C ) ∗ A + ) via the obvious map. Then conjugating ( M N ( ψ A ) ◦ φ )( a ) by U gives e ⊗ e ⊗ a for all a ∈ A . It follows that [ ψ ] ◦ [ φ ] ◦ [ j A ] = [ j A ], where we write j A for the restriction of j A + to A ⊂ A + .Next we calculate( M N ( ψ C ) ◦ φ )(1) = M N ( ψ C )(1 ⊗
1) = 1 ⊗ e ⊗ e . Conjugating this with the unitary U from above, pushed into the first and thirdtensor factors, gives e ⊗ e ⊗
1. Hence [ ψ ] ◦ [ φ ] ◦ [ j C ] = [ j C ], where j C denotes therestriction of j A + to C ⊂ A + . Combining these two observations gives [ ψ ] ◦ [ φ ] ◦ [ j A + ] = [ j A + ].Finally, we have( M N ( ψ C ) ◦ φ )( e ij ) = M N ( ψ C )( e ij ⊗
1) = e ij ⊗ e ⊗ e , so that conjugating ( M N ( ψ C ) ◦ φ )( e ij ) by U in the first and third tensor factorsgives e ⊗ e ⊗ e ij for all i, j . We conclude [ ψ ] ◦ [ φ ] ◦ [ j M N ( C ) ] = [ j M N ( C ) ], and thisfinishes the proof. (cid:3) With these preparations in place let us now present our main result on amplifiedquantum Cuntz-Krieger algebras.
Theorem 5.3.
Assume that E = ( E , E ) is a finite directed simple graph andlet G = ( C ( E ) , tr , B E ) be the corresponding directed quantum graph. Then thefollowing holds.a) We have an isomorphism M N ( F O ( M N ( G )) + ) ∼ = M N ( C ) ∗ ( F O ( G ) + ) .b) F O ( M N ( G )) is KK -equivalent to the classical Cuntz-Krieger algebra O B E .Proof. a ) In the sequel we shall write C = M N ( C ) ∗ ( F O ( G ) + ) and D = F O ( M N ( G )).We define a ∗ -homomorphism g : D → C by g ( S eij ) = X k e ki S e e jk on generators. To check that this is well-defined we use Lemma 5.1 to calculate X kl g ( S eik ) g ( S elk ) ∗ g ( S elj ) = X rstkl e ri S e e kr e sk S ∗ e e ls e tl S e e jt = X rkl e ri S e e kk S ∗ e e ll S e e jr = X r e ri S e S ∗ e S e e jr = X r e ri S e e jr = g ( S eij ) and X k g ( S eki ) ∗ g ( S ekj ) = X rsk e ri S ∗ e e kr e sk S e e js = X r e ri S ∗ e S e e jr = X r X f ∈ E B E ( e, f ) e ri S f S ∗ f e jr = X k X f ∈ E B E ( e, f ) g ( S fik ) g ( S fjk ) ∗ for e ∈ E and 1 ≤ i, j ≤ N .Let g + : D + → C be the unital extension of g . It is easy to see that the imageof g + is contained in the relative commutant M N ( C ) ′ of M N ( C ) inside the freeproduct. In fact, we have g ( S eij ) e kl = X r e ri S e e jr e kl = e ki S e e jl = X r e kl e ri S e e jr = e kl g ( S eij )for all i, j, k, l . We can thus extend g + to a unital ∗ -homomorphism G : M N ( D + ) → C by setting G ( e ij ) = e ij and G ( x ) = g ( x ) for x ∈ D + .Let us also define a unital ∗ -homomorphism F : C → M N ( D + ) = D + ⊗ M N ( C )by F ( e ij ) = 1 ⊗ e ij F ( S e ) = X ij S eij ⊗ e ij . To see that this is well-defined we only need to check that these formulas defineunital ∗ -homomorphisms from M N ( C ) and F O ( G ) + into M N ( D + ), respectively.For M N ( C ) this is obvious. For F O ( G ) + we need to check the free Cuntz-Kriegerrelations for the elements F ( S e ). In fact, each F ( S e ) is a partial isometry byconstruction, and using Lemma 5.1 we calculate F ( S e ) ∗ F ( S e ) = X ijkl S ∗ eij S ekl ⊗ e ji e kl = X ijl S ∗ eij S eil ⊗ e jl = X ijl X f ∈ E B E ( e, f ) S fji S ∗ fli ⊗ e jl = X f ∈ E B E ( e, f ) X ijkl S fji S ∗ flk ⊗ e ji e kl = X f ∈ E B E ( e, f ) F ( S f ) F ( S f ) ∗ as required.Next observe that F ◦ G : M N ( D + ) → M N ( D + ) satisfies( F ◦ G )( S eij ⊗
1) = X k F ( e ki ) F ( S e ) F ( e jk )= X krs (1 ⊗ e ki )( S ers ⊗ e rs )(1 ⊗ e jk ) = S eij ⊗ UANTUM CUNTZ-KRIEGER ALGEBRAS 21 for all e ∈ E and ( F ◦ G )( e ij ) = e ij for all i, j . This implies F ◦ G = id. Similarly,we have ( G ◦ F )( S e ) = X ij G ( S eij ⊗ e ij ) = X ij e ii S e e jj = S e for all e ∈ E , and ( G ◦ F )( e ij ) = e ij for all i, j . We conclude that G ◦ F = id. b ) Clearly M N ( F O ( M N ( G )) + ) is KK -equivalent to F O ( M N ( G )) + . Accordingto Proposition 5.2, we also know that M N ( C ) ∗ ( F O ( G ) + ) is KK -equivalent to F O ( G ) + . It is easy to check that these equivalences are both compatible with thecanonical augmentation morphisms to C . Hence F O ( M N ( G )) is KK -equivalentto F O ( G ). Finally, recall from Theorem 2.6 that the free Cuntz-Krieger algebra F O ( G ) = F O B E is KK -equivalent to O B E . (cid:3) Under some mild extra assumptions, Theorem 5.3 allows one to compute the K -theory of F O ( M N ( G )) in terms of the graph E , see [10] and chapter 7 in [29].Finally, remark that if E is the graph with one vertex and one self-loop thenwe have F O ( G ) = F O B E = C ( S ), and F O ( M N ( G )) = F O ( T M N ) is the quantumCuntz-Krieger algebra of the trivial quantum graph on M N ( C ). Therefore Theorem5.3 implies Theorem 4.7.6. Quantum symmetries of quantum Cuntz-Krieger algebras
In this section we study how quantum symmetries and quantum isomorphismsof directed quantum graphs induce symmetries of their associated quantum Cuntz-Krieger algebras. This will be useful in particular to exhibit relations between the C ∗ -algebras corresponding to quantum isomorphic quantum graphs.6.1. Gauge actions.
Before discussing quantum symmetries, let us first show thatthere is a canonical gauge action on quantum Cuntz-Krieger algebras, thus provid-ing very natural classical symmetries. This is analogous to the well-known gaugeaction on Cuntz-Krieger algebras and graph C ∗ -algebras, which plays a crucial rolein the analysis of the structure of these C ∗ -algebras, compare [29].Let G = ( B, ψ, A ) be a directed quantum graph, and let F O ( G ) be the corre-sponding quantum Cuntz-Krieger algebra. For λ ∈ U (1) consider the linear map S λ : B → F O ( G ) given by S λ ( b ) = λS ( b ) , where S : B → F O ( G ) is the canonical linear map. Then we have S ∗ λ ( b ) =( λS ( b ∗ )) ∗ = λS ∗ ( b ) for all b ∈ B , and using this relation it is easy to check that S λ : B → F O ( G ) is a quantum Cuntz-Krieger G -family. By the universal propertyof F O ( G ) we obtain a corresponding automorphism α λ ∈ Aut( F O ( G )), and theseautomorphisms combine to a strongly continuous action of U (1) on F O ( G ).In terms of the generators of F O ( G ) as in Proposition 3.9 the gauge action isgiven by α λ ( S ( a ) ij ) = λS ( a ) ij , from which it is easy to determine the action on arbitrary noncommutative poly-nomials in the generators, and the decomposition into spectral subspaces.In some cases one may define more general gauge type actions. For instance,for the complete quantum graph K ( M N ( C ) , tr) from paragraph 4.2 and the trivialquantum graph T M N from paragraph 4.3 we have an action of U (1) × U (1) N , givenby α λµ ( S ij ) = λ µ i µ j S ij on generators. In fact, one may even extend this to an action of U (1) × U ( N ) bysetting α λU ( S ) = λU SU ∗ , where S = ( S ij ) is the generating matrix partial isometry.However, none of the above actions seems to suffice to obtain structural informa-tion about quantum Cuntz-Krieger algebras in the same way as for classical graphalgebras. In particular, the corresponding fixed point algebras tend to have a morecomplicated structure than in the classical setting.It turns out that this deficiency can be compensated to some extent by consider-ing actions of compact quantum groups instead, and in particular symmetries aris-ing from suitable monoidal equivalences between quantum automorphism groupsof directed quantum graphs. We will explain these constructions in the followingparagraphs.6.2. Compact quantum groups.
Let us first give a quick review of the definitionof compact quantum groups and their action on C ∗ -algebras. For more backgroundand further information we refer to [36], [28].A compact quantum group G is given by a unital C ∗ -algebra C ( G ) togetherwith a unital ∗ -homomorphism ∆ : C ( G ) → C ( G ) ⊗ C ( G ) such that (∆ ⊗ id)∆ =(id ⊗ ∆)∆ and the density conditions[∆( C ( G ))( C ( G ) ⊗ C ( G ) ⊗ C ( G ) = [∆( C ( G ))(1 ⊗ C ( G ))]hold.We will mainly work with the canonical dense Hopf ∗ -algebra O ( G ) ⊂ C ( G ),consisting of the matrix coefficients of all finite dimensional unitary representationsof G . For the definition of unitary representations and their intertwiners see [28].The collection of all finite dimensional unitary representations of G forms naturallya C ∗ -tensor category Rep ( G ).On the C ∗ -level we will only consider the universal completions of O ( G ) in thesequel, and always denote them by C ( G ). With this in mind, a morphism H → G of compact quantum groups is nothing but a ∗ -homomorphism C ( G ) → C ( H )compatible with the comultiplications. Equivalently, such a morphism is given by ahomomorphism O ( G ) → O ( H ) of Hopf ∗ -algebras. One says that H is a quantumsubgroup of G if there exists a morphism H → G such that the correspondinghomomorphism of Hopf ∗ -algebras O ( G ) → O ( H ) is surjective.By definition, an action of a compact quantum group G on a C ∗ -algebra A is a ∗ -homomorphism α : A → A ⊗ C ( G ) satisfying ( α ⊗ id) α = (id ⊗ ∆) α andthe density condition [(1 ⊗ C ( G )) α ( A )] = A ⊗ C ( G ). A C ∗ -algebra A equippedwith an action of G will also be called a G - C ∗ -algebra. Every G - C ∗ -algebra A contains a canonical dense ∗ -subalgebra A ⊂ A , given by the algebraic direct sumof the spectral subspaces of the action. Moreover, the map α restricts to a ∗ -homomorphism α : A → A ⊗ O ( G ), and this defines an algebra coaction in thesense of Hopf algebras. In particular, one has (id ⊗ ǫ ) α ( a ) = a for all a ∈ A , where ǫ : O ( G ) → C is the counit.If A is a G - C ∗ -algebra then the fixed point subalgebra of A is defined by A G = { a ∈ A | α ( a ) = a ⊗ } , and a unital G - C ∗ -algebra A is called ergodic if A G = C
1. The same terminologyis also used for ∗ -algebras equipped with algebra coactions of O ( G ).Let us now review the definition of quantum automorphism groups of finite quan-tum spaces in the sense of Definition 3.1. These quantum groups were introduced byWang [33] and studied further by Banica [2] and others. If G is a compact quantum UANTUM CUNTZ-KRIEGER ALGEBRAS 23 group and ω : A → C a state on a G - C ∗ -algebra A with action α : A → A ⊗ C ( G ),then we say that the action preserves ω if( ω ⊗ id) α ( a ) = ω ( a )1for all a ∈ A . Definition 6.1.
Let (
B, ψ ) be a finite quantum space. The quantum automorphismgroup of (
B, ψ ) is the universal compact quantum group G + ( B, ψ ) equipped withan action β : B → B ⊗ C ( G + ( B, ψ )) which preserves ψ .In other words, if G is a compact quantum group and γ : B → B ⊗ C ( G ) an actionof G preserving ψ , then there exists a unique ∗ -homomorphism π : C ( G + ( B, ψ )) → C ( G ), compatible with the comultiplications, such that the diagram B β / / γ & & ▼▼▼▼▼▼▼▼▼▼▼▼ B ⊗ C ( G + ( B, ψ )) id ⊗ π (cid:15) (cid:15) B ⊗ C ( G )is commutative.The most prominent example of a quantum automorphism group is the quantumpermutation group S + N . This is the quantum automorphism group of B = C N withits unique δ -form. The corresponding C ∗ -algebra C ( S + N ) = C ( G + ( C N , tr)) is theuniversal C ∗ -algebra generated by projections u ij for 1 ≤ i, j ≤ N such that X k u ik = 1 = X k u kj for all i, j . These relations can be phrased by saying that the matrix u = ( u ij ) is amagic unitary. The comultiplication ∆ : C ( S + N ) → C ( S + N ) ⊗ C ( S + N ) is defined by∆( u ij ) = n X k =1 u ik ⊗ u kj on generators. Remark 6.2.
Quantum automorphism groups can always be described explicitlyin terms of generators and relations, see Proposition 2.10 in [26]. More precisely,let us assume that (
B, ψ ) is a finite quantum space in standard form as in section3.1, so that B = d M a =1 M N a ( C ) , ψ ( x ) = d X a =1 Tr( Q ( a ) x a )for x = ( x , . . . , x d ) ∈ B . Then the Hopf ∗ -algebra O ( G + ( B, ψ )) is generated byelements v rsbija for 1 ≤ a, b ≤ d and 1 ≤ i, j ≤ N a , ≤ r, s ≤ N b , satisfying therelations(A1a) P w v xwckla v wycrsb = δ ab δ lr v xycksa (A1b) P w ( Q ( c ) ) − ww v srbywc v lkawxc = δ lr δ ab ( Q ( a ) ) − ll v skayxc (A2) ( v xyckla ) ∗ = v yxclka (A3a) P xb ( Q ( b ) ) xx v xxbkla = δ kl ( Q ( a ) ) kk (A3b) P ka v xybkka = δ xy .In terms of the standard matrix units e ( a ) ij for B and the generators v rsbija , thedefining action β : B → B ⊗ O ( G + ( B, ψ )) is given by β ( e ( a ) ij ) = X bkl e ( b ) kl ⊗ v klbija , and the matrix v = ( v rsbija ) is also called the fundamental matrix of G + ( B, ψ ).We will reobtain the above description of the ∗ -algebra O ( G + ( B, ψ )) as a specialcase of Proposition 6.10 below.6.3.
Quantum symmetries of quantum graphs.
In this paragraph we discussthe quantum automorphism group of a directed quantum graph, and also quantumisomorphisms relating a pair of directed quantum graphs.Recall first that if E = ( E , E ) is a simple finite graph then the automorphismgroup Aut( E ) consists of all bijections of E which preserves the adjacency relationin E . If | E | = N and A ∈ M N ( Z ) is the adjacency matrix of E , then this can beexpressed as Aut( E ) = { σ ∈ S N | σA = Aσ } ⊂ S N , where one views elements of the symmetric group as permutation matrices. In [4],Banica defined the quantum automorphism group G + ( E ) of the graph E via the C ∗ -algebra C ( G + ( E )) = C ( S + N ) / h uA = Au i , where u = ( u ij ) ∈ M N ( C ( S + N )) denotes the defining magic unitary matrix. Thisyields a quantum subgroup of S + N , which contains the classical automorphism groupAut( E ) as a quantum subgroup.If G = ( B, ψ, A ) is a directed quantum graph we shall say that an action β : B → B ⊗ C ( G ) of a compact quantum group G is compatible with A : B → B if β ◦ A = ( A ⊗ id) ◦ β . Motivated by the considerations in [4], we define the quantumautomorphism group of a directed quantum graph as follows, compare [7]. Definition 6.3.
Let G = ( B, ψ, A ) be a directed quantum graph. The quantumautomorphism group G + ( G ) of G is the universal compact quantum group equippedwith a ψ -preserving action β : B → B ⊗ C ( G + ( G )) which is compatible with thequantum adjacency matrix A .That is, if G is a compact quantum group and γ : B → B ⊗ C ( G ) an actionof G which preserves ψ and is compatible with A , then there exists a unique ∗ -homomorphism π : C ( G + ( G )) → C ( G ), compatible with the comultiplications,such that the diagram B β / / γ % % ▲▲▲▲▲▲▲▲▲▲▲ B ⊗ C ( G + ( G )) id ⊗ π (cid:15) (cid:15) B ⊗ C ( G )is commutative.Comparing this with Definition 6.1, it is straightforward to check that C ( G + ( G ))can be identified with the quotient of C ( G + ( B, ψ )) obtained by imposing the rela-tion (1 ⊗ A ) v = v (1 ⊗ A ) on the fundamental matrix v of G + ( B, ψ ). Remark 6.4. If G = K ( M N ( C ) , tr) or G = T M N is the complete or the trivialquantum graph on M N ( C ), then it is easy to see that compatibility with the quan-tum adjacency matrix is in fact automatic. That is, we have G + ( G ) = G + ( M N ( C ) , tr)in either case.Let us recall that two compact quantum groups G , G are called monoidallyequivalent if their representation categories Rep ( G ) and Rep ( G ) are unitarilyequivalent as C ∗ -tensor categories [6], [28]. A monoidal equivalence is a unitarytensor functor F : Rep ( G + ( G )) → Rep ( G + ( G )) whose underlying functor is anequivalence.Assume that G i = ( B i , ψ i , A i ) are directed quantum graphs for i = 1 ,
2. Thenthe quantum automorphism group G + ( G i ) is a quantum subgroup of G + ( B i , ψ i ) UANTUM CUNTZ-KRIEGER ALGEBRAS 25 such that the quantum adjacency matrix A i is an intertwiner for the definingrepresentation B i = L ( B i ) of G + ( G i ). Note also that the multiplication map m i : B i ⊗ B i → B i and the unit map u i : C → B i are intertwiners for the action of G + ( G i ), so that B i becomes a monoid object in the tensor category Rep ( G + ( G i )).In analogy to [7] we give the following definition. Definition 6.5.
Two directed quantum graphs G i = ( B i , ψ i , A i ) for i = 1 , F : Rep ( G + ( G )) → Rep ( G + ( G )) such thata) F maps the monoid object B to the monoid object B .b) F ( A ) = A .We will write G ∼ = q G in this case.From Definition 6.5 it is easy to see that the notion of quantum isomorphismis an equivalence relation on isomorphism classes of directed quantum graphs. Forconcrete computations it is however more convenient to describe quantum isomor-phisms in terms of bi-Galois objects [7], sometimes also called linking algebras.Concretely, if G i = ( B i , ψ i , A i ) for i = 1 , O ( G + ( G , G )) is the bi-Galois object generated by the coefficients of a unital ∗ -homomorphism β ji : B i → B j ⊗ O ( G + ( G j , G i ))satisfying the conditions ( ψ j ⊗ id) β ji ( x ) = ψ i ( x )1for all x ∈ B i and ( A j ⊗ id) β ji = β ji A i . Note that these conditions generalize the requirements on the action of the quantumautomorphism group of a quantum graph to be state-preserving and compatiblewith the quantum adjacency matrix, respectively.We write C ( G + ( G j , G i )) for the universal enveloping C ∗ -algebra of O ( G + ( G j , G i )).In exactly the same way as in [7] one then arrives at the following characterizationof quantum isomorphisms. Theorem 6.6.
Let G , G be directed quantum graphs. Then the following condi-tions are equivalent.a) G and G are quantum isomorphic.b) O ( G + ( G , G )) is non-zero.c) O ( G + ( G , G )) admits a nonzero faithful state.d) C ( G + ( G , G )) is non-zero. If the equivalent conditions in Theorem 6.6 are satisfied then O ( G + ( G , G ))is a O ( G + ( G ))- O ( G + ( G )) bi-Galois object in a natural way [30]. In particular,there exist ergodic left and right actions of G + ( G ) and G + ( G ) on O ( G + ( G , G )),respectively. Moreover, O ( G + ( G , G )) is equipped with a unique faithful statewhich is invariant with respect to both actions.For G = G and the identity monoidal equivalence, the ∗ -algebra O ( G + ( G , G ))equals O ( G + ( G )) = O ( G + ( G )), both actions are implemented by the comultipli-cation, and the invariant faithful state is nothing but the Haar state. Remark 6.7.
The abelianization of O ( G + ( G , G )) is the algebra of coordinatefunctions on the space of “classical isomorphisms” between the quantum graphs G and G , that is, the space of unital ∗ -isomorphisms ϕ : B → B satisfying ψ ◦ ϕ = ψ , A ◦ ϕ = ϕ ◦ A . If moreover each G i is associated with a classical directed graph E i = ( E i , E i ) asin paragraph 4.1, then by Gelfand duality such a map ϕ corresponds precisely to a graph isomorphism ϕ ∗ : E → E via ϕ ( f ) = f ◦ ϕ ∗ for f ∈ C ( E ). This is thereason for the ordering of the quantum graphs in our notation O ( G + ( G , G )). Remark 6.8.
There is a canonical algebra isomorphism S : O ( G + ( G , G )) →O ( G + ( G , G )) op , which can be viewed as a generalization of the antipode of theHopf ∗ -algebra associated to a compact quantum group. More precisely, if ( e m ) and( f n ) are orthonormal bases for B and B , respectively, and we write β ( e m ) = P n f n ⊗ u nm , then u = ( u ij ) ∈ End( B , B ) ⊗ O ( G + ( G , G )) is a unitary matrix,and there is an algebra isomorphism S : O ( G + ( G , G )) → O ( G + ( G , G )) op givenby (id ⊗ S )( u ) = u ∗ = u − , (id ⊗ S )( u ∗ ) = ( J t ¯ u ( J − ) t ) , where (¯ u ) kl = ( u ∗ kl ) , J i : B i → B i is the anti-linear involution map given by J i ( b ) = b ∗ and t denotes the transpose map. We refer to [7] for more details. Remark 6.9.
We have a wealth of examples quantum isomorphisms between thecomplete quantum graphs K ( B, ψ ) introduced in paragraph 4.2, and also betweenthe trivial quantum graphs T ( B, ψ ) introduced in paragraph 4.3. Recall that K ( B, ψ ) (resp. T ( B, ψ )) is defined by equipping the finite quantum space (
B, ψ )with the quantum adjacency matrix A : L ( B ) → L ( B ) given by A ( b ) = δ ψ ( b )1(resp. A = id). More precisely, if ( B i , ψ i ) are finite quantum spaces for i = 1 , δ i -forms ψ i , then K ( B , ψ ) ∼ = q K ( B , ψ ) ⇐⇒ T ( B , ψ ) ∼ = q T ( B , ψ ) ⇐⇒ δ = δ . These equivalences follow from work of DeRijdt and Vander Vennet in [13], whereunitary fiber functors on quantum automophism groups of finite quantum spacesequipped with δ -forms were classified.Let G i = ( B i , ψ i , A i ) be directed quantum graphs in standard form, in the senseexplained in paragraph 3.1. Explicitly, we fix multimatrix decompositions B i = d i M a =1 M N ia ( C )and diagonal positive invertible matrices Q i ( a ) implementing ψ i . Let us express thequantum adjacency matrices relative to the standard matrix units e ( a ) kl ∈ M N ia ( C ),so that A i ( e ( a ) kl ) = X brs ( A i ) rsbkla e ( b ) rs . We then obtain the following result, compare [26] for the case G = G . Proposition 6.10.
Let G and G be directed quantum graphs given as above. Then O ( G + ( G , G )) is the universal unital ∗ -algebra with generators v klbija for ≤ i, j ≤ N a , ≤ k, l ≤ N b , ≤ a ≤ d , ≤ b ≤ d , satisfying the relations (A1a) P w v xwckla v wycrsb = δ ab δ lr v xycksa (A1b) P l ( Q a ) ) − ll v xwbmla v zyclka = δ bc δ wz ( Q c ) ) − zz v xycmka (A2) ( v xybkla ) ∗ = v yxblka (A3a) P xb ( Q b ) ) xx v xxbkla = δ kl ( Q a ) ) kk (A3b) P ka v xybkka = δ xy (A4) P rsb ( A ) xycrsb v rsbkla = P rsb ( A ) rsbkla v xycrsb for all admissible indices.Proof. The following argument is analogous to the one for Proposition 2.10 in [26],compare [33]. Expressing the universal morphism β : B → B ⊗ O ( G + ( G , G )) UANTUM CUNTZ-KRIEGER ALGEBRAS 27 relative to the bases chosen as above, we can write β ( e ( a ) kl ) = X xyb e ( b ) xy ⊗ v xybkla . Then O ( G + ( G , G )) is generated as a ∗ -algebra by the elements v xybkla . Now theconditions on this implementing a bi-Galois object are equivalent to the equationslisted above. More precisely, we have • ( A a ) ⇐⇒ β is an algebra homomorphism . This follows from β ( e ( a ) kl ) β ( e ( b ) rs ) = X xwcmyd e ( c ) xw e ( d ) my ⊗ v xwckla v mydrsb = X xwcn e ( c ) xy ⊗ v xwckla v wycrsb and β ( e ( a ) kl e ( b ) rs ) = δ ab δ lr β ji ( e ( a ) ks ) = X xyc δ ab δ lr e ( c ) xy ⊗ v xycksa . • ( A b ) ⇐⇒ S : O ( G + ( G , G )) → O ( G + ( G , G )) given by S ( v klarsb ) = ( Q b ) ) ss ( Q a ) ) − ll v srblka defines an algebra anti-isomorphism . Indeed, we have X l S ( v lmawxb ) S ( v klayzc ) = X l ( Q b ) ) xx ( Q a ) ) − mm v xwbmla ( Q c ) ) zz ( Q a ) ) − ll v zyclka and δ bc δ wz S ( v kmayxc ) = δ bc δ wz ( Q c ) ) xx ( Q a ) ) − mm v xycmka , so this statement follows in combination with ( A a ). • ( A ⇐⇒ β is involutive . This follows immediately from ( e ( a ) kl ) ∗ = e ( a ) lk . • ( A a ) ⇐⇒ ( ψ ⊗ id) ◦ β ( b ) = ψ ( b )1 for all b ∈ B . This follows from( ψ ⊗ id) ◦ β ( e ( a ) kl ) = X xyb ψ ( e ( b ) xy ) v xybkla = X xb ( Q b ) ) xx v xxbkla and ψ ( e ( a ) kl )1 = ( Q a ) ) kk δ kl . • ( A b ) ⇐⇒ β is unital . This follows from β (1) = X ak β ( e ( a ) kk ) = X xybak e ( b ) xy ⊗ v xybkka . • ( A ⇐⇒ β ◦ A = ( A ⊗ id) ◦ β . This follows from( β ◦ A )( e ( a ) kl ) = X rsb ( A ) rsbkla β ( e ( b ) rs ) = X rsbxyc ( A ) rsbkla e ( c ) xy ⊗ v xycrsb and( A ⊗ id) ◦ β ( e ( a ) kl ) = X rsb A ( e ( b ) rs ) ⊗ v rsbkla = X rsbxyc ( A ) xycrsb e ( c ) xy ⊗ v rsbkla . Combining these observations yields the claim. (cid:3)
Quantum symmetries of quantum Cuntz-Krieger algebras.
We shallnow show that quantum automorphisms and quantum isomorphisms of directedquantum graphs lift naturally to the level of their associated C ∗ -algebras.Firstly, we have the following lifting result for quantum symmetries, comparethe work in [31] on classical graph C ∗ -algebras. Theorem 6.11.
Let G = ( B, ψ, A ) be a directed quantum graph. Then the canonicalaction β : B → B ⊗ C ( G + ( G )) of the quantum automorphism group of G inducesan action ˆ β : F O ( G ) → F O ( G ) ⊗ C ( G + ( G )) such that ˆ β ( S ( b )) = ( S ⊗ id) β ( b ) for all b ∈ B . The proof of Theorem 6.11 will be obtained as a special case of the more generalTheorem 6.13 on quantum isomorphisms below. Nonetheless, for the sake of claritywe have decided to state this important special case separately.
Remark 6.12.
There are typically plenty of quantum automorphisms of F O ( G ),and in fact, even ∗ -automorphisms, which do not arise from quantum automor-phisms as in Theorem 6.11. For instance, the gauge action on the free Cuntz-Krieger algebra associated with a classical directed graph cannot be described thisway, compare paragraph 6.1.Now assume that G , G are quantum isomorphic directed quantum graphs instandard form, with corresponding linking algebras O ( G + ( G j , G i )). The associated ∗ -homomorphisms β ji : B i → B j ⊗ O ( G + ( G j , G i )) for 1 ≤ i, j ≤ β ji ( e ( a ) kl ) = X xyb e ( b ) xy ⊗ v xybkla in terms of the standard matrix units. Here v xybkla are the generators of O ( G + ( G j , G i ))as in Proposition 6.10. Theorem 6.13.
Let G i = ( B i , ψ i , A i ) for i = 1 , be directed quantum graphs andassume that G ∼ = q G . Then there exists ∗ -homomorphisms ˆ β ji : F O ( G i ) → F O ( G j ) ⊗ C ( G + ( G j , G i )) for ≤ i, j ≤ such that ˆ β ji ( S i ( b )) = ( S j ⊗ id) β ji ( b ) for all b ∈ B i .Proof. Observe first that for i = j we are precisely in the situation of Theorem6.11, so that Theorem 6.11 is indeed a special case of the claim at hand.Let us write m O : O ⊗ O → O for the multiplication in O = O ( G + ( G j , G i )). Weclaim that ( m ∗ j ⊗ id) β ji = (id ⊗ id ⊗ m O )(id ⊗ σ ⊗ id)( β ji ⊗ β ji ) m ∗ i , where m i : B i → B i → B i denotes multiplication in B i and σ is the flip map.Indeed, rewriting Lemma 3.2 in terms of the standard matrix units yields m ∗ i ( e ( a ) kl ) = X r ( Q i ( a ) ) − rr e ( a ) kr ⊗ e ( a ) rl , UANTUM CUNTZ-KRIEGER ALGEBRAS 29 and using relation ( A b ) from Proposition 6.10 we get( m ∗ j ⊗ id) β ji ( e ( a ) kl ) = X xnb m ∗ j ( e ( b ) xn ) ⊗ v xnbkla = X xybn ( Q j ( b ) ) − yy e ( b ) xy ⊗ e ( b ) yn ⊗ v xnbkla = X xybmnc ( Q j ( b ) ) − yy δ bc δ ym e ( b ) xy ⊗ e ( c ) mn ⊗ v xnbkla = X wxybmnc ( Q i ( a ) ) − ww e ( b ) xy ⊗ e ( c ) mn ⊗ v xybkwa v mncwla = X w ( Q i ( a ) ) − ww (id ⊗ id ⊗ m O )(id ⊗ σ ⊗ id)( β ji ⊗ β ji )( e ( a ) kw ⊗ e ( a ) wl )= (id ⊗ id ⊗ m O )(id ⊗ σ ⊗ id)( β ji ⊗ β ji ) m ∗ i ( e ( a ) kl )as required.Now consider the linear map s : B i → F O ( G j ) ⊗ C ( G + ( G j , G i )) = D given by s = ( S j ⊗ id) β ji . Then s ∗ ( b ) = s ( b ∗ ) ∗ = ( S j ⊗ id) β ji ( b ∗ ) ∗ = ( S ∗ j ⊗ id) β ji ( b ) , and we claim that s is a quantum Cuntz-Krieger G i -family in D . Writing µ for themultiplication in F O ( G j ) and µ D for the one in D , our above considerations yield µ D (id ⊗ µ D )( s ⊗ s ∗ ⊗ s )(id ⊗ m ∗ i ) m ∗ i = µ D (id ⊗ µ D )( S j ⊗ id ⊗ S ∗ j ⊗ id ⊗ S j ⊗ id)( β ji ⊗ β ji ⊗ β ji )(id ⊗ m ∗ i ) m ∗ i = µ D (id ⊗ id ⊗ µ ⊗ id)( S j ⊗ id ⊗ S ∗ j ⊗ S j ⊗ m O ) σ ( β ji ⊗ β ji ⊗ β ji )(id ⊗ m ∗ i ) m ∗ i = µ D (id ⊗ id ⊗ µ ⊗ id)( S j ⊗ id ⊗ S ∗ j ⊗ S j ⊗ id)(id ⊗ id ⊗ m ∗ j ⊗ id)( β ji ⊗ β ji ) m ∗ i = ( µ ⊗ id)(id ⊗ µ ⊗ id)( S j ⊗ S ∗ j ⊗ S j ⊗ m O )(id ⊗ m ∗ j ⊗ id)(id ⊗ σ ⊗ id)( β ji ⊗ β ji ) m ∗ i = ( µ ⊗ id)(id ⊗ µ ⊗ id)( S j ⊗ S ∗ j ⊗ S j ⊗ id)(id ⊗ m ∗ j ⊗ id)( m ∗ j ⊗ id) β ji = ( S j ⊗ id) β ji = s, and similarly µ D ( s ∗ ⊗ s ) m ∗ i = ( µ ⊗ m O ) σ ( S ∗ j ⊗ id ⊗ S j ⊗ id)( β ji ⊗ β ji ) m ∗ i = ( µ ⊗ id)( S ∗ j ⊗ S j ⊗ m O ) σ ( β ji ⊗ β ji ) m ∗ i = ( µ ⊗ id)( S ∗ j ⊗ S j ⊗ id)( m ∗ j ⊗ id) β ji = ( µ ⊗ id)( S j ⊗ S ∗ j ⊗ id)( m ∗ j ⊗ id)( A j ⊗ id) β ji = ( µ ⊗ id)( S j ⊗ S ∗ j ⊗ id)( m ∗ j ⊗ id) β ji A i = µ D ( s ⊗ s ∗ ) m ∗ i A i , using the quantum Cuntz-Krieger relation for S j . Hence the universal property of F O ( G i ) yields the claim. (cid:3) Remark 6.14.
If we denote by C i ⊂ F O ( G i ) the dense ∗ -subalgebra generatedby S i ( B i ), then the restriction of the map ˆ β ji in Theorem 6.13 to C i is injective.Indeed, there exists a canonical unital ∗ -isomorphism θ ji : O ( G + ( G i )) → O ( G + ( G i , G j )) (cid:3) O ( G j ) O ( G + ( G j , G i )) , where O ( G + ( G i , G j )) (cid:3) O ( G j ) O ( G + ( G j , G i )) = { x | ( ρ j ⊗ id)( x ) = (id ⊗ λ j )( x ) }⊂ O ( G + ( G i , G j )) ⊗ O ( G + ( G j , G i )) , and ρ j : O ( G + ( G i , G j )) → O ( G + ( G i , G j )) ⊗ O ( G + ( G j )) λ j : O ( G + ( G j , G i )) → O ( G + ( G j )) ⊗ O ( G + ( G j , G i ))are the canonical ergodic actions of G + ( G j ) on the linking algebras. The map θ ji satisfies ( ˆ β ij ⊗ id) ˆ β ji ( x ) = (id ⊗ θ ji ) ˆ β ii ( x )for all x ∈ C i . If ǫ i : O ( G + ( G i , G j )) (cid:3) O ( G j ) O ( G + ( G j , G i )) ∼ = O ( G + ( G i )) → C is thecharacter given by the counit of O ( G + ( G i )), this implies(id ⊗ ǫ i )( ˆ β ij ⊗ id) ˆ β ji ( x ) = x for x ∈ C i . Hence the restriction of ˆ β ji to C i is indeed injective.However, it is not clear whether the map ˆ β ji : F O ( G i ) → F O ( G j ) ⊗ C ( G + ( G j , G i ))itself is injective. In the following section we show that this is at least sometimesthe case.7. Unitary error bases and finite dimensional quantum symmetries
In this section we apply the general results of the previous section to certainpairs of complete quantum graphs and trivial quantum graphs, respectively. Moreprecisely, we fix N ∈ N and consider G K ( N ) = K N = K ( C N , tr) G K ( N ) = K ( M N ( C ) , tr)and G T ( N ) = T N = T ( C N , tr) G T ( N ) = T ( M N ( C ) , tr) = T M N , compare section 4. The similarity between these pairs stems from the fact that wehave canonical identifications G + ( G K ( N )) = G + ( G T ( N )) = S + N ,G + ( G K ( N )) = G + ( G T ( N )) = G + ( M N ( C ) , tr) , respectively. We will therefore also use the short hand notation G + ( G ( N )) and G + ( G ( N )) for these quantum automorphism groups.We recall that G + ( G ( N )) and G + ( G ( N )) are monoidally equivalent, and thatwe have quantum isomorphisms G K ( N ) ∼ = q G K ( N ) and G T ( N ) ∼ = q G T ( N ), see theremarks in paragraph 6.3. This means in particular that there exists a bi-Galoisobject O ( G + ( G ( N ) , G ( N ))) linking G + ( G ( N )) and G + ( G ( N )). If X is a set ofcardinality N , then this ∗ -algebra can be described in terms of generators v rsx with1 ≤ r, s ≤ N and x ∈ X , satisfying the relations as in Proposition 6.10.7.1. Representations from unitary error bases.
With some inspiration fromquantum information theory, we shall now construct unital ∗ -homomorphisms fromthe linking algebra O ( G + ( G ( N ) , G ( N ))) to M N ( C ). The key tool in this con-struction is the notion of a unitary error basis [35]. Definition 7.1.
Let N ∈ N and let X be a finite set of cardinality N . A unitaryerror basis for M N ( C ) is a basis W = { w x } x ∈ X for M N ( C ) consisting of unitarymatrices that are orthonormal with respect to the normalized trace inner product,so that tr( w ∗ x w y ) = δ xy for all x, y ∈ X . UANTUM CUNTZ-KRIEGER ALGEBRAS 31
Unitary error bases play a fundamental role in quantum information theory. Inparticular, they form a one-to-one correspondence with “tight” quantum teleporta-tion and superdense coding schemes [35].
Proposition 7.2.
Let N ∈ N and assume that W = { w x } x ∈ X is a unitary errorbasis for M N ( C ) . With the notation as above, there exists a unital ∗ -representation π W : O ( G + ( G ( N ) , G ( N ))) → M N ( C ) such that π W ( v rsx ) = 1 N w ∗ x e rs w x for all r, s, x .Proof. Recalling that we write e rs ∈ M N ( C ) for the standard matrix units, let usdefine V rsx = 1 N w ∗ x e rs w x for all 1 ≤ r, s ≤ N and x ∈ X . It suffices to check that the elements V rsx ∈ M N ( C )satisfy the relations in Proposition 6.10.In order to do this, we recall from Theorem 1 in [35] that a unitary error basis canbe equivalently characterized by the following properties for a family of unitaries W = { w x } x ∈ X ⊂ M N ( C ):a) ( Depolarizing property ): P x ∈ X w ∗ x aw x = N Tr( a )1 for a ∈ M N ( C ).b) ( Maximally entangled basis property ): If Ω = √ N P Ni =1 e i ⊗ e i ∈ C N ⊗ C N is amaximally entangled state and Ω x = ( w x ⊗ { Ω x } x ∈ X is an orthonormalbasis for C N ⊗ C N .Observing that Q = N − id and Q = N − id we therefore we have to verifythe following relations: • ( A a ) ⇐⇒ P w V rwx V wsy = δ xy V rsx . This follows from X t V rtx V tsy = N − X t w ∗ x e rt w x w ∗ y e ts w y = N − Tr( w x w ∗ y ) w ∗ x ( e rs ) w y = δ xy N − w ∗ x ( e rs ) w x = δ xy V rsx . • ( A b ) ⇐⇒ V jix V srx = δ is N − V jrx . This follows directly from( w ∗ x e ji w x )( w ∗ x e sr w x ) = δ is w ∗ x e jr w x . • ( A ⇐⇒ ( V ijx ) ∗ = V jix . This is immediate. • ( A a ) ⇐⇒ P i N V iix = 1. This follows from X i N V iix = X i w ∗ x e ii w x = w ∗ x w x = 1 . • ( A b ) ⇐⇒ P z V ijz = δ ij
1. This is the depolarizing property of W . • ( A ⇐⇒ P rs ( A ) ijrs V rsx = P y ( A ) yx V ijy . For the trivial quantum graphsthis is obvious. In the case of complete quantum graphs we have ( A ) xy =1 , ( A ) ijkl = N δ ij δ kl for all x, y, i, j, k, l . Combining this with relations (A3a)and (A3b) yields the claim. More precisely, using (A3a) we obtain(A4) ⇐⇒ δ ij X s N V ssx = X y V ijy ⇐⇒ δ ij X y V ijy ⇐⇒ (A3b)as required.This completes the proof. (cid:3) Remark 7.3.
It is easy to construct examples of unitary error bases. Let
X, Z ∈ M N ( C ) be the generalized Pauli matrices given by their action on the standardbasis | i , . . . , | N − i of C N according to the formulas X | j i = ω j | j i , Z | j i = | j + 1 i , where we write ω = e πiN and calculate modulo N . Then W = { X j Z k } ≤ j,k ≤ N − is a unitary error basis for M N ( C ).7.2. Applications.
We shall use Proposition 7.2 to study the structure of thequantum Cuntz-Krieger algebras F O ( G K ( N )) = F O ( K ( M N , tr)) and F O ( G T ( N )) = F O ( T M N ), by comparing them with F O ( G K ( N )) and F O ( G T ( N )), respectively.Recall from paragraph 4.2 that F O ( G K ( N )) identifies canonically with the Cuntzalgebra O N . Note also that F O ( G T ( N )) = ∗ N C ( S ) is the non-unital free productof N copies of C ( S ), compare Proposition 4.10. Proposition 7.4.
There are injective ∗ -homomorphisms π KN : O N ֒ → M N ( F O ( K ( M N , tr))) σ KN : F O ( K ( M N , tr)) ֒ → M N ( O N ) and π TN : ∗ N C ( S ) ֒ → M N ( F O ( T M N )) σ TN : F O ( T M N ) ֒ → M N ( ∗ N C ( S )) for all N ∈ N .Proof. The construction of these maps for trivial quantum graphs is virtually iden-tical to the one for complete quantum graphs. In order to treat both cases simul-taneously we will therefore write G ( N ) and G ( N ) to denote either G K ( N ) and G K ( N ), or G T ( N ) and G T ( N ), respectively. Our task is then to define injective ∗ -homomorphisms π N : F O ( G ( N )) ֒ → M N ( F O ( G ( N ))) σ N : F O ( G ( N )) ֒ → M N ( F O ( G ( N )))for N ∈ N .Since the quantum graphs G ( N ) and G ( N ) are quantum isomorphic, Theorem6.13 yields natural ∗ -homomorphismsˆ β : F O ( G ( N )) → F O ( G ( N )) ⊗ C ( G + ( G ( N ) , G ( N )))ˆ γ : F O ( G ( N )) → F O ( G ( N )) ⊗ C ( G + ( G ( N ) , G ( N ))) op , taking into account that C ( G + ( G ( N ) , G ( N ))) ∼ = C ( G + ( G ( N ) , G ( N ))) op .In order to give explicit formulas for these maps let X be a set of cardinality N and denote the standard generators of F O ( G ( N )) by S x = S ( N e x ) for x ∈ X . Similarly, write S rs = S ( N e rs ) for the standard generators of F O ( G ( N )) = F O ( G ). Here S : C N → F O ( G ( N )) and S : M N ( C ) → F O ( G ( N )) are thecanonical linear maps. Then we calculateˆ β ( S x ) = N X rs S rs ⊗ v rsx , ˆ γ ( S rs ) = X x S x ⊗ v srx , where v rsx for 1 ≤ r, s ≤ N, x ∈ X are the standard generators of the linking algebra O ( G + ( G ( N ) , G ( N ))), see Proposition 6.10. UANTUM CUNTZ-KRIEGER ALGEBRAS 33
Consider now the ∗ -homomorphism π W : O ( G + ( G ( N ) , G ( N ))) → M N ( C ) ob-tained in Proposition 7.2. Slicing the maps ˆ β, ˆ γ with π W , we define the desired ∗ -homomorphims π N , σ N by π N = (id ⊗ π W ) ˆ β,σ N = (id ⊗ t )(id ⊗ π op W )ˆ γ, using the isomorphism t : M N ( C ) op ∼ = M N ( C ) given by sending a matrix Y to itstranspose Y t . Concretely, if we let V rsx be constructed out of a unitary error basisas in Proposition 7.2 then we have π N ( S x ) = N X rs S rs ⊗ V rsx ,σ N ( S rs ) = X x S x ⊗ ( V srx ) t . Let us denote by m : M N ( C ) op ⊗ M N ( C ) → M N ( C ) , m ( a op ⊗ b ) = ab the multipli-cation map. Using the relations in Proposition 7.2 we readily see that(id ⊗ m )(id ⊗ t ⊗ id)( σ N ⊗ id) π N ( a ) = a ⊗ a contained in the ∗ -algebra generated by the elements S x for x ∈ X . Simi-larly, we have (id ⊗ m )(id ⊗ t ⊗ id)( π N ⊗ id) σ N ( b ) = b ⊗ b contained in the ∗ -algebra generated by the elements S rs . Since m iscompletely bounded, it follows by continuity that π N and σ N are injective. (cid:3) Let us continue to use the notation from above and denote the embeddingsobtained in Proposition 7.4 by π N and σ N , referring to either the trivial or completequantum graphs G ( N ) , G ( N ). Following ideas of Gao, Harris and Junge [19], weshall refine these embeddings and realize each of F O ( G ( N )) and F O ( G ( N )) as aniterated crossed product of the other algebra with respect to certain Z N -actions,up to tensoring with matrices. This is indeed very much related to the work in [19],which exhibited a similar connection between free group C ∗ -algebras and Brown’suniversal non-commutative unitary algebras.In the sequel we write again ω = e πiN and calculate modulo N . We relabelthe generators of F O ( G i ( N )) in the proof of Proposition 7.4 by S ( i ) kl for i = 1 , ≤ k, l ≤ N −
1, and let
X, Z be the generalized Pauli matrices fromRemark 7.3.Let us consider the following order N automorphisms α j ∈ Aut( F O ( G ( N ))) for j = 1 , α ( S (2) kl ) = ω k − l S (2) kl α ( S (2) kl ) = S (2) k − ,l − . Note that α and α can be viewed as examples of gauge automorphisms as inparagraph 6.1. More precisely, they are the gauge automorphisms associated withthe unitaries X, Z in the sense that( α ⊗ id)( S ) = (1 ⊗ X ) S (1 ⊗ X ∗ ) ( α ⊗ id)( S ) = (1 ⊗ Z ∗ ) S (1 ⊗ Z )for S = ( S (2) ij ) ∈ F O ( G ( N )) ⊗ M N ( C ). Similarly, we define order N automorphisms β j ∈ Aut( F O ( G ( N ))) for j = 1 , β ( S (1) kl ) = S (1) k − ,l β ( S (1) kl ) = S (1) k,l − . Clearly, all these automorphisms define actions of Z N on F O ( G ( N )) and F O ( G ( N )),respectively. From the relation XZ = ωZX it follows that both pairs of actions α , α and β , β mutually commute. Let us now consider the iterated crossed products F O ( G ( N )) ⋊ α Z N ⋊ α Z N , F O ( G ( N )) ⋊ β Z N ⋊ β Z N , where α , β are naturally extended to the crossed products by letting Z N act onitself through appropriate dual actions. More precisely, given a ∈ F O ( G ( N )) , b ∈ F O ( G ( N )) and g ∈ Z N ∼ = { , . . . , N − } , we let α ( au g ) = α ( a ) ω g u g , β ( bu g ) = β ( g ) ω g u g . Abstractly, the algebra F O ( G ( N )) ⋊ α Z N ⋊ α Z N is the universal C ∗ -algebraspanned by elements of the form x = N − X j,k =0 a jk v j w k , where a jk ∈ F O ( G ( N )) and v, w are unitaries, such that the relations v N = w N = 1 , va jk = α ( a jk ) v, wa jk = α ( a jk ) w = a jk , wv = ωvw are satisfied. A similar description holds for F O ( G ( N )) ⋊ β Z N ⋊ β Z N .Our aim is to establish the following description of the iterated crossed productsobtained in this way. Theorem 7.5.
For the double crossed products with respect to the actions of Z N introduced above one obtains ∗ -isomorphisms M N ( F O ( K ( M N ( C ) , tr))) ∼ = O N ⋊ β Z N ⋊ β Z N M N ( O N ) ∼ = F O ( K ( M N ( C ) , tr)) ⋊ α Z N ⋊ α Z N and M N ( F O ( T M N )) ∼ = ( ∗ N C ( S )) ⋊ β Z N ⋊ β Z N M N ( ∗ N C ( S )) ∼ = F O ( T M N )) ⋊ α Z N ⋊ α Z N for all N ∈ N . In order to prove Theorem 7.5 we will construct the required isomorphisms ex-plicitly, using again uniform notation to treat the cases of trivial and completequantum graphs simultaneously.Consider the unitary error basis W = { X j Z k } ≤ j,k ≤ N − for M N ( C ) describedin Remark 7.3. Moreover let π N : F O ( G ( N )) → F O ( G ( N )) ⊗ M N ( C ) and σ N : F O ( G ( N )) → F O ( G ( N )) ⊗ M N ( C ) be the corresponding embeddings constructedin the proof of Proposition 7.4. That is, if we set V rslm = 1 N ( X l Z m ) ∗ e rs ( X l Z m ) = 1 N ω − ( r − s ) l e r − m,s − m , and use our previous notation for the generators of F O ( G j ( N )), then we obtain π N ( S (1) lm ) = N X ≤ r,s ≤ N − S (2) rs ⊗ V rslm = X ≤ r,s ≤ N − S (2) rs ⊗ ω − ( r − s ) l e r − m,s − m and σ N ( S (2) jk ) = X ≤ l,m ≤ N − S (1) lm ⊗ ( V kjlm ) t = 1 N X ≤ l,m ≤ N − S (1) lm ⊗ ω ( j − k ) l e j − m,k − m . From the above formulas we can easily see that(1 ⊗ X ) σ N ( S (2) jk )(1 ⊗ X ∗ ) = σ N ( α ( S (2) jk ))(1 ⊗ X ) π N ( S (1) jk )(1 ⊗ X ∗ ) = π N ( β ( S (1) jk )) . UANTUM CUNTZ-KRIEGER ALGEBRAS 35
Hence ( σ N , (1 ⊗ X )) defines a covariant representation of the dynamical system( F O ( G ( N )) , Z N , α ), and similarly ( π N , ⊗ X ) defines a covariant representationof ( F O ( G ( N )) , Z N , β ). As a consequence, we obtain ∗ -homomorphisms σ ′ N : F O ( G ( N )) ⋊ α Z N → F O ( G ( N )) ⊗ M N ( C ) π ′ N : F O ( G ( N )) ⋊ β Z N → F O ( G ( N )) ⊗ M N ( C )satisfying σ ′ N ( a ) = σ N ( a ) , σ ′ N ( v ) = 1 ⊗ X,π ′ N ( b ) = π N ( b ) , π ′ N ( v ) = 1 ⊗ X, where a ∈ F O ( G ( N )) and b ∈ F O ( G ( N )), respectively. Similarly, we compute(1 ⊗ Z ∗ ) σ N ( S (2) jk )(1 ⊗ Z ) = σ N ( α ( S (2) jk )) , (1 ⊗ Z ∗ )(1 ⊗ X ) = ω (1 ⊗ X )(1 ⊗ Z ∗ ) , (1 ⊗ Z ∗ ) π N ( S (1) jk )(1 ⊗ Z ) = π N ( β ( S (1) jk )) . Hence ( σ ′ N , (1 ⊗ Z ∗ )) defines a covariant representation of the dynamical system( F O ( G ( N )) ⋊ α Z N , Z N , α ), and ( π ′ N , (1 ⊗ Z ∗ )) defines a covariant representationof ( F O ( G ( N )) ⋊ β Z N , Z N , β ). In the same way as before we obtain associated ∗ -homomorphisms σ ′′ N : F O ( G ( N )) ⋊ α Z N ⋊ α Z N → F O ( G ( N )) ⊗ M N ( C ) π ′′ N : F O ( G ( N )) ⋊ β Z N ⋊ β Z N → F O ( G ( N )) ⊗ M N ( C ) , satisfying σ ′′ N ( a ) = σ N ( a ) , σ ′′ N ( v ) = σ ′ N ( v ) = 1 ⊗ X, σ ′′ N ( w ) = 1 ⊗ Z ∗ ,π ′′ N ( b ) = π N ( b ) , π ′′ N ( v ) = π ′ N ( v ) = 1 ⊗ X, π ′′ N ( w ) = 1 ⊗ Z ∗ , respectively, where a ∈ F O ( G ( N )) , b ∈ F O ( G ( N )).With these constructions in place, Theorem 7.5 is a consequence of the followingassertion. Theorem 7.6.
The maps σ ′′ N and π ′′ N are isomorphisms.Proof. Using C ∗ ( X, Z ∗ ) = M N ( C ) and the description of σ ′′ N given above it is easyto see that σ ′′ N is surjective. Explicitly, the range of σ ′′ N contains σ N ( F O ( G ( N )))and 1 ⊗ M N ( C ), and these two algebras generate F O ( G ( N )) ⊗ M N ( C ).To show that σ ′′ N is injective consider the ∗ -homomorphism( π N ⊗ id) σ N : F O ( G ( N )) → F O ( G ( N )) ⊗ M N ( C ) ⊗ M N ( C ) . If we denote by {| ξ jk i} ≤ j,k ≤ N − ⊂ C N ⊗ C N the orthonormal basis of maximallyentangled vectors given by ξ jk = 1 √ N N − X r =0 X j Z k | r i ⊗ | r i , then one obtains( π N ⊗ id) σ ′′ N ( S (2) jk ) = 1 N X lmrs S (2) rs ⊗ ω − ( r − s ) l e r − m,s − m ⊗ ω ( j − k ) l e j − m,k − m = X sm S (2) s + j − k,s ⊗ e s + j − k − m,s − m ⊗ e j − m,k − m = X nm S (2) n + j,n + k ⊗ e n + j − m,n + k − m ⊗ e j − m,k − m = X ln α − l α − n ( S (2) jk ) ⊗ | ξ l,n ih ξ l,n | . Next, we define a unitary V on C N ⊗ C N by setting V ( | j i ⊗ | k i ) = ω − jk | ξ jk i . Thenwe have V ∗ (1 ⊗ X ) V = Z ⊗ , V ∗ (1 ⊗ Z ∗ ) V = X ⊗ Z. Thus, if we consider the ∗ -homomorphismΦ : F O ( G ( N )) ⋊ α Z N ⋊ α Z N → F O ( G ( N )) ⊗ M N ( C ) ⊗ M N ( C )given by Φ = ad(1 ⊗ V ∗ )( π N ⊗ id) σ ′′ N , then we getΦ( a ) = X l,n α − l α − n ( a ) ⊗ e ll ⊗ e nn for all a ∈ F O ( G ( N )), and alsoΦ( v ) = (1 ⊗ V ∗ )( π N ⊗ id) σ ′′ N ( v )(1 ⊗ V ) = 1 ⊗ V ∗ (1 ⊗ X ) V = 1 ⊗ Z ⊗ w ) = (1 ⊗ V ∗ )( π N ⊗ id) σ ′′ N ( w )(1 ⊗ V ) = 1 ⊗ V ∗ (1 ⊗ Z ∗ ) V = 1 ⊗ X ⊗ Z. From these formulas it follows that the image of Φ is exactly the reduced crossedproduct F O ( G ( N )) ⋊ α ,r Z N ⋊ α ,r Z N , and Φ is none other than the canonicalquotient map from the full crossed product to the reduced crossed product. Since Z N is finite, and hence amenable, the map Φ is an isomorphism, forcing σ ′′ N to beinjective. This proves the claim for σ ′′ N .For π ′′ N one proceeds in a similar way, essentially by swapping the roles of themaps π N and σ N and repeating the above arguments. (cid:3) Remark 7.7.
The first pair of isomorphisms in Theorem 7.5 should not come as agreat surprise, given that Theorem 4.5 in section 4 already asserts an isomorphism F O ( K ( M N ( C ) , tr)) ∼ = O N . In fact, the latter isomorphism can be verified byconsidering the injective ∗ -homomorphism σ KN : F O ( K ( M N ( C ) , tr)) → M N ( O N )obtained in Proposition 7.4 and inspecting the relations in the proof of Proposition7.2. In the next section we will prove Theorem 4.5 in full generality. Remark 7.8.
Taking into account the identification F O ( K ( M N ( C ) , tr)) ∼ = O N ,the statement for complete quantum graphs in Theorem 7.5 is reminiscent of Takesaki-Takai duality. However, the isomorphisms are slightly different. Note also that the C ∗ -algebras ∗ N C ( S ) and F O ( T M N ) are not even Morita equivalent, compareTheorem 4.7. Remark 7.9.
Using the isomorphism from Theorem 4.7 we see that π TN inducesan embedding ∗ N C ( S ) → M N ( C ) ∗ ( C ( S ) ⊕ C ). In the notation used above thismaps the generators S (1) kl to P rs ω k ( s − r ) e r − l,r Se s,s − l , where S denotes the standardgenerator of C ( S ) ⊂ C ( S ) ⊕ C . UANTUM CUNTZ-KRIEGER ALGEBRAS 37
Remark 7.10.
It seems natural to look at pairs of quantum Cuntz-Krieger algebrasassociated to quantum isomorphic quantum graphs beyond the cases considered inTheorem 7.5. Finding “small” representations of linking algebras could potentiallyallow one to transfer properties like unitality, nuclearity, or existence of tracesfrom one algebra to the other, without a priori knowing whether the algebras areisomorphic or not.8.
The structure of complete quantum Cuntz-Krieger algebras
In this final section we discuss our main result on the structure of completequantum Cuntz-Krieger algebras, that is, we provide the proof of Theorem 4.5stated in section 4.Let us begin with a simple lemma.
Lemma 8.1.
Let A be a non-zero unital C ∗ -algebra and let n , n ∈ N . Moreoverassume that u = ( u xy ) ∈ M n ,n ( A ) is a rectangular unitary matrix with coefficientsin A . Let s x for ≤ x ≤ n be the standard generators of O n and define elements ˆ s y ∈ O n ⊗ A for ≤ y ≤ n by ˆ s y = n X x =1 s x ⊗ u xy . Then the elements ˆ s y satisfy the defining relations of O n and C ∗ (ˆ s , . . . , ˆ s n ) ∼ = O n . Proof.
In order to verify the Cuntz relations we calculateˆ s ∗ z ˆ s y = X x ,x s ∗ x s x ⊗ u ∗ x z u x y = X x ⊗ u ∗ x z u x y = δ y,z (1 ⊗ X y ˆ s y ˆ s ∗ y = X y,x ,x s x s ∗ x ⊗ u x y u ∗ x y = X x ,x s x s ∗ x ⊗ δ x ,x
1= 1 ⊗ . Since O n is simple this yields the claim. (cid:3) Now let us fix a complete quantum graph K ( B, ψ ) satisfying the hypotheses ofTheorem 4.5, that is, (
B, ψ ) is a finite quantum space in standard form such that ψ : B → C is a δ -form with δ ∈ N . We shall use the same notation that asafter Definition 3.1, so that B = L da =1 M N a ( C ) and ψ ( x ) = P da =1 Tr( Q ( a ) x i ) for x = ( x , . . . , x d ).By Remark 6.9, we have a quantum isomorphism K ( B, ψ ) ∼ = q K δ . Denoteby v xija for 1 ≤ a ≤ d, ≤ i, j ≤ N a , ≤ x ≤ δ the standard generators ofthe C ∗ -algebra A = C ( G + ( K δ , K ( B, ψ ))) given in Proposition 6.10. Moreover let n = dim( B ) and consider the rectangular matrix u = ( u xija ) ∈ M δ ,n ( A ) given by u xija = ( Q ( a ) ) − / jj δ − v xija . Using the relations in Proposition 6.10 one obtains( u ∗ u ) ija,klb = X x ( Q ( a ) ) − / jj δ − ( v xija ) ∗ ( Q ( b ) ) − / ll δ − v xklb = X x ( Q ( a ) ) − / jj ( Q ( b ) ) − / ll δ − v xjia v xklb = X x ( Q ( a ) ) − / jj ( Q ( a ) ) − / ll δ − δ ab δ ik v xjla = X x ( Q ( a ) ) − / jj ( Q ( a ) ) − / ll ( Q ( a ) ) jj δ ab δ ik δ jl = δ ab δ ik δ jl and ( uu ∗ ) xy = X ija ( Q ( a ) ) − / jj δ − v xija ( Q ( a ) ) − / jj δ − ( v yija ) ∗ = X ija ( Q ( a ) ) − jj δ − v xija v yjia = X ia δ xy v xiia = δ xy . We conclude that u ∗ u = 1 M n ( A ) and uu ∗ = 1 M δ ( A ) , or equivalently, that u isunitary.Next, we consider the ∗ -homomorphism ˆ β : F O ( K ( B, ψ )) → O δ ⊗ A fromTheorem 6.13, which satisfiesˆ β ( S ( e ( a ) ij )) = X x S ( e x ) ⊗ v xjia in terms of the standard matrix units. Equivalently, if we write S ( a ) ij = S ( f ( a ) ij ),where f ( a ) ij = ( Q ( a ) ) − / ii e ( a ) ij ( Q ( a ) ) − / jj are the adapted matrix units for ( B, ψ ), and s x = S ( δ e x ) for the canonical Cuntz isometries generating O δ , thenˆ β ( S ( a ) ij ) = ( Q ( a ) ) − / ii ( Q ( a ) ) − / jj δ − X x s x ⊗ v xija = ( Q ( a ) ) − / ii δ − X x s x ⊗ u xija . Hence the unitarity of the matrix u = ( u xija ) combined with Lemma 8.1 impliesthat the elements ( Q ( a ) ) / ii δ ˆ β ( S ( a ) ij ) form an n -tuple of Cuntz isometries in O δ ⊗ A .According to Remark 6.14, the restriction of ˆ β to the ∗ -algebra generated by the S ( a ) ij is injective. This shows that F O ( K ( B, ψ )) is unital with unit e = X ija ( Q ( a ) ) ii δ S ( a ) ij ( S ( a ) ij ) ∗ , and that the elements ( Q ( a ) ) / ii δS ( a ) ij form an n -tuple of Cuntz isometries generating F O ( K ( B, ψ )). This completes the proof of Theorem 4.5.
Remark 8.2.
It seems reasonable to expect that F O ( K ( B, ψ )) ∼ = O n for all choicesof δ -forms ψ , but we are unable to supply a proof. Note that when δ / ∈ N , we nolonger have a quantum isomorphism between K ( B, ψ ) and a classical completegraph, and therefore a different approach would be needed.
UANTUM CUNTZ-KRIEGER ALGEBRAS 39
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Michael Brannan, Department of Mathematics, Texas A&M University, College Sta-tion, TX 77840, USA
E-mail address : [email protected] Kari Eifler, Department of Mathematics, Texas A&M University, College Station,TX 77840, USA
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