Crossed products for interactions and graph algebras
aa r X i v : . [ m a t h . OA ] M a y CROSSED PRODUCTS FOR INTERACTIONSAND GRAPH ALGEBRAS
B. K. KWAŚNIEWSKI
Abstract.
We consider Exel’s interaction ( V , H ) over a unital C ∗ -algebra A , such that V ( A ) and H ( A ) are hereditary subalgebras of A . For the associated crossed product,we obtain a uniqueness theorem, ideal lattice description, simplicity criterion and aversion of Pimsner-Voiculescu exact sequence. These results cover the case of crossedproducts by endomorphisms with hereditary ranges and complemented kernels. Asmodel examples of interactions not coming from endomorphisms we introduce andstudy in detail interactions arising from finite graphs.The interaction ( V , H ) associated to a graph E acts on the core F E of the graphalgebra C ∗ ( E ) . By describing a partial homeomorphism of c F E dual to ( V , H ) wefind the fundamental structure theorems for C ∗ ( E ) , such as Cuntz-Krieger uniquenesstheorem, as results concerning reversible noncommutative dynamics on F E . We alsoprovide a new approach to calculation of K -theory of C ∗ ( E ) using only an inducedpartial automorphism of K ( F E ) and the six-term exact sequence. Contents
1. Introduction 21.1. Preliminaries on Hilbert bimodules 52. Complete interactions and their crossed products 82.1. Interactions and C ∗ -dynamical systems 82.2. Crossed product for corner interactions 112.3. Topological freeness, ideal structure and simplicity criteria 132.4. K -theory 153. Graph C ∗ -algebras via interactions 163.1. Graph C ∗ -algebra C ∗ ( E ) and its AF-core 173.2. Interactions arising from graphs 183.3. Dynamical systems dual to graph interactions. 223.4. Topological freeness of graph interactions. 273.5. K -theory 30References 33 Mathematics Subject Classification.
Primary 46L55, Secondary 46L80.
Key words and phrases. interaction, graph algebra, endomorphism, topological freeness, crossedproduct, Hilbert bimodule, K -theory, Pimsner-Voiculescu exact sequence.This work was in part supported by Polish National Science Centre grant number DEC-2011/01/D/ST1/04112. Introduction
In [12] Exel extended celebrated Pimsner’s construction [38] of the (nowadays called)Cuntz-Pimsner algebras by introducing an intriguing new concept of a generalized C ∗ -correspondence. The leading example in [12] arises from an interaction - a pair ( V , H ) of positive linear maps on a C ∗ -algebra A that are mutual generalized inverses and suchthat the image of one map is in the multiplicative domain of the other. An interactioncan be considered a ‘symmetrized’ generalization of a C ∗ -dynamical system , i.e. a pair ( α, L ) consisting of an endomorphism α : A → A and its transfer operator L : A → A ,[11]. One can think of many examples of interactions naturally appearing in variousproblems, cf. [13], [17], [14]. However, at present there is only one significant applicationof an interaction ( V , H ) which is not a C ∗ -dynamical system. Namely, in the recentpaper [14] Exel showed that the C ∗ -algebra O n,m introduced in [3], is Morita equivalentto the crossed product C ∗ ( A, V , H ) for an interaction ( V , H ) , over a commutative C ∗ -algebra A , where neither V nor H is multiplicative. Moreover, for crossed products underconsideration general structure theorems known so far concern only the case when theinitial object is an injective endomorphism, cf. [35], [36], [43], [18], [7]. In particular,there are no such theorems for genuine interactions, i.e when both V and H are notmultiplicative.The purpose of the present article is two fold.Firstly, we establish general tools to study the structure of C ∗ ( A, V , H ) for an ac-cessible and, as the C ∗ -dynamical system case indicates, important class of interactions ( V , H ) . Thus this might be a considerable step in understanding these new objects. Moreprecisely, crossed products associated with C ∗ -dynamical systems ( α, L ) on a unital C ∗ -algebra A boast their greatest successes in the case α ( A ) is a hereditary subalgebra of A , cf. [2], [11], [35], [36], [42]. Then L is a corner retraction , see [43, page 424], [26].It is uniquely determined by α and it is called a complete transfer operator in [2], see[23], [26]. In the present paper we focus on interactions ( V , H ) for which both V ( A ) and H ( A ) are hereditary subalgebras of A . Then V ( A ) and H ( A ) are automatically cornersin A . We call such interactions corner interactions . It turns out that each mapping insuch an interaction ( V , H ) is completely determined by the other. This plus the obviousconnotation to complete transfer operators make it tempting to call ( V , H ) a completeinteraction [28], but we resist this temptation here.We show that for a corner interaction ( V , H ) the crossed product C ∗ ( A, V , H ) definedin [12] is the universal C ∗ -algebra generated by a copy of A and a partial isometry s subject to relations V ( a ) = s ( a ) s ∗ , H ( a ) = s ∗ ( a ) s, a ∈ A. As a consequence C ∗ ( A, V , H ) can be modeled as the crossed product A ⋊ X Z , [1], of A bya Hilbert bimodule X = AsA . It also follows that C ∗ ( A, V , H ) ∼ = C ∗ ( A, V ) ∼ = C ∗ ( A, H ) where C ∗ ( A, V ) (resp. C ∗ ( A, H ) ) is the crossed product of A by the completely positivemapping V (resp. H ), as introduced in [26]. We study C ∗ ( A, V , H ) by applying generalmethods developed for Hilbert bimodules [25] and C ∗ -correspondences [22]. For instance,we have a naturally defined partial homeomorphism b V of b A dual to ( V , H ) . Identifying itwith the inverse to the induced partial homeomorphism X -Ind studied in [25] we obtain: NTERACTIONS AND GRAPH ALGEBRAS 3 uniqueness theorem - topological freeness of b V implies faithfulness of every representationof C ∗ ( A, V , H ) which is faithful on A ; ideal lattice description via b V -invariant open setswhen b V is free; and the simplicity of C ∗ ( A, V , H ) when b V is minimal and topologically free(see Theorem 2.20 below). Similarly, identifying the abstract morphisms in Katsura’sversion of Pimsner-Voiculescu exact sequence [22] we get a natural cyclic exact sequencefor K -groups of C ∗ ( A, V , H ) (Theorem 2.25). It generalizes the corresponding exactsequence obtained by Paschke for injective endomorphisms [36], which plays a crucialrole, for instance, in [42].Secondly, we provide a detailed analysis of nontrivial corner interactions with aninteresting noncommutative dynamics related to Markov shifts, and graph C ∗ -algebrasas crossed products. More specifically, already in [9] Cuntz considered his C ∗ -algebras O n as crossed products of the core UHF-algebras by injective endomorphisms implementedby one of the generating isometries. As noticed by Rørdam [42, Example 2.5], a similarreasoning can be performed for Cuntz-Krieger algebras O A by considering an isometrygiven by the sum of all generating partial isometries with properly restricted initialspaces. An analogous isometry in O A , but in a sense canonically associated with theunderlying dynamics of Markov shifts, was found in [11, proof of Theorem 4.3], cf. [2,formula (4.18)]. For the graph C ∗ -algebra C ∗ ( E ) associated with a row-finite graph E with no sources the corresponding isometry appears implicitly in [7, Theorem 5.1] andexplicitly in [19, Theorem 5.2], see formula (3.2) below. In particular, if we assume E is finite, i.e. the sets of vertices and edges are finite, and E has no sources, weknow from [19, Theorem 5.2] states that C ∗ ( E ) is naturally isomorphic to the crossedproduct of the AF-core C ∗ -algebra F E by an injective endomorphism with hereditaryrange implemented by the aforementioned isometry s . Thus we have C ∗ ( E ) = C ∗ ( F E ∪ { s } ) , s F E s ∗ ⊂ F E , s ∗ F E s ⊂ F E . Moreover, one can notice that the above picture remains valid for arbitrary finite graphs,possibly with sources. The only difference is that s may be no longer an isometrybut a partial isometry. Hence the mapping F E ∋ a → sas ∗ ∈ F E may be no longermultiplicative (at least not on its whole domain) and then a natural framework for C ∗ ( E ) is the crossed product for an interaction ( V , H ) over F E where V ( · ) := s ( · ) s ∗ , H ( · ) := s ∗ ( · ) s . We call the pair ( V , H ) arising in this way a graph interaction . It canbe viewed from many different perspectives as a model example illustrating and givingnew insight, for instance, to the following objects and issues that we hope to be pursuedin the future. • Interactions with nontrivial algebras and not multiplicative dynamics . The crossedproduct C ∗ ( F E , V , H ) is naturally isomorphic to the graph C ∗ -algebra C ∗ ( E ) (Propo-sition 3.2). In general, ( V , H ) is not a C ∗ -dynamical system and is not a part of agroup interaction [13]. We precisely identify the values of n ∈ N for which ( V n , H n ) isan interaction (see Proposition 3.5), and it turns out that such n ’s might have almostarbitrary distribution. Moreover, ( V n , H n ) is an interaction for all n ∈ N if and only if ( V , H ) is a C ∗ -dynamical system (which may happen even if E has sources). We follow here the original conventions of [29], [4] and hence in the context of representations ofgraphs we consider different orientation of edges than in [39], [7], [19]
B. K. KWAŚNIEWSKI • Noncommutative Markov shifts . The main motivation in [11] for introducing C ∗ -dynamical systems ( α, L ) was to realize Cuntz-Krieger algebras O A as crossed productsof the underlying Markov shifts, which was in turn suggested by [10, Proposition 2.17].In terms of graph C ∗ -algebras the relevant statement, see [7, Theorem 5.1], says thatwhen E is finite and has no sinks, then C ∗ ( E ) is isomorphic to Exel crossed product D E ⋊ φ E , L N where D E ∼ = C ( E ∞ ) is a canonical masa in F E . The spectrum of D E isidentified with the space of infinite paths E ∞ , φ E is a transpose to the Markov shifton E ∞ and L is its classical Ruelle-Perron-Frobenious operator. Both φ E and L extendnaturally to completely positive maps on C ∗ ( E ) and the extension of φ E is called thenoncommutative Markov shift, cf. e.g. [20]. However, from the point of view of thecrossed product construction the predominant role is played by L , see [26]. In particular, L = H where ( V , H ) is the graph interaction, and F E is a minimal C ∗ -algebra invariantunder V and containing D E . Thus there are good reasons to regard the graph interaction ( V , H ) as an alternative candidate for the noncommutative counterpart of the Markovshift. Our dual and K -theoretic pictures of ( V , H ) (see Theorem 3.9 and Proposition3.22, respectively) support this point of view. • Graph C ∗ -algebras . The structure of graph algebras was originally studied viagrupoids [29], [30], and K -theory was calculated using a dual Pimsner-Voiculescu exactsequence and skew products of initial graphs [40], [39]. The corresponding results canalso be achieved in the realm of partial actions of free groups on certain commutative C ∗ -algebras, see [15], [16]. We present here another approach, based on interactions. Weshow that the partial homeomorphism b V dual to V is topologically free if and only if E satisfies the so-called condition ( L ) [4]. Hence we derive the Cuntz-Krieger uniquenesstheorem [30], [4], [39] from our general uniqueness theorem for interactions. Similarly,we see that freeness of b V is equivalent to condition (K) for E [30], [4]. Thus minimalityand freeness of b V is equivalent to the known simplicity criteria for C ∗ ( E ) . Moreover, itturns out that pure infiniteness of C ∗ ( E ) , as defined in [31], [29], is equivalent to a verystrong version of topological freeness of b V (see Remark 3.20), which therefore might beconsidered an instance of a noncommutative version of local boundary action, see [31].Finally, our approach to calculation of K -groups for C ∗ ( E ) seems to be the most directupon the existing ones; it uses only direct limit description of the AF-core F E and thecyclic six-term exact sequence. • Topological freeness . The condition known as topological freeness was for the firsttime explicitly stated in [32] where the author use it to show, what we call here, unique-ness theorem. Namely, he proved that topological freeness of a homeomorphism dual toan automorphisms α of a C ∗ -algebra A implies that any representation of A ⋊ α Z whoserestriction to A is injective, is automatically faithful. The converse implication (equiva-lence between topological freeness and the aforementioned uniqueness property) in thecase A is noncommutative turned out to be a difficult problem. It was proved in [34,Theorem 10.4] combined with [33, Theorem 2.5], see also [33, Remark 4.8], under theassumption that A is separable. The proof is nontrivial and passes through conditionsinvolving such notions as Connes spectrum, inner derivations, or proper outerness. Sinceit is known that condition (L) is necessary for Cuntz-Krieger uniqueness theorem to hold,our explicit characterization of topological freeness for graph interactions (see Theorem NTERACTIONS AND GRAPH ALGEBRAS 5 • Dilations of completely positive maps . Let us consider a C ∗ -algebra C ∗ ( A ∪ { s } ) generated by a C ∗ -algebra A and a partial isometry s such that sAs ∗ ⊂ A . Also assumethat A and C ∗ ( A ∪ { s } ) have a common unit. Then V ( · ) = s ( · ) s ∗ is a completely positivemap on A sending the unit to an idempotent (this is a general form of such mappings,cf. [26]). We may put H ( · ) := s ∗ ( · ) s and then one can see that B := span { a s ∗ a s ∗ a ...s ∗ a n sb sb ...sb n : a i , b i ∈ A, n ∈ N } is the smallest C ∗ -algebra preserved by H and containing A . Plainly, the pair ( V , H ) isa corner interaction on B . Hence, potentially, our results could be applied to study thestructure of C ∗ ( A ∪ { s } ) = C ∗ ( B ∪ { s } ) . Nevertheless, the dilation of V from A to B is anontrivial procedure and in general depends on the initial representation of V via s . Thecore algebras B arising in this way are studied in detail for instance in [21], [24], [27].Our analysis of the graph interaction ( V , H ) can be viewed as a case study of the abovesituation when A ∼ = C N is a finite dimensional commutative C ∗ -algebra, see Remark3.10. In particular, Theorem 3.9 can be interpreted as that the partial homeomorphismdual to a dilation of the Ruelle-Perron-Frobenius operator H = L (from A to B = F E )is a quotient of the Markov shift.We begin by presenting relevant notions and statements concerning Hilbert bimodulesand briefly clarifying their relationship with generalized C ∗ -correspondences. Generalcorner interactions are studied in Section 2. Section 3 is devoted to analysis of graphinteractions.1.1. Preliminaries on Hilbert bimodules.
Throughout A is a C ∗ -algebra which(starting from Section 2) will always be unital. By homomorphisms, epimorphisms,etc. between C ∗ -algebras we always mean ∗ -preserving maps. All ideals in C ∗ -algebrasare assumed to be closed and two sided. We adhere to the convention that β ( A, B ) = span { β ( a, b ) ∈ C : a ∈ A, b ∈ B } for maps β : A × B → C such as inner products, multiplications or representations.As in [25] we say that a partial homeomorphism ϕ of a topological space M , i.e. ahomeomorphism whose domain ∆ and range ϕ (∆) are open subsets of M , is topologicallyfree if for any n > the set of fixed points for ϕ n (on its natural domain) has emptyinterior. A set V is ϕ - invariant if ϕ ( V ∩ ∆) = V ∩ ϕ (∆) . If there are no nontrivial closedinvariant sets, then ϕ is called minimal , and ϕ is said to be (residually) free , if it istopologically free on every closed invariant set (in the Hausdorff space case this amountsto requiring that ϕ has no periodic points).Following [6, 1.8] and [1] by a Hilbert bimodule over A we mean X which is botha left Hilbert A -module and a right Hilbert A -module with respective inner products h· , ·i A and A h· , ·i satisfying the so-called imprimitivity condition : x · h y, z i A = A h x, y i · z ,for all x, y, z ∈ X . A covariant representation of X is a pair ( π A , π X ) consisting of ahomomorphism π A : A → B ( H ) and a linear map π X : X → B ( H ) such that(1.1) π X ( ax ) = π A ( a ) π X ( x ) , π X ( xa ) = π X ( x ) π A ( a ) , B. K. KWAŚNIEWSKI (1.2) π A ( h x, y i A ) = π X ( x ) ∗ π X ( y ) , π A ( A h x, y i ) = π X ( x ) π X ( y ) ∗ , for all a ∈ A , x, y ∈ X . The crossed product A ⋊ X Z is a C ∗ -algebra generated by acopy of A and X universal with respect to covariant representations of X , see [1]. It isequipped with the circle gauge action γ = { γ z } z ∈ T given on generators by γ z ( a ) = a and γ z ( x ) = zx , for a ∈ A , x ∈ X , z ∈ T = { z ∈ C : | z | = 1 } .As it is standard, we abuse the language and denote by π both an irreducible represen-tation of A and its equivalence class in the spectrum b A of A . It should not cause confusionwhen we consider induced representations, as for a Hilbert bimodule X over A the in-duced representation functor X -Ind preserves such classes. We briefly recall, and refer to[41] for all necessary details, that X -Ind maps a representation π : A → B ( H ) to a rep-resentation X -Ind( π ) : A → B ( X ⊗ π H ) where the Hilbert space X ⊗ π H is generated bysimple tensors x ⊗ π h , x ∈ X , h ∈ H , satisfying h x ⊗ π h , x ⊗ π h i = h h , π ( h x , x i A ) h i , and X -Ind( π )( a )( x ⊗ π h ) = ( ax ) ⊗ π h, a ∈ A. The spaces h X, X i A and A h X, X i are ideals in A and the bimodule X implements aMorita equivalence between them. Hence X -Ind : \ h X, X i A → \ A h X, X i is a homeomor-phism which we may naturally treat as a partial homeomorphism of b A , see [25].The results of [25] can be summarized as follows. Theorem 1.1.
Let X -Ind be a partial homeomorphism of b A , as described above. i) If X -Ind is topologically free, then every faithful covariant representation ( π A , π X ) of X ‘integrates’ to the faithful representation of A ⋊ X Z . ii) If X -Ind is free, then J \ J ∩ A is a lattice isomorphism between ideals in A ⋊ X Z and open invariant sets in b A . iii) If X -Ind is topologically free and minimal, then A ⋊ X Z is simple.Remark . The map X -Ind is a lift of the so-called Rieffel homeomorphism h X : Prim h X, X i A → Prim A h X, X i , cf. [41, Corollary 3.33], [25, Remark 2.3]. Plainly,topological freeness of ( Prim ( A ) , h X ) implies topological freeness of ( b A, X -Ind) , but theconverse is not true and as we will see, cf. Example 3.4 below, Cuntz algebras O n providean excellent example of this phenomenon. Remark . In [43] Schweizer showed that if X is a full nondegenerate C ∗ -correspondenceover a unital C ∗ -algebra A , then the Cuntz-Pimsner algebra O X , defined as in [38], issimple if and only if X is minimal and aperiodic [43, Definition 3.7]. Clearly, if X is a Hilbert bimodule, minimality of X -Ind is equivalent to the minimality of X andtopological freeness of X -Ind implies the aperiodicity of X . Moreover, the algebras O X and A ⋊ X Z coincide if and only if A h X, X i is an essential ideal in A (which in turn isequivalent to injectivity of the left action of A on X ). In particular, if the ideal A h X, X i is essential in A and h X, X i A = A is unital, then [43, Theorem 3.9] implies that A ⋊ X Z is simple iff X is minimal and aperiodic.Let us fix a Hilbert bimodule X over A . We notice that it is naturally equipped withthe ternary ring operation [ x, y, z ] := x h y, z i A = A h x, y i z, x, y, z ∈ X, NTERACTIONS AND GRAPH ALGEBRAS 7 making it into a generalized correspondence over A , as defined in [12, Definition 7.1].Alternatively, this generalized correspondence could be described in terms of [12, Propo-sition 7.6] as the triple ( X, λ, ρ ) where we consider X as a A h X, X i - h X, X i A -Hilbertbimodule and define homomorphisms λ : A → A h X, X i and ρ : A → h X, X i A to be(necessarily unique) extensions of the identity maps.The following fact should be compared with [12, Proposition 7.13]. Proposition 1.4.
The crossed product A ⋊ X Z of the Hilbert bimodule X is naturallyisomorphic to the covariance algebra C ∗ ( A, X ) , as defined in [12, 7.12] , for X treated asa generalized correspondence.Proof. The Toeplitz algebra T ( A, X ) for the generalized correspondence X , see [12, page57], is a universal C ∗ -algebra generated by a copy of A and X subject to all A - A -bimodulerelations plus the ternary ring relations:(1.3) xy ∗ z = x h y, z i A = A h x, y i z, x, y, z ∈ X. The C ∗ -algebra C ∗ ( A, X ) is the quotient T ( A, X ) / ( J ℓ + J r ) where J ℓ (respectively J r )is an ideal in T ( A, X ) generated by the elements a − k such that a ∈ (ker λ ) ⊥ , k ∈ XX ∗ (resp. a ∈ (ker ρ ) ⊥ , k ∈ X ∗ X ) and(1.4) ax = kx ( or resp. xa = xk ) for all x ∈ X. Note that (ker λ ) ⊥ = A h X, X i and (ker ρ ) ⊥ = h X, X i A . By (1.3), XX ∗ and X ∗ X are C ∗ -subalgebras of T ( A, X ) . Hence using approximate units argument we see that when a is fixed relations (1.4) determine k uniquely. It follows that J ℓ = span { A h x, y i − xy ∗ : x, y ∈ X } , J r = span {h x, y i A − x ∗ y : x, y ∈ X } , because if (for instance) a − k ∈ J ℓ where a = P ni =1 A h x i , y i i ∈ (ker λ ) ⊥ and k ∈ X ∗ X ,then by (1.3), ax = P ni =1 x i y ∗ i x for all x ∈ X and thus k = P ni =1 x i y ∗ i .Accordingly, both C ∗ ( A, X ) and A ⋊ X Z are universal C ∗ -algebras generated by copiesof A and X subject to the same relations. (cid:3) Katsura obtained in [22] a version of the Pimsner-Voiculescu exact sequence for gen-eral C ∗ -correspondences and their C ∗ -algebras. We recall it in the case X is a Hilbertbimodule and in a form suitable for our purposes. We consider the linking algebra D X = K ( X ⊕ A ) in the following matrix representation D X = (cid:18) K ( X ) X e X A (cid:19) , where e X is the dual Hilbert bimodule of X , cf. e.g. [41, pages 49, 50], and let ι : A h X, X i → A , ι : K ( X ) → D X and ι : A → D X be inclusion maps; ι ( a ) = (cid:18) a
00 0 (cid:19) , ι ( a ) = (cid:18) a (cid:19) . By [22, Proposition B.3], ( ι ) ∗ : K ∗ ( A ) → K ∗ ( D X ) is an isomorphism B. K. KWAŚNIEWSKI and by [22, Theorem 8.6] the following sequence is exact:(1.5) K ( A h X, X i ) ι ∗ − ( X ∗ ◦ φ ∗ ) / / K ( A ) ( i A ) ∗ / / K ( A ⋊ X Z ) (cid:15) (cid:15) K ( A ⋊ X Z ) O O K ( A ) ( i A ) ∗ o o K ( A h X, X i ) ι ∗ − ( X ∗ ◦ φ ∗ ) o o where φ : A → L ( X ) is the homomorphism implementing left action of A on X , and X ∗ : K ∗ ( A h X, X i ) → K ∗ ( A ) is the composition of ( ι ) ∗ : K ∗ ( A h X, X i ) → K ∗ ( D X ) andthe inverse to the isomorphism ( ι ) ∗ : K ∗ ( A ) → K ∗ ( D X ) .2. Complete interactions and their crossed products
In this section, following closely the relationship between C ∗ -dynamical systems andinteractions, we introduce corner interactions, describe the structure of the associatedcrossed product and establish fundamental tools for its analysis (Theorems 2.20, 2.25).2.1. Interactions and C ∗ -dynamical systems. It is instructive to consider interac-tions as generalization of pairs ( α, L ) , sometimes called Exel systems [19], consistingof an endomorphism α : A → A and its transfer operator , i.e. a positive linear map L : A → A such that L ( α ( a ) b ) = a L ( b ) , a, b ∈ A , see [11]. Then L is automaticallycontinuous, ∗ -preserving, and we also have: L ( bα ( a )) = L ( b ) a , a, b ∈ A . We say that atransfer operator L is regular if α ( L (1)) = α (1) , or equivalently [11, Proposition 2.3], if E ( a ) := α ( L ( a )) is a conditional expectation from A onto α ( A ) . We note that originally[11] Exel called such transfer operators non-degenerate. However, the use of the latterterm is a bit unfortunate. For instance, it is used in the related context to mean adifferent property in [12, page 60], and also there are historical reasons to change thisname, see [26].It is important, see [23], that the range of a regular transfer operator L coincides withthe annihilator (ker α ) ⊥ of the kernel of α and L (1) is the unit in L ( A ) = (ker α ) ⊥ , soin particular the latter is a complemented ideal. Definition 2.1.
A pair ( α, L ) where L : A → A is a regular transfer operator for anendomorphism α : A → A will be called a C ∗ -dynamical system .A dissatisfaction concerning asymmetry in the C ∗ -dynamical system ( α, L ) ; α is mul-tiplicative while L is ‘merely’ positive linear, lead the author of [12] to the followingmore general notion. Definition 2.2 ([12], Definition 3.1) . The pair ( V , H ) of positive linear maps V , H : A → A is called an interaction over A if(i) V ◦ H ◦ V = V ,(ii) H ◦ V ◦ H = H ,(iii) V ( ab ) = V ( a ) V ( b ) , if either a or b belong to H ( A ) ,(iv) H ( ab ) = H ( a ) H ( b ) , if either a or b belong to V ( A ) . Remark . An interaction ( V , H ) , or even a C ∗ -dynamical system ( α, L ) , in generaldoes not generate a semigroup of interactions [28] and all the more is not an element NTERACTIONS AND GRAPH ALGEBRAS 9 of a group interaction in the sense of [13]. This will be a generic case in our examplearising from graphs, cf. Proposition 3.5 below. Accordingly, in general the facts provedin [28], [13], can not be applied in our present context.Let ( V , H ) be an interaction. By [12, Propositions 2.6, 2.7], V ( A ) and H ( A ) are C ∗ -subalgebras of A , E V := V ◦ H is a conditional expectation onto V ( A ) , E H := H ◦ V is aconditional expectation onto H ( A ) , and the mappings V : H ( A ) → V ( A ) , H : V ( A ) → H ( A ) are isomorphisms, each being the inverse of the other. Actually we have Proposition 2.4.
The relations E V = V ◦ H , E H = H ◦ V , θ = V| E H ( A ) yield a one-to-one correspondence between interactions ( V , H ) and triples ( θ, E V , E H ) consisting of twoconditional expectations E V , E H and an isomorphism θ : E H ( A ) → E V ( A ) .Proof. It suffices to verify that if ( θ, E V , E H ) is as in the assertion, then V ( a ) := θ ( E H ( a )) and H ( a ) := θ − ( E V ( a )) form an interaction. This is straightforward. (cid:3) Recall that the C ∗ -algebra A has the unit . It follows that the algebras involved inan interaction are automatically also unital. Lemma 2.5. If ( V , H ) is an interaction, then V (1) = E V (1) and H (1) = E H (1) are unitsin V ( A ) and H ( A ) , respectively (in particular, they are projections).Proof. Let us observe that E V (1) = V ( H (1)) = V ( H (1)1) = V ( H (1)) V (1) = V ( H (1)) V (cid:0) H ( V (1)) (cid:1) = V (cid:0) H (1) H ( V (1)) (cid:1) = V (cid:0) H (1 V (1)) (cid:1) = V (cid:0) H ( V (1)) (cid:1) = V (1) . Therefore we have V ( a ) = E V ( V ( a )) = E V (1 V ( a )) = E V (1) V ( a ) = V (1) V ( a ) for arbitrary a ∈ A . It follows that V (1) is the unit in V ( A ) and a similar argument works for H . (cid:3) The following statement generalizes [12, Proposition 3.4].
Proposition 2.6.
Any C ∗ -dynamical system ( α, L ) is an interaction.Proof. Consider the conditions (i)-(iv) in Definition 2.2. Since α ◦ L ◦ α = E ◦ α = α , (i)is satisfied. To see (ii) recall that L (1) is the unit in L ( A ) , cf. [23, Proposition 1.5], andtherefore L ( α ( L ( a ))) = L (1 α ( L ( a ))) = L (1) L ( a ) = L ( a ) . Condition (iii) is trivial for ( α, L ) , and (iv) holds because L ( aα ( b )) = L ( a ) b = L ( a ) L (1) b = L ( a ) L (1 α ( b )) = L ( a ) L ( α ( b )) , and by passing to adjoints we also get L ( α ( b ) a ) = L ( α ( b )) L ( a ) . (cid:3) As shown in [2], in the case the conditional expectation E = α ◦ L is given by(2.1) E ( a ) = α (1) aα (1) , a ∈ A, there is a very natural crossed product associated to the C ∗ -dynamical system ( α, L ) .This crossed product coincides with the one introduced in [11] and is sufficient to covermany classic constructions, see [2]. A transfer operator for which (2.1) holds is called complete [2], [23]. It is a cornerretraction [43], [26]. By [23] a given endomorphism α admits a complete transfer operator L if and only if ker α is a complemented ideal and α ( A ) is a hereditary subalgebra of A . In this case L is a unique regular transfer operator for α , see [23], [2], [43], [26]. Wenaturally generalize the aforementioned concepts to interactions, cf. also [28]. Definition 2.7.
An interaction ( V , H ) will be called a corner interaction if V ( A ) and H ( A ) are hereditary subalgebras of A . Proposition 2.8.
An interaction ( V , H ) is corner if and only if V ( A ) = V (1) A V (1) and H ( A ) = H (1) A H (1) are corners in A . Moreover, for a corner interaction ( V , H ) thefollowing conditions are equivalent i) ( V , H ) is a C ∗ -dynamical system (with a complete transfer operator), ii) V is multiplicative, iii) ker V is an ideal in A , iv) H ( A ) is an ideal in A , v) H (1) lies in the center of A .Proof. For the first part of the assertion apply Lemma 2.5 and notice that if B is ahereditary subalgebra of A and B has a unit P , then B = P AP . To show the secondpart of assertion let us suppose that ( V , H ) is a corner interaction.The implications i) ⇒ ii) ⇒ iii) and the equivalence iv) ⇔ v) are clear.iii) ⇒ v). By the first part of the assertion V is isometric on H (1) A H (1) and thus ker V ∩ H (1) A H (1) = { } . In view of Lemma 2.5, for any a ∈ A we have a (1 − H (1)) ∈ ker V . Hence if ker V is an ideal, then H (1) a (cid:0) − H (1) (cid:1) a ∗ H (1) ∈ (ker V ) ∩ H (1) A H (1) = { } , that is H (1) a (1 − H (1)) = 0 which means that H (1) a = a H (1) .v) ⇒ i). By the first part of the assertion E H ( a ) = H (1) a H (1) . Thus, for any a, b ∈ A ,we have V ( ab ) = V ( E H ( ab )) = V ( H (1) ab H (1)) = V ( a H (1) b H (1))= V ( a E H ( b )) = V ( a ) V ( E H ( b )) = V ( a ) V ( b ) . Hence V is an endomorphism of A . The map H is a transfer operator for V because H ( a V ( b )) = H ( a ) H ( V ( b ))) = H ( a ) H (1) b H (1) = H ( a ) b. (cid:3) As it is indicated by the uniqueness of the complete transfer operator, it turns outthat each mapping in a corner interaction determines the other.
Proposition 2.9.
A positive linear map V : A → A is a part of a non-zero cornerinteraction ( V , H ) if and only if kV (1) k = 1 , V ( A ) is a hereditary subalgebra of A andthere is a projection P ∈ A such that V : P AP → V ( A ) is an isomorphism.Moreover, in the above equivalence P and H are uniquely determined by V , and wehave (2.2) H ( a ) := V − ( V (1) a V (1)) , a ∈ A, where V − is the inverse to V : P AP → V ( A ) . NTERACTIONS AND GRAPH ALGEBRAS 11
Proof.
The necessity of the stated conditions follows from Proposition 2.8 and Lemma2.5. For the sufficiency note that V ( P ) is a unit in V ( A ) and therefore V ( A ) = V ( P ) A V ( P ) , as V ( A ) is hereditary in A . In particular, E V ( a ) := V ( P ) a V ( P ) is aconditional expectation onto V ( A ) . We define E H ( a ) := V − ( V ( a )) where V − is theinverse to V : P AP → V ( A ) . Then E H is an idempotent map of norm one because kE H k = kVk = kV (1) k = 1 . Hence E H is a conditional expectation onto P AP . By Propo-sition 2.4, the triple ( V , E V , E H ) yields a (necessarily corner) interaction ( V , H ) where H ( a ) = V − ( V ( P ) a V ( P )) . In particular, it follows from Lemma 2.5 that V ( P ) = V (1) ,that is H is given by (2.2).What remains to be shown is the uniqueness of P . Suppose then that ( V , H i ) , i = 1 , ,are two corner interactions and consider projections P := H (1) and P := H (1) . Wehave V ( P P P ) = V ( P ) = V (1) = V ( P ) = V ( P P P ) , and as V is injective on H i ( A ) = P i AP i , i = 1 , , it follows that P P P = P and P = P P P . This implies P = P . (cid:3) Crossed product for corner interactions.
From now on ( V , H ) will alwaysstand for a corner interaction. We define the corresponding crossed product in universalterms. Definition 2.10. A covariant representation of ( V , H ) is a pair ( π, S ) consisting of anon-degenerate representation π : A → B ( H ) and an operator S ∈ B ( H ) (which isnecessarily a partial isometry) such that Sπ ( a ) S ∗ = π ( V ( a )) and S ∗ π ( a ) S = π ( H ( a )) for all a ∈ A. The crossed product for the interaction ( V , H ) is the C ∗ -algebra C ∗ ( A, V , H ) generatedby i A ( A ) and s where ( i A , s ) is a universal covariant representation of ( V , H ) . It isequipped with the circle gauge action determined by γ z ( i A ( a )) = i A ( a ) , a ∈ A , and γ z ( s ) = zs .Obviously, the above definition generalizes the crossed product studied in [2]. Inother words C ∗ ( A, V , H ) coincides with Exel’s crossed product [11] when ( V , H ) is a C ∗ -dynamical system. To show it is essentially the same algebra as the one associated to(general) interactions in [12], we realize C ∗ ( A, V , H ) as the crossed product for a Hilbertbimodule. To this end, we conveniently adopt Exel’s construction of his generalizedcorrespondence associated to ( V , H ) , [12, Section 5].Let X = A ⊙ A be the algebraic tensor product over the complexes, and let h· , ·i A and A h· , ·i be the A -valued sesqui-linear functions defined on X × X by h a ⊙ b, c ⊙ d i A = b ∗ H ( a ∗ c ) d, A h a ⊙ b, c ⊙ d i = a V ( bd ∗ ) c ∗ . We consider the linear space X as an A - A -bimodule with the natural module operations: a · ( b ⊙ c ) = ab ⊙ c , ( a ⊙ b ) · c = a ⊙ bc . Lemma 2.11.
A quotient of X becomes naturally a pre-Hilbert A - A -bimodule. Moreprecisely, i) the space X with a function h· , ·i A (respectively A h· , ·i ) becomes a right (respec-tively left) semi-inner product A -module; ii) the corresponding semi-norms k x k A := kh x, x i A k and A k x k := k A h x, x ik coincide on X and thus the quotient space X / k · k obtained by modding out thevectors of length zero with respect to the seminorm k x k := k x k A = A k x k is botha left and a right pre-Hilbert module over A ; iii) denoting by a ⊗ b the canonical image of a ⊙ b in the quotient space X / k · k wehave ac ⊗ b = a ⊗ H ( c ) b, if c ∈ V ( A ) , a ⊗ cb = a V ( c ) ⊗ b, if c ∈ H ( A ) , and a ⊗ b = a V (1) ⊗ H (1) b for all a, b ∈ A ; iv) the inner-products in X / k · k satisfy the imprimitivity condition.Proof. i) All axioms of A -valued semi-inner products for h· , ·i A and A h· , ·i except thenon-negativity are straightforward, and to show the latter one may rewrite the proof of[12, Proposition 5.2] (just erase the symbol e H or put e H = H (1) ).ii) Similarly, the proof of [12, Proposition 5.4] implies that for x = P ni =1 a ∗ i ⊙ b i , a i , b i ∈ A ,we have(2.3) k x k A = kH ( aa ∗ ) H ( V ( bb ∗ )) k = kV ( H ( aa ∗ )) V ( bb ∗ ) k = A k x k where a = ( a , ..., a n ) T and b = ( b , ..., b n ) T are viewed as column matrices.iii) For the first part consult the proof of [12, Proposition 5.6]. The second part can beproved analogously. Namely, for every x , y , a , b ∈ A we have h x ⊗ y, a ⊗ b i A = y ∗ H ( x ∗ a ) b = y ∗ H ( x ∗ a V (1)) H (1) b = h x ⊗ y, a V (1) ⊗ H (1) b i A , which implies that k a ⊗ b − a V (1) ⊗ H (1) b k = 0 .iv) The form of imprimitivity condition allows us to check it only on simple tensors.Using iii), for a, b, c, d, e, f ∈ A , we have a ⊗ b h c ⊗ d, e ⊗ f i A = a ⊗ bd ∗ H ( c ∗ e ) f = a ⊗ H (1) bd ∗ H ( c ∗ e ) f = a V (cid:16) H (1) bd ∗ H ( c ∗ e ) (cid:17) ⊗ f = a V ( H (1) bd ∗ ) V ( H ( c ∗ e )) ⊗ f = a V ( bd ∗ ) V (1) c ∗ e V (1) ⊗ f = a V ( bd ∗ ) c ∗ e ⊗ f = A h a ⊗ b, c ⊗ d i e ⊗ f. (cid:3) Definition 2.12.
We call the completion X of the pre-Hilbert bimodule X describedin Lemma 2.11 a Hilbert bimodule associated to ( V , H ) . Remark . The Hilbert bimodule X could be obtained directly from the imprimitivity K V - K H -bimodule X constructed in [12, Section 5]. Indeed, by (2.3), X and X coincideas Banach spaces, and since h X, X i A = A H (1) A, A h X, X i = A V (1) A,X can be considered as an imprimitivity A V (1) A - A H (1) A -bimodule. Furthermore, themappings λ V : A → K V , λ H : A → K V , the author of [12] uses to define an A - A -bimodulestructure on X , when restricted respectively to A V (1) A and A H (1) A are isomorphisms. NTERACTIONS AND GRAPH ALGEBRAS 13
Hence we may use them to assume the identifications K V = A V (1) A and K H = A H (1) A ,and then Exel generalized correspondence and the Hilbert bimodule X coincide.Now we are ready to identify the structure of C ∗ ( A, V , H ) as the Hilbert bimodulecrossed product. Proposition 2.14.
We have a one-to-one correspondence between covariant representa-tions ( π, S ) of the interaction ( V , H ) and covariant representations ( π, π X ) of the Hilbertbimodule X associated to ( V , H ) . It is given by relations π X ( a ⊗ b ) = π ( a ) Sπ ( b ) , x ∈ X, S = π X (1 ⊗ . In particular, C ∗ ( A, V , H ) ∼ = A ⋊ X Z and the isomorphism is gauge-invariant.Proof. Let ( π, S ) be a covariant representation of ( V , H ) . Since k X i π ( a i ) Sπ ( b i ) k = k X i,j π ( a i ) Sπ ( b i b ∗ j ) S ∗ π ( a ∗ j ) k = k π (cid:16) X i,j a i V ( b i b ∗ j ) a ∗ j (cid:17) k≤ k X i a i ⊗ b i k , we see that π X ( P i a i ⊗ b i ) := P i π ( a i ) Sπ ( b i ) defines a contractive linear mapping on X / k · k . Clearly, it satisfies (1.1) and (1.2). Hence by continuity it extends uniquelyto X in a way that ( π, π X ) is a covariant representation of X . Conversely suppose that ( π, π X ) is a covariant representation of the Hilbert bimodule X and put S := π X (1 ⊗ .Then for a ∈ A we have Sπ ( a ) S ∗ = π X ((1 ⊗ a ) π X (1 ⊗ ∗ = π ( A h ⊗ a, ⊗ i ) = π ( V ( a )) . Similarly, S ∗ π ( a ) S = π X (1 ⊗ ∗ π X ( a (1 ⊗ π ( h ⊗ , a ⊗ i A ) = π ( H ( a )) . (cid:3) Remark . The Hilbert bimodule X is nothing but the GNS C ∗ -correspondence deter-mined by the completely positive map H , cf. [26]. In particular, the above propositionshows that C ∗ ( A, V , H ) is isomorphic to the crossed product of A by the completelypositive map H (or V , depending on preferences), see [26].Finally, by Remark 2.13 and Propositions 1.4, 2.14 we get Corollary 2.16.
Let X be the generalized correspondence constructed out of ( V , H ) asin [12, Section 5] . The crossed product C ∗ ( A, V , H ) for the interaction ( V , H ) and thecovariance algebra C ∗ ( A, X ) for X are naturally isomorphic. Topological freeness, ideal structure and simplicity criteria.
Let ( V , H ) bea corner interaction. Since V ( A ) and H ( A ) are hereditary subalgebras of A we havea standard way, cf. e.g. [37, Proposition 4.1.9], of identifying their spectra with opensubsets of b A . Namely, we assume that(2.4) [ V ( A ) = { π ∈ b A : π ( V (1)) = 0 } , \ H ( A ) = { π ∈ b A : π ( H (1)) = 0 } . The isomorphisms V : H ( A ) → V ( A ) and H : V ( A ) → H ( A ) induce mutually inversehomeomorphisms b V : [ V ( A ) → \ H ( A ) and b H : \ H ( A ) → [ V ( A ) , which under identifications(2.4) become partial homeomorphisms of b A . Definition 2.17.
We refer to b V and b H as partial homeomorphisms dual to ( V , H ) . Remark . For an irreducible representation π : A → B ( H ) with π ( H (1)) = 0 theelement b H ( π ) ∈ b A is given by the (unique up to unitary equivalence) extension of therepresentation b H ( π ) | V ( A ) = π ◦ H : V ( A ) → B ( π ( H (1)) H ) , Moreover, in the case ( V , H ) is a C ∗ -dynamical system H ( A ) is an ideal and then π ( H (1)) H = H . Proposition 2.19. If X is the Hilbert bimodule associated to ( V , H ) and X -Ind is thepartial homeomorphism of b A associated to X , then X -Ind = b H . Proof.
Let π : A → B ( H ) be an irreducible representation with π ( H (1)) = 0 . For ( a ⊗ b ) ⊗ π h ∈ X ⊗ π H , a, b ∈ A, h ∈ H , using Lemma 2.11 iii) we have X -Ind( π )( V (1))( a ⊗ b ) ⊗ π h = (cid:0) V (1) a ⊗ b (cid:1) ⊗ π h = (cid:0) V (1) a V (1) ⊗ b (cid:1) ⊗ π h = (1 ⊗ H ( a ) b ) ⊗ π h = (1 ⊗ ⊗ π π ( H ( a ) b ) h. Hence we see that the space H := X -Ind( π ) (cid:0) V (1) (cid:1) ( X ⊗ π H ) consists of the vectors ofthe form (1 ⊗ ⊗ π h , h ∈ π ( H (1)) H . Moreover, for h ∈ π ( H (1)) H we have k (1 ⊗ ⊗ π h k = h (1 ⊗ ⊗ π h, (1 ⊗ ⊗ π h i = h h, π ( h ⊗ , ⊗ i A ) h i = h h, π ( H (1)) h i = k h k , and thus the mapping (1 ⊗ ⊗ π h h is a unitary U from H onto π ( H (1)) H . For a ∈ V ( A ) we have X -Ind( π )( a )(1 ⊗ ⊗ π h = ( a ⊗ ⊗ π h = (1 ⊗ H ( a )) ⊗ π h = (1 ⊗ ⊗ π π ( H ( a )) h, that is X -Ind( π )( a ) U ∗ h = U ∗ π ( H ( a )) h . It follows that U establishes unitary equivalencebetween X -Ind( π ) : V ( A ) → B ( H ) and π ◦ H : V ( A ) → B ( π ( H (1)) H ) . Hence X -Ind = b H , cf. Remark 2.18. (cid:3) As b H = b V − , our preference for b V in the sequel is totally a subjective choice. Theorem 2.20.
Let ( V , H ) a corner interaction and b V the partial homeomorphism dualto V . i) If b V is topologically free, then every representation of C ∗ ( A, V , H ) which is faithfulon A is automatically faithful on C ∗ ( A, V , H ) . ii) If b V is free, then J \ J ∩ A is a lattice isomorphism between ideals in C ∗ ( A, V , H ) and open b V -invariant sets in b A . iii) If b V is topologically free and minimal, then C ∗ ( A, V , H ) is simple.Proof. Combine Propositions 2.14, 2.19 and Theorem 1.1. (cid:3)
Remark . Our simplicity criterion (Theorem 2.20 iii)) have an intersection with thecriteria in [43, Theorems 4.1, 4.6] only in the case of a C ∗ -dynamical system ( α, L ) where α is an isomorphism from A onto a full corner α (1) Aα (1) in A , cf. Remark 1.3and Corollary 2.23 below. In this case topological freeness implies that no power of α or L is inner (i.e. implemented by an isometry in A ). NTERACTIONS AND GRAPH ALGEBRAS 15
In general, one can deduce from Propositions 2.14, 2.19, see [25, discussion before The-orem 2.5], that open b V -invariant sets in b A are in a one-to-one correspondence with gaugeinvariant ideals in C ∗ ( A, V , H ) . Therefore it is useful to have a convenient descriptionof the former. Lemma 2.22.
Let I be an ideal in A . The following conditions are equivalent: i) The set b I ⊂ b A is b V -invariant, ii) V ( I ) = V (1) I V (1) , iii) V ( I ) ⊂ I and H ( I ) ⊂ I .Proof. Notice that V (1) I V (1) = I ∩ V ( A ) and H (1) I H (1) = I ∩ H ( A ) . Hence V ( I ) = V ( H (1) I H (1)) = V ( I ∩ H ( A )) and ii) reads as V ( I ∩ H ( A )) = I ∩ V ( A ) . Now equivalencei) ⇔ ii) is clear.ii) ⇒ iii). We have V ( I ) = V (1) I V (1) ⊂ I and H ( I ) = H ( V (1) I V (1)) = H ( V ( I )) = H (1) I H (1) ⊂ I .iii) ⇒ ii). The inclusion V ( I ) ⊂ I implies V ( I ) ⊂ V (1) I V (1) and H ( I ) ⊂ I implies that V (1) I V (1) = V ( H ( I )) ⊂ V ( I ) . (cid:3) Corollary 2.23.
Suppose ( α, L ) is a corner C ∗ -dynamical system. The partial home-omorphism b α is minimal if and only if there is no nontrivial ideal I in A such that α ( I ) ⊂ I .Proof. The if part follows immediately from Lemma 2.22. If we suppose that α ( I ) ⊂ I and α ( I ) = α (1) Iα (1) for a certain nontrivial ideal I in A , then one sees (by inductionon n ) that the closure J of elements of the form P nk =0 L k ( a k ) , a k ∈ I , k = 0 , ..., n , n ∈ N , is a nontrivial ideal in A (it does not contain the unit) such that α ( J ) ⊂ J and L ( J ) ⊂ J . Hence by Lemma 2.22, b α is not minimal. (cid:3) K -theory. We retain the notation from page 8 with the additional assumptionthat X is the Hilbert bimodule associated to a corner interaction ( V , H ) . In particular, A h X, X i = A V (1) A . Lemma 2.24.
The following diagram commutes and the horizontal map is an isomor-phism K ∗ ( V ( A )) ι ∗ / / ( ι ◦H ) ∗ & & ▼▼▼▼▼▼▼▼▼▼ K ∗ ( A V (1) A ) ( ι ◦ φ ) ∗ w w ♦♦♦♦♦♦♦♦♦♦♦ K ∗ ( D X ) Proof.
Since V ( A ) is a full corner in A V (1) A it is known that the inclusion ι : V ( A ) → A V (1) A yields isomorphisms of K -groups, cf. e.g. [22, Proposition B.5]. We claim thatthe map M ( H ( A )) ∋ (cid:18) a a a a (cid:19) Φ (cid:18) φ ( V ( a )) 1 ⊗ a ♭ (1 ⊗ a ∗ ) a (cid:19) ∈ D X , where ♭ : X → e X is the canonical antilinear isomorphism, is a homomorphism of C ∗ -algebras. Plainly, it is linear, ∗ -preserving, and the reader easily checks that Φ( ab ) = Φ( a )Φ( b ) , for a = [ a ij ] , b = [ b ij ] ∈ M ( H ( A )) , using the following calculations (1 ⊗ a ) · ♭ (1 ⊗ b ∗ ) x ⊗ y = Θ (1 ⊗ a ) , (1 ⊗ b ∗ ) x ⊗ y = 1 ⊗ a b H ( x ) y = V ( a b ) V ( H ( x )) ⊗ y = V ( a ) V ( b ) x ⊗ y = φ ( V ( a b ))( x ⊗ y ) ,φ ( V ( a ))(1 ⊗ b ) = V ( a ) ⊗ b = 1 ⊗ a b ,♭ (1 ⊗ a ∗ ) · (1 ⊗ b ) = h ⊗ a ∗ , ⊗ b i A = a H (1) b = a b . This shows our claim. The following diagram commutes (it commutes on the level of C ∗ -algebras) K ∗ ( V ( A )) ι ∗ / / ( ι ◦H ) ∗ (cid:15) (cid:15) K ∗ ( A V (1) A ) ( ι ◦ φ ) ∗ (cid:15) (cid:15) K ∗ ( M ( H ( A ))) Φ ∗ / / K ∗ ( D X ) . However, since for any C ∗ -algebra B the homomorphisms ι ii : B → M ( B ) , i = 1 , ,induce the same mappings on the level of K -theory, the mappings ( ι ◦ H ) ∗ , ( ι ◦H ) ∗ : K ∗ ( V ( A )) → K ∗ ( M ( H ( A ))) coincide. Moreover, by the form of Φ we see that Φ ◦ ι ◦ H = ι ◦ H on V ( A ) . Hence ( ι ◦ φ ) ∗ ◦ ι ∗ = Φ ∗ ◦ ( ι ◦ H ) ∗ = Φ ∗ ◦ ( ι ◦ H ) ∗ = ( ι ◦ H ) ∗ . (cid:3) Using the above lemma we see that in sequence (1.5) we may replace K ∗ ( A h X, X i ) = K ∗ ( A V (1) A ) with K ∗ ( V ( A )) and then X ∗ turns into ( ι ) − ∗ ◦ ( ι ◦ H ) ∗ = H ∗ . Hence weget the following version of Pimsner-Voiculescu exact sequence, cf. [36], [42]. Theorem 2.25.
For any corner interaction ( V , H ) we have the following exact sequence K ( V ( A )) ι ∗ −H ∗ / / K ( A ) ( i A ) ∗ / / K ( C ∗ ( A, V , H )) (cid:15) (cid:15) K ( C ∗ ( A, V , H )) O O K ( A ) ( i A ) ∗ o o K ( V ( A )) ι ∗ −H ∗ o o . Graph C ∗ -algebras via interactions In this section we introduce and study properties of graph interactions. We show thatTheorem 2.20 applied to graph interactions is equivalent to the Cuntz-Krieger uniquenesstheorem and its consequences. We use Theorem 2.25 to calculate K -theory for graphalgebras straight from the dynamics on their AF-cores. NTERACTIONS AND GRAPH ALGEBRAS 17
Graph C ∗ -algebra C ∗ ( E ) and its AF-core. Throughout we let E = ( E , E , r, s ) to be a fixed finite directed graph. Thus E is a set of vertices, E is a set of edges, r, s : E → E are range, source maps, and we assume that both sets E , E are fi-nite. We write E n , n > , for the set of paths µ = µ . . . µ n , µ i ∈ E , r ( µ i ) = s ( µ i +1 ) , i = 1 , ..., n − , of length n . The maps r , s naturally extend to E n , so that ( E , E n , s, r ) is the graph, and s extends to the set E ∞ of infinite paths µ = µ µ µ ... . We alsoput s ( v ) = r ( v ) = v for v ∈ E . The elements of E sinks := E \ s ( E ) and re-spectively E sources := E \ r ( E ) are called sinks and sources. We also consider sets E nsinks = { µ ∈ E n : r ( µ ) ∈ E sinks } , n ∈ N .We adhere to conventions of [29], [4]. In our setting a Cuntz-Krieger E -family composeof non-zero pair-wise orthogonal projections { P v : v ∈ E } and partial isometries { S e : e ∈ E } satisfying S ∗ e S e = P r ( e ) and P v = X e ∈ s − ( v ) S e S ∗ e for all v ∈ s ( E ) , e ∈ E . Having such a family we put S µ := S µ S µ · · · S µ n for µ = µ ...µ n ( S µ = 0 ⇒ µ ∈ E n )and S v := P v for v ∈ E . The above Cuntz-Krieger relations extend to operators S µ , see[29, Lemma 1.1], as follows S ∗ ν S µ = S µ ′ , if µ = νµ ′ , µ ′ / ∈ E ,S ∗ ν ′ if ν = µν ′ , ν ′ / ∈ E , otherwise.In particular, C ∗ ( { P v : v ∈ E } ∪ { S e : e ∈ E } ) is the closure of the linear span ofelements S µ S ∗ ν , µ ∈ E n , ν ∈ E m , n, m ∈ N .The graph C ∗ -algebra C ∗ ( E ) of E is a universal C ∗ -algebra generated by a universalCuntz-Krieger E -family { s e : e ∈ E } , { p v : v ∈ E } . It is equipped with the natural circle gauge action γ : T → Aut C ∗ ( E ) established by relations γ λ ( p v ) = p v , γ λ ( s e ) = λs e , for v ∈ E , e ∈ E , λ ∈ T . The fixed point C ∗ -algebra for γ is called the core . It isan AF-algebra of the form F E := span { s µ s ∗ ν : µ, ν ∈ E n , n = 0 , , . . . } . We recall the standard Bratteli diagram for F E . For each vertex v and N ∈ N we set F N ( v ) := span { s µ s ∗ ν : µ, ν ∈ E N , r ( µ ) = r ( ν ) = v } , which is a simple I n factor with n = |{ µ ∈ E N : r ( µ ) = v }| (if n = 0 we put F N ( v ) := { } ). The spaces F N := (cid:16) ⊕ v / ∈ E sinks F N ( v ) (cid:17) ⊕ (cid:16) ⊕ w ∈ E sinks ⊕ Ni =0 F i ( w ) (cid:17) , N ∈ N , form an increasing family of finite-dimensional algebras, cf. e.g. [4], and(3.1) F E = [ N ∈ N F N . We denote by Λ( E ) the corresponding Bratteli diagram for F E . If E has no sinks wecan view Λ( E ) as an infinite vertical concatenation of E : on the n -th level we have the vertices r ( E n ) , n ∈ N , and multiplicities are given by the number of edges withcorresponding endings and sources. If E has sinks, one has to attach to every sink oneach level an infinite tail, so on the n -th level of Λ( E ) we have r ( E n ) ∪ S N − k =0 { v ( k ) : v ∈ r ( E ksinks ) } and each v ( k ) descends to v ( k ) with multiplicity one. We adopt theconvention that if V is a subset of E we treat it as a full subgraph of E and Λ( V ) stands for the corresponding Bratteli diagram for F V . In particular, if V is hereditary ,i.e. s ( e ) ∈ V = ⇒ r ( e ) ∈ V for all e ∈ E , and saturated , i.e. every vertex which feedsinto V and only V is in V , then the subdiagram Λ( V ) of Λ( E ) yields an ideal in F E whichis naturally identified with F V . In general, viewing Λ( E ) as an infinite directed graphthe hereditary and saturated subgraphs (subdiagrams) of Λ( E ) correspond to ideals in F E , see [5, 3.3].3.2. Interactions arising from graphs.
For each vertex v ∈ E we let n v := | r − ( v ) | be the number of the edges received by v . We define an operator s in C ∗ ( E ) as thesum of the partial isometries { s e : e ∈ E } "averaged" on the spaces corresponding toprojections { p v : v ∈ r ( E ) } that are not sources:(3.2) s := X e ∈ E √ n r ( e ) s e = X v ∈ r ( E ) √ n v X e ∈ r − ( v ) s e . Since s ∗ s = P v ∈ r ( E ) p v is a projection the operator s is a partial isometry. It is anisometry iff E has no sources. We use s to define(3.3) V ( a ) := sas ∗ , H ( a ) := s ∗ as, a ∈ C ∗ ( E ) . Plainly, ( V , H ) is a corner interaction over C ∗ ( E ) . Moreover, one sees that V and H areunique bounded linear maps on C ∗ ( E ) satisfying the following formulas(3.4) V (cid:16) s µ s ∗ ν (cid:17) = √ n s ( µ ) n s ( ν ) P e,f ∈ E s eµ s ∗ fν , n s ( µ ) n s ( ν ) = 0 , , n s ( µ ) n s ( ν ) = 0 , (3.5) H (cid:16) s eµ s ∗ fν (cid:17) = 1 √ n s ( µ ) n s ( ν ) s µ s ∗ ν , H (cid:16) p v (cid:17) = P e ∈ s − ( v ) p r ( e ) n r ( e ) , v / ∈ E sinks , , v ∈ E sinks , where µ ∈ E n , ν ∈ E m , n, m ∈ N , e, f ∈ E , v ∈ E . It follows that V and H preserve the core algebra F E . Hence ( V , H ) defines a corner interaction over F E .We note, however, that V hardly ever preserves the canonical diagonal algebra D E :=span (cid:8) s µ s ∗ µ : µ ∈ E n , n ∈ N (cid:9) ⊂ F E . Definition 3.1.
We call the pair ( V , H ) of continuous linear maps on F E satisfying(3.4), (3.5) a (corner) interaction of the graph E or simply a graph interaction .The following statement is one of the facts justifying the above definition. Proposition 3.2.
We have a one-to-one correspondence between Cuntz-Krieger E -families { P v : v ∈ E } , { S e : e ∈ E } for E and faithful covariant representations NTERACTIONS AND GRAPH ALGEBRAS 19 ( π, S ) of the graph interaction ( V , H ) . It is given by the relations S = X e ∈ E √ n r ( e ) S e , P v = π ( p v ) , S e = √ n r ( e ) π ( s e s ∗ e ) S. In particular, we have a gauge-invariant isomorphism C ∗ ( E ) ∼ = C ∗ ( F E , V , H ) .Proof. A Cuntz-Krieger E -family { P v : v ∈ E } , { S e : e ∈ E } yields a representation π of C ∗ ( E ) which is well known to be faithful on F E . By the definition of ( V , H ) thepair ( π | F E , S ) where S := π ( s ) = P e ∈ E √ n r ( e ) S e is a covariant representation of ( V , H ) .Conversely, let ( π, S ) be a faithful representation of ( V , H ) and put P v := π ( p v ) and S e := √ n r ( e ) π ( s e s ∗ e ) S . We claim that { P v : v ∈ E } , { S e : e ∈ E } is a Cuntz-Krieger E -family such that S = P e ∈ E S e √ n r ( e ) . Indeed, for e ∈ E we have S ∗ e S e = n r ( e ) π ( p r ( e ) ) π ( H ( s e s ∗ e )) π ( p r ( e ) ) = π ( p r ( e ) ) = P r ( v ) , and for v ∈ s ( E ) we have X e ∈ s − ( v ) S e S ∗ e = X e ∈ s − ( v ) n r ( e ) π ( s e s ∗ e ) π ( V (1)) π ( s e s ∗ e )= X e ∈ s − ( v ) ,e ,e ∈ E n r ( e ) √ n r ( e ) n r ( e ) π ( s e s ∗ e ( s e s ∗ e ) s e s ∗ e )= X e ∈ s − ( v ) π ( s e s ∗ e ) = π ( p v ) = P v . Now note that S ∗ S = π ( H (1)) = P v ∈ r ( E ) π ( p v ) and thus S = P e ∈ E Sπ ( p v ) . Moreover,for each v ∈ r ( E ) we have X e ∈ r − ( v ) π ( s e s ∗ e ) Sπ ( p v ) S ∗ = X e ∈ r − ( v ) π ( s e s ∗ e V ( p v ))= X e,e ,e ∈ r − ( v ) π ( s e s ∗ e s e s ∗ e ) n v = X e ,e ∈ r − ( v ) π ( s e s ∗ e ) n v = π ( V ( p v ))= Sπ ( p v ) S ∗ . Hence the final space of the partial isometry Sπ ( p v ) decomposes into the orthogonal sumof ranges of the projections π ( s e s ∗ e ) , e ∈ r − ( v ) , and consequently X e ∈ E S e √ n r ( e ) = X e ∈ E π ( s e s ∗ e ) Sπ ( p r ( e ) ) = X v ∈ E X e ∈ r − ( v ) π ( s e s ∗ e ) Sπ ( p v ) = S. (cid:3) Remark . If E has no sources then s is an isometry and V is an injective endomorphismwith hereditary range. In this case C ∗ ( E ) coincides with various crossed products byendomorphisms that involve isometries, cf. [2], [11], [35]. In particular, Proposition3.2 has a nontrivial intersection with [19, Theorem 5.2] proved for locally finite graphswithout sources. Remark . The canonical completely positive map φ E : C ∗ ( E ) → C ∗ ( E ) is given bythe formula φ E ( x ) = X e ∈ E s e xs ∗ e . This map (unlike V but like H ) always preserves both F E and D E and the pair ( φ E , H ) is a C ∗ -dynamical on D E , cf. Proposition 3.5 below. Moreover, if E has no sinks thesame relations as in Proposition 3.2 yield an isomorphism between C ∗ ( E ) and the Exel’scrossed product D E ⋊ ( φ E , H ) N , see [7, Theorem 5.1]. The advantage of D E ⋊ ( φ E , H ) N over C ∗ ( F E , V , H ) is that it starts from a commutative C ∗ -algebra D E . The disadvantagesare that the dynamics in ( φ E , H ) is irreversible and involves two mappings (at leastimplicitly, see [26]), while in essence ( V , H ) is a single map (recall Proposition 2.9)possessing a natural generalized inverse.A natural question to ask is when the graph interaction ( V , H ) is a C ∗ -dynamicalsystem. It is somewhat surprising that this holds only if ( V , H ) is a part of a groupinteraction. We take up the rest of this subsection to clarify this issue in detail. Tothis end we will use a partially-stochastic matrix P = [ p v,w ] arising from the adjacencymatrix A E = [ A E ( v, w )] v,w ∈ E of the graph E . Namely, we let(3.6) p v,w := ( A E ( v,w ) n w , A E ( v, w ) = 0 , , A E ( v, w ) = 0 , where A E ( v, w ) = |{ e ∈ E : s ( e ) = v, r ( e ) = w }| . By a partially-stochastic matrix wemean a non-negative matrix in which each non-zero column sums up to one. Proposition 3.5.
Let s be the operator given by (3.2) and let n ≥ . The followingconditions are equivalent: i) ( V n , H n ) is an interaction over F E , ii) ( φ nE , H n ) is a C ∗ -dynamical system on D E , iii) operator s n is a partial isometry, iv) n -th power of the matrix P = { p v,w } v,w ∈ E is partially-stochastic, v) for any µ ∈ E n and ν ∈ E k , k < n , such that r ( µ ) = r ( ν ) we have s ( ν ) / ∈ E sources .Proof. i) ⇔ iii). As V n ( · ) = s n ( · ) s ∗ n and H n ( · ) = s ∗ n ( · ) s n one readily checks that iii)implies i), and if we assume i) then s n is a partial isometry because H n (1) is a projectionby Lemma 2.5.iii) ⇔ iv). Operator s n is a partial isometry iff H n (1) is a projection. Since H ( p v ) = P w ∈ E p v,w p w , cf. (3.5), we get H n (1) = X v ,...,v n ∈ E p v ,v · p v ,v · ... · p v n − ,v n p v n = X v,w ∈ E p ( n ) v,w p w NTERACTIONS AND GRAPH ALGEBRAS 21 where P n = { p ( n ) v,w } v,w ∈ E stands for the n -th power of P . By the orthogonality ofprojections p w , it follows that H n (1) is a projection iff P v ∈ E p ( n ) v,w ∈ { , } for all w ∈ E ,that is iff P n is partially-stochastic.ii) ⇔ iv). We know that φ E : D E → D E is an endomorphism and H is its transferoperator. Moreover, it is a straight forward fact that an iteration of an endomorphismand its transfer operator gives again an endomorphism and its transfer operator. Thus ( φ nE , H n ) is a C ∗ -dynamical system iff the transfer operator H n is regular, that is iff φ nE ( H n (1)) = φ nE (1) . However, as φ nE ( H n (1)) = X v,w ∈ E p ( n ) v,w φ nE ( p w ) = X v ∈ E ,µ ∈ E n p ( n ) v,r ( µ ) s µ s ∗ µ and φ nE (1) = P µ ∈ E n s µ s ∗ µ we see that φ nE ( H n (1)) = φ nE (1) if and only if P n = { p ( n ) v,w } v,w ∈ E is partially-stochastic.iv) ⇒ v). Assume that v) is not true, that is let µ ∈ E n and ν ∈ E k , k < n , besuch that r ( µ ) = r ( ν ) and s ( ν ) ∈ E sources . Notice that the condition P v ∈ E p ( n ) v,w > isequivalent to existence of η ∈ E n such that w = r ( µ ) . Hence putting w := r ( µ ) = r ( ν ) and v := s ( ν ) we have P v ∈ E p ( n ) v,w > and p ( k ) v ,w > . Then P v ∈ E p ( n − k ) v,v = 0 (because v ∈ E sources ) and therefore < X v ∈ E p ( n ) v,w = X v ∈ E ,vn − k ∈ E \{ v } p ( n − k ) v,v n − k p ( k ) v n − k ,w ≤ X v n − k ∈ E \{ v } p ( k ) v n − k ,w < , that is P n is not partially-stochastic.v) ⇒ iv). Suppose that P v ∈ E p ( n ) v,w > . By our assumption for each < k < n the condition p ( n − k ) v k ,w = 0 implies that v k / ∈ E sources . However, relation v k / ∈ E sources is equivalent to P v k − ∈ E p (1) v k − ,v k = 1 (because P is partially-stochastic). Thereforeproceeding inductively for k = 1 , , ..., n − we get X v ∈ E p ( n ) v,w = X v ,v ∈ E p (1) v ,v p ( n − v ,w = X v ∈ E p ( n − v ,w = ... = X v n − ∈ E p (1) v n − ,w = 1 . (cid:3) Example.
It follows from Proposition 3.5 that if we consider a graph interaction ( V , H ) arising from the following graph q v q w . . .. . . q w n − q v n − q v n q v ✏✏✏✏✮ ✛✛ ✛PPPP✐ ✛✛ then ( V , H ) has the property that its k -th power ( V k , H k ) , for k > , is an interactionunless k = n . Hence by considering a disjoint sum of graphs of the above form onecan obtain a graph interaction with an arbitrary finite distribution of powers beinginteractions.In our specific situation of graph interactions we may prolong the list of equivalentconditions in Proposition 2.8 as follows. Corollary 3.6.
Let ( V , H ) be the interaction associated to the graph E . The followingconditions are equivalent: i) ( V , H ) is a C ∗ -dynamical system, ii) ( V n , H n ) is an interaction for all n ∈ N , iii) ( φ nE , H n ) is a C ∗ -dynamical system for all n ∈ N , iv) operator s given by (3.2) is a power partial isometry, v) every power of the matrix P = { p v,w } v,w ∈ E is partially-stochastic, vi) every two paths in E that have the same length and the same ending either bothstarts in sources or not in sources.Proof. Item vi) holds if and only if item v) in Proposition 3.5 holds for all n ∈ N . Henceby Proposition 3.5 we get the equivalence between all the items from ii) to vi) in thepresent assertion. Furthermore, we recall that H (1) = s ∗ s = P v ∈ r ( E ) p v , and item i)is equivalent to H (1) being a central element in F E , see Proposition 2.8. Hence theequivalence i) ⇔ vi) follows from the relations H (1) s µ s ∗ ν = ( , if s ( µ ) / ∈ r ( E ) s µ s ∗ ν , otherwise , s µ s ∗ ν H (1) = ( , if s ( ν ) / ∈ r ( E ) s µ s ∗ ν , otherwise , which hold for all µ, ν ∈ E n , n ∈ N . (cid:3) A natural question to ask is when H is multiplicative. We rush to say that it is hardlythe case. Proposition 3.7.
The pair ( H , V ) , where ( V , H ) is the interaction of E , is a C ∗ -dynamical system if and only if the mapping r : E → E is injective.Proof. By Proposition 2.8 multiplicativity of H is equivalent to V (1) being a centralelement in F E . If r : E → E is injective, then F E = D E is commutative and ( H , V ) is a C ∗ -dynamical system because V (1) ∈ F E . Conversely, let us assume that theprojection V (1) = ss ∗ = P v ∈ r ( E ) 1 n v P e,f ∈ r − ( v ) s e s ∗ f is central in F E and let g, h ∈ E besuch that r ( g ) = r ( h ) = v . Since V (1) s g s ∗ h = 1 n v X e ∈ r − ( v ) s e s ∗ h , and s g s ∗ h V (1) = 1 n v X f ∈ r − ( v ) s g s ∗ f we have P e ∈ r − ( v ) s e s ∗ h = P f ∈ r − ( v ) s g s ∗ f , which implies g = h . Hence r : E → E isinjective. (cid:3) Dynamical systems dual to graph interactions.
Let ( V , H ) be the interactionof the graph E . We obtain a satisfactory picture of the system dual to ( V , H ) usinga Markov shift (Ω E , σ E ) dual to the commutative system ( D E , φ E ) . Namely, we put Ω E = S ∞ N =0 E Nsinks ∪ E ∞ and let σ E : Ω E \ E sinks → { µ ∈ Ω E : s ( µ ) / ∈ E sources } be theshift defined by the formula σ E ( µ ) = ( µ µ ... if µ = µ µ ... ∈ S ∞ N =2 E Nsinks ∪ E ∞ r ( µ ) if µ ∈ E sinks . NTERACTIONS AND GRAPH ALGEBRAS 23
There is a natural ‘product’ topology on Ω E with the basis formed by the cylinder sets U ν = { νµ : νµ ∈ Ω E } , ν ∈ E n , n ∈ N . Equipped with this topology Ω E is a compactHausdorff space and σ E is a local homeomorphism whose both domain and codomainare clopen. Moreover, the standard argument, cf. e.g. [20, Lemma 3.2], shows that s ν s ∗ ν χ U ν , ν ∈ E n , n ∈ N , establishes an isomorphism D E ∼ = C (Ω E ) which intertwines φ E : D E → D E with the operator of composition with σ E .Let us consider the relation of ‘eventual equality’ defined on Ω E as follows: µ ∼ ν def ⇐⇒ ( ν, µ ∈ E ∞ and µ N µ N +1 ... = ν N ν N +1 ... for some N ∈ N ,ν, µ ∈ E Nsinks for some N ∈ N and r ( µ N ) = r ( ν N ) . Plainly, ∼ is an equivalence relation. We denote by [ µ ] the equivalence class of µ ∈ Ω E ,and view Ω E / ∼ as a topological space equipped with the quotient topology. Lemma 3.8.
The quotient map q : Ω E Ω E / ∼ is open and the sets (3.7) U v,n := { [ µ ] : ∃ η ∈ E k ,k ∈ N s ( η ) = v, r ( η ) = µ n + k } , v ∈ r ( E n ) , n ∈ N , form a basis for the quotient topology of Ω E / ∼ . Moreover, the formula (3.8) [ σ E ][ µ ] := [ σ E ( µ )] defines a partial homeomorphism of Ω E / ∼ with natural domain and codomain: { [ µ ] : µ ∈ Ω E \ E sinks } = [ v ∈ E \ E sinks U v, , { [ µ ] ∈ Ω E : s ( µ ) / ∈ E sources } = [ v ∈ E \ E sources U v, . Proof.
A moment of thought yields that if ν ∈ E n is such that r ( ν ) = v , then q ( U ν ) = U v,n . In particular, one sees that q − ( U v,n ) = { µ ∈ Ω E : ∃ η ∈ E k ,k ∈ N s ( η ) = v, r ( η ) = µ n + k } = [ k ∈ N [ η ∈ Ek,s ( η )= v [ ν ∈ En + kr ( ν )= r ( η ) U ν , which means that U v,n is open in Ω E / ∼ . We conclude that (3.7) defines a basis for thetopology of Ω E / ∼ and q is an open map.Now, it is straightforward to check that (3.8) gives a well defined mapping whose domainand codomain are open sets of the form described in the assertion. The map [ σ E ] isinvertible as for µ ∈ Ω E such that s ( µ ) / ∈ E sources its inverse can be described by theformula [ σ E ] − [ µ ] = [ eµ ] for an arbitrary edge e ∈ E such that r ( e ) = s ( µ ) , where eµ := e when µ ∈ E sinks is a vertex. Since [ σ E ]( U v,n +1 ) = U v,n and [ σ E ] − ( U v,n ) = U v,n +1 for v ∈ E n , n ∈ N , we see that [ σ E ] is a partial homeomorphism. (cid:3) We show that the quotient partial reversible dynamical system (Ω E / ∼ , [ σ E ]) embedsas a dense subsystem into ( c F E , b V ) . Under this embedding the relation ∼ coincides with the unitary equivalence of GNS-representations associated to pure extensions of the purestates of D E = C (Ω E ) . More precisely, for any path µ ∈ Ω E the formula(3.9) ω µ ( s ν s ∗ η ) = ( ν = η = µ ...µ n otherwise , for ν, η ∈ E n , n ∈ N , determines a pure state ω µ : F E → C (a pure extension of the point evaluation δ µ actingon D E = C (Ω E ) ). Indeed, the functional ω µ is a pure state on each F k , k ∈ N , andthus it is also a pure state on F E = S k ∈ N F k , cf. e.g. [5, 4.16]. We denote by π µ theGNS-representation associated to ω µ and take up the rest of the subsection to prove thefollowing Theorem 3.9 (Partial homeomorphism dual to a graph interaction) . Under the abovenotation [ µ ] π µ is a topological embedding of Ω E / ∼ as a dense subset into c F E .This embedding intertwines [ σ E ] and b V . Accordingly, the space c F E admits the followingdecomposition into disjoint sets c F E = ∞ [ N =0 b G N ∪ b G ∞ where the sets b G N = { π µ : [ µ ] ∈ E Nsinks / ∼} are open discrete and b G ∞ = { π µ : [ µ ] ∈ E ∞ / ∼} is a closed subset of c F E . The set ∆ = c F E \ b G is the domain of b V , and b V is uniquely determined by the formula b V ( π µ ) = π σ E ( µ ) , µ ∈ Ω E \ E sinks . In particular, π µ ∈ b V (∆) , for µ ∈ E Nsinks , iff there is ν ∈ E N +1 sinks such that r ( µ ) = r ( ν ) ,and then b H ( π µ ) = π ν . Similarly, π µ ∈ b V (∆) , for µ ∈ E ∞ , iff there is ν ∼ µ suchthat s ( ν ) is not a source, and then for any ν ∈ E such that ν ν ν ... ∈ E ∞ we have b H ( π µ ) = π ν ν ν ,... .Remark . One may verify that if we put A := span( { p v : v ∈ E sinks } ∪ { s e s ∗ e : e ∈ E } ) ∼ = C | E sinks | + | E | , then H preserves A and the smallest C ∗ -algebra containing A and invariant under V is F E . In this sense H : F E → F E is a natural dilation of the positive linear map H : A → A . This explains the similarity of assertions in Theorem 3.9 and in [24,Theorem 3.5]; both of these results describe dual partial homeomorphisms obtained inthe process of dilations. The essential difference is that a dilation of a multiplicative mapon a commutative algebra always leads a commutative C ∗ -algebra, cf. [27], [24], while astochastic factor manifested by a lack of multiplicativity of the initial mapping inevitablyleads to noncommutative objects after a dilation. Significantly, our dual picture of thegraph interaction ‘collapses’ to the non Hausdorff quotient similar to that of Penrosetilings [8, 3.2]. NTERACTIONS AND GRAPH ALGEBRAS 25
We start by noting that the infinite direct sum ⊕ ∞ N =0 ⊕ w ∈ E sinks F N ( w ) yields an ideal I sinks in F E generated by the projections p w , w ∈ E sinks . We rewrite it in the form I sinks = M N ∈ N G N , where G N := (cid:16) ⊕ w ∈ E sinks F N ( w ) (cid:17) . Plainly, F N ( w ) = { } for w ∈ E sinks iff there is µ ∈ E Nsinks such that r ( µ ) = w andthen (since F N ( w ) is a finite factor) π µ is a unique up to unitary equivalence irreduciblerepresentation of F E such that ker π µ ∩ F N ( w ) = { } . Consequently, we see that b I sinks = ∞ [ N =0 b G N ∋ π µ [ µ ] ∈ ∞ [ N =0 E Nsinks / ∼ establishes a homeomorphism between the corresponding discrete spaces. The comple-ment of b I sinks = S ∞ N =0 b G N in c F E is a closed set which we identify in a usual way withthe spectrum of the quotient algebra G ∞ := F E /I sinks . We will describe a dense subset of b G ∞ exploiting the fact that states ω µ arising from µ ∈ E ∞ can be considered as analogs of Glimm’s product states for UHF-algebras, cf.e.g. [37, 6.5]. Lemma 3.11.
For infinite paths µ, ν ∈ E ∞ the representations π µ and π ν are unitarilyequivalent if and only if µ ∼ ν . In particular, [ µ ] π µ is a well defined embedding of E ∞ / ∼ into b G ∞ .Proof. We mimic the proof of the corresponding result for UHF-algebras, cf. [37,6.5.6]. Note that if ( µ N +1 , µ N +2 , ... ) = ( ν N +1 , ν N +2 , ... ) , then both s µ ...µ N s ∗ µ ...µ N and s ν ,...,ν N s ∗ ν ,...,ν N are in F N ( v ) where v = r ( µ N ) and since F N ( v ) ∼ = M n ( C ) there is aunitary u ∈ F N ( v ) such that ω µ ( a ) = ω ν ( u ∗ au ) for a ∈ F N ( v ) . Then automatically ω µ ( a ) = ω ν ( u ∗ au ) for all a ∈ F E and hence π µ ∼ = π ν . Conversely, suppose that π µ ∼ = π ν ,then, cf. [37, 3.13.4], there is a unitary u ∈ F E such that ω µ ( a ) = ω ν ( u ∗ au ) for all a ∈ F E .For sufficiently large n there is x ∈ F n with k u − x k < and k x k ≤ . To get the con-tradiction we assume that µ k = ν k for some k > n . The element a := s µ ...µ k s ∗ µ ...µ k ∈ F k commutes with all the elements from F n . Indeed, if b = s α s ∗ β ∈ F n , α, β ∈ E n , theneither s αf s ∗ βf = 0 for all f ∈ E k − n and then ab = ba = 0 or b = P f ∈ E k − n s αf s ∗ βf and then ab = s αµ n +1 ...µ k s ∗ βµ n +1 ...µ k = ba. From this it also follows that ω ν ( ab ) = 0 for all b ∈ F n .Accordingly, ω ν ( x ∗ ax ) = ω ν ( ax ∗ x ) = 0 and since k ( u ∗ − x ∗ ) au k = k x ∗ a ( u − x ) k < / we get > ω ν (( u ∗ − x ∗ ) au ) + ω ν ( x ∗ a ( u − x )) = ω ν ( u ∗ au ) − ω ν ( x ∗ ax ) = ω µ ( a ) = 1 , an absurd. (cid:3) Remark . The C ∗ -algebra G ∞ is a graph algebra arising from a graph which hasno sinks. Indeed, the saturation E sinks of E sinks (the minimal saturated set containing E sinks ) is the hereditary and saturated set corresponding to the ideal I sinks in F E . Hence I sinks = F E sinks and G ∞ ∼ = F E sinkless where E sinkless := E \ E sinks . Let us now treat µ ∈ E ∞ as the full subdiagram of the Bratelli diagram Λ( E ) wherethe only vertex on the n -th level is r ( µ n ) . Similarly, we treat µ ∈ E Nsinks as the fullsubdiagram of Λ( E ) where on the n -th level for n ≤ N is r ( µ n ) and for n > N is r ( µ ) ( N ) , cf. notation in subsection 3.1. For any µ ∈ Ω E we denote by W ( µ ) the fullsubdiagram of Λ( E ) consisting of all ancestors of the base vertices of µ ⊂ Λ( E ) . Lemma 3.13.
For any µ ∈ Ω E the Bratteli subdiagram Λ(ker π µ ) of Λ( E ) correspondingto ker π µ is Λ( E ) \ W ( µ ) .Proof. The assertion follows immediately from the form of primitive ideal subdiagrams,see [5, 3.8], definition (3.9) of ω µ and the fact that ker π µ is the largest ideal containedin ker ω µ . (cid:3) Lemma 3.14.
The mapping [ µ ] π µ ∈ c F E is a homeomorphism from Ω E / ∼ onto itsimage.Proof. We already know that [ µ ] π µ is injective and restricts to homeomorphismbetween discrete spaces (Ω E \ E ∞ ) / ∼ and c F E \ b G ∞ . Hence it suffices to prove that [ µ ] π µ is continuous and open when considered as a mapping from E ∞ / ∼ onto { π µ : [ µ ] ∈ E ∞ / ∼} ⊂ b G ∞ . To this end, we may assume that E has no sinks, cf.Remark 3.12. Suppose then that E has no sinks.Any open set in c F E is of the form b J = { π ∈ c F E : ker π + J } = { π ∈ c F E :Λ( J ) \ Λ(ker π ) = ∅} where J is an ideal in F E or equivalently Λ( J ) is a hereditaryand saturated subdiagram of Λ( E ) . It follows that if we denote by Λ v,n the smallesthereditary and saturated subdiagram of Λ( E ) which on the n -th level contains vertex v ,then the sets b J v,n := { π ∈ c F E : Λ v,n \ Λ(ker π ) = ∅} , v ∈ E , n ∈ N , form a basis for the topology of c F E . Moreover, in view of Lemma 3.13, definitions of Λ v,n , W ( µ ) and form of U v,n , see (3.7), the preimage of b J v,n under the map [ µ ] π µ is { [ µ ] ∈ Ω E / ∼ : Λ v,n ∩ W ( µ ) = ∅} = { [ µ ] ∈ Ω E : ∃ ν ∈ E k ,k ∈ N s ( ν ) = v, r ( ν ) = µ n + k } = U v,n .Thus, in view of Lemma 3.8, we see that [ µ ] π µ establishes one-to-one corre-spondence between the topological bases for its domain and codomain and hence is ahomeomorphism onto codomain. (cid:3) Now, to obtain Theorem 3.9 we only need the following
Lemma 3.15.
The mapping [ µ ] π µ ∈ c F E intertwines [ σ E ] and b V .Proof. To see that \ V ( F E ) = { π ∈ c F E : π ( V (1)) = 0 } coincides with ∆ = c F E \ b G let π ∈ c F E and note that π ( V (1)) = 0 ⇐⇒ ∀ v ∈ s ( E ) π ( p v ) = 0 ⇐⇒ ∃ w ∈ E sinks π ∼ = π w . NTERACTIONS AND GRAPH ALGEBRAS 27
Furthermore, by (3.4) and (3.5), we have(3.10) V ( F N ( v )) = V (1) F N +1 ( v ) V (1) , H ( F N +1 ( v )) = F N ( v ) , N ∈ N , and H ( F ( v )) ⊂ P w ∈ r ( s − ( v )) F ( w ) . In particular, for µ ∈ E Nsinks , N > , we have π µ ∈ ∆ and ( π µ ◦ V )( F N − ( w )) = π µ ( V (1) F N ( w ) V (1)) = 0 . Hence b V ( π µ ) ∼ = π σ E ( µ ) . Let us now fix µ = µ µ µ ... ∈ E ∞ . Let H µ be the Hilbert spaceand ξ µ ∈ H µ the cyclic vector associated to the pure state ω µ via GNS-construction. For ν, η ∈ E n , using (3.4) and (3.9), we get ω µ ( V ( s ν s ∗ η )) = √ n s ( ν ) n s ( η ) P e,f ∈ E ω µ ( s eν s ∗ fη ) , n s ( ν ) n s ( η ) = 0 , , n s ( ν ) n s ( η ) = 0 , = ( n r ( µ , ν = η = µ ...µ n +1 otherwise = 1 n r ( µ ) ω σ E ( µ ) ( s ν s ∗ η ) . It follows that ω µ ◦ V = n r ( µ ω σ E ( µ ) and therefore b V ( π µ ) ∼ = π σ E ( µ ) , cf. [37, Corollary3.3.8]. (cid:3) Topological freeness of graph interactions.
We will now use Theorem 3.9to identify the relevant properties of the partial homeomorphism b V dual to the graphinteraction ( V , H ) . We recall that the condition (L) introduced in [29] requires thatevery loop in E has an exit. For convenience, by loops we will mean simple loops, thatis paths µ = µ ...µ n such that s ( µ ) = r ( µ n ) and s ( µ ) = r ( µ k ) for k = 1 , ..., n − . Aloop µ is said to have an exit if there is an edge e such that s ( e ) = s ( µ i ) and e = µ i forsome i = 1 , ..., n . Proposition 3.16.
Suppose that every loop in E has an exit. Then every open setintersecting b G ∞ contains infinitely many non-periodic points for b V and if E has no sinksthe number of this non-periodic points is uncountable. In particular, b V is topologicallyfree.Proof. By Theorem 3.9 and Lemma 3.8 it suffices to consider the dynamical system (Ω E / ∼ , [ σ E ]) and an open set of the form U n,v = { [ µ ] : ∃ η ∈ E k ,k ∈ N s ( η ) = v, r ( η ) = µ n + k } which contains [ µ ] for µ = µ µ ... ∈ E ∞ . Since E is finite there must be a vertex v whichappears as a base point of µ infinitely many times. Namely, there exists an increasingsequence { n k } k ∈ N ⊂ N such that r ( µ n k ) = v for all k ∈ N . Moreover, since every loop in E has an exit, the vertex v has to be connected either to a sink or to a vertex lying ontwo different loops. Let us consider these two cases:
1) Suppose ν is a finite path such that v = s ( ν ) and w := r ( ν ) ∈ E sinks . Consider thefamily of finite, and hence non-periodic, paths µ ( n k ) := µ ...µ n k ν ∈ E n + | ν | sinks , k ∈ N . Plainly, all except finitely many of elements [ µ ( n k ) ] belong to U n,v (and they are alldifferent).2) Suppose ν is a finite path such that v = s ( ν ) and the vertex w := r ( ν ) is abase point for two different loops µ and µ . We put µ ǫ = µ ǫ µ ǫ µ ǫ ... ∈ E ∞ for ǫ = { ǫ i } ∞ i =1 ∈ { , } N \{ } . Since there is an uncountable number of non-periodic sequencesin { , } N \{ } which pair-wisely do not eventually coincide the paths µ ǫ correspondingto these sequences give rise to the uncountable family of non-periodic elements [ µ ǫ ] in Ω E / ∼ . Moreover, one readily sees that for sufficiently large n k all the equivalent classesof paths µ ( ǫ ) := µ ...µ n k νµ ǫ ∈ E ∞ , ǫ = { ǫ i } ∞ i =1 ∈ { , } N \{ } belong to U n,v . This proves our assertion. (cid:3) Example.
In the case C ∗ ( E ) = O n is the Cuntz algebra, that is when E is the graph witha single vertex and n edges, n ≥ , then F E is an UHF-algebra and the states ω µ aresimply Glimm’s product states. In particular, it is well known that Prim ( F E ) = { } and c F E is uncountable, cf. [37, 6.5.6]. Hence, on one hand the Rieffel homeomorphism givenby the imprimitivity F E -bimodule X = F E s F E associated with the graph interaction ( V , H ) is the identity on Prim ( F E ) , and thereby it is not topologically free ([28, Theorem6.5] can not be applied). On the other hand, we have just shown that c F E containsuncountably many non-periodic points for X -Ind = b V − , cf. Proposition 2.19, andhence it is topologically free.Suppose now that µ is a loop in E . Let µ ∞ ∈ E ∞ be the path obtained by the infiniteconcatenation of µ . Then Λ( E ) \ W ( µ ∞ ) is a Bratteli diagram for a primitive ideal in F E , which we denote by I µ . In other words, see Lemma 3.13, we have I µ = ker π µ ∞ where π µ ∞ is the irreducible representation associated to µ ∞ . Proposition 3.17.
If the loop µ has no exits, then up to unitary equivalence π µ ∞ is theonly representation of F E whose kernel is I µ and the singleton { π µ ∞ } is an open set in c F E .Proof. The quotient F E /I µ is an AF-algebra with the diagram W ( µ ∞ ) . The path µ ∞ treated as a subdiagram of W ( µ ∞ ) is hereditary and its saturation µ ∞ yields an ideal K in F E /I µ . Since µ ∞ has no exits, K is isomorphic to the ideal of compact operators K ( H ) on a separable Hilbert space H (finite or infinite dimensional). Therefore every faithfulirreducible representation of F E /I µ is unitarily equivalent to the representation given bythe isomorphism K ∼ = K ( H ) ⊂ B ( H ) . This shows that π µ ∞ is determined by its kernel.Moreover, since W ( µ ∞ ) contains all its ancestors, the subdiagram µ ∞ is hereditary andsaturated not only in W ( µ ∞ ) but also in Λ( E ) . Therefore we let now K stand for theideal in F E , corresponding to µ ∞ . Let P ∈ Prim ( F E ) . As K is simple P + K implies NTERACTIONS AND GRAPH ALGEBRAS 29
K ∩ P = { } . By the form of W ( µ ∞ ) and hereditariness of Λ( P ) , K ∩ P = { } implies Λ( P ) ⊂ Λ( F E ) \ W ( µ ∞ ) = Λ( I µ ) . However, if P ⊂ I µ , we must have P = I µ becauseno part of Λ( I µ ) is not connected to W ( µ ∞ ) (consult the form of diagrams of primitiveideals [5, 3.8]). Concluding, we get { P ∈ Prim ( F E ) : P + K} = { P ∈ Prim ( F E ) : K ∩ P = { }} = { I µ } , which means that { I µ } is open in Prim ( F E ) . Accordingly, { π µ ∞ } is open in c F E . (cid:3) We have the following characterization of minimality of b V . Proposition 3.18.
The map V \ F Λ( V ) is a one-to-one correspondence between thehereditary saturated subsets of E and b V -invariant open subsets of c F E . In particular, b V is minimal if and only if there are no nontrivial hereditary saturated subsets of E .Proof. Recall that for a hereditary and saturated subset V of E we treat Λ( V ) as asubdiagram of Λ( E ) where on the n -th level we have ( r ( E n ) ∩ V ) ∪ S N − k =0 { v ( k ) : v ∈ r ( E ksinks ) ∩ V } . Now, using condition iii) of Lemma 2.22 and relations (3.10) one readilysee that the open set b I for an ideal I in F E is b V -invariant if and only if the correspondingBratteli diagram for I is of the form Λ( V ) where V ⊂ E is hereditary and saturated. (cid:3) Combining the above results we not only characterize freeness and topological freenessof ( c F E , b V ) but also spot out an interesting dichotomy concerning its core subsystem ( b G ∞ , b V ) , cf. Remark 3.20 below. Theorem 3.19.
Let ( c F E , b V ) be a partial homeomorphism dual to the graph interaction ( V , H ) . We have the following dynamical dichotomy: a) either every open set intersecting b G ∞ contains infinitely many nonperiodic pointsfor b V ; this holds if every loop in E has an exit, or b) there are b V -periodic orbits O = { π µ ∞ , π σ E ( µ ∞ ) ..., π σ n − E ( µ ∞ ) } in b G ∞ forming opendiscrete sets in c F E ; they correspond to loops without exits µ .In particular, I) b V is topologically free if and only if every loop in E has an exit (satisfies condition(L)), II) b V is free if and only if every loop has an exit connected to this loop (satisfies theso-called condition (K) introduced in [30] , see also [4] ).Proof. In view of Propositions 3.16, 3.17 only item II) requires a comment. By Proposi-tion 3.18 every closed b V -invariant set is of the form c F E \ c F V = [ F E \ V for a hereditary andsaturated subset V ⊂ E . Hence b V is free if and only if every loop outside a hereditarysaturated set V has an exit outside V . The latter condition is clearly equivalent to thecondition that every loop has an exit connected to this loop, cf. [4, page 318]. (cid:3) Remark . Since E is finite, by [29, Theorem 3.9], C ∗ ( E ) is purely infinite in the sensof [31], [29] if and only if E has no sinks and every loop in E has an exit. In view ofProposition 3.16 we conclude that C ∗ ( E ) is purely infinite if and only if every nonempty open set in c F E contains uncountable number of nonperiodic points for b V . In particular,every b V -periodic orbit O = { π µ ∞ , π σ E ( µ ∞ ) ..., π σ n − E ( µ ∞ ) } yields a gauge invariant ideal J O in C ∗ ( E ) (generated by T π ∈ d F E \ O ker π ) which is not purely infinite. Indeed, if v = s ( µ ) is the source of a loop µ witch has no exit, then p v C ∗ ( E ) p v = p v J O p v = C ∗ ( s µ ) ∼ = C ( T ) because s µ is a unitary in C ∗ ( s µ ) with the full spectrum, cf. [29, proof of Theorem 2.4].Concluding, we deduce from our general results for corner interactions the followingfundamental classic results for graph algebras, cf. [39], [29], [30], [4]. Corollary 3.21.
Consider the graph C ∗ -algebra C ∗ ( E ) of the finite directed graph E . i) If every loop in E has an exit, then any Cuntz-Krieger E -family { P v : v ∈ E } , { S e : e ∈ E } generates a C ∗ -algebra isomorphic to C ∗ ( E ) , via s e S e , p v P v , e ∈ E , v ∈ E . ii) If every loop in E has an exit connected to this loop, then there is a latticeisomorphism between hereditary saturated subsets of E and ideals in C ∗ ( E ) ,given by V J V , where J V is an ideal generated by p v , v ∈ V . iii) If every loop in E has an exit and E has no nontrivial hereditary saturated sets,then C ∗ ( E ) is simple.Proof. Apply Propositions 3.2, 3.18 and Theorems 2.20, 3.19. (cid:3) K -theory. We now turn to description of K -groups for C ∗ ( E ) . As K groups forAF-algebras are trivial, using Pimsner-Voiculescu sequence from Theorem 2.25 appliedto the graph interaction ( V , H ) associated to E we have K ( C ∗ ( E )) ∼ = ker( ι ∗ − H ∗ ) ,K ( C ∗ ( E )) ∼ = coker( ι ∗ − H ∗ ) = K ( F E ) / im( ι ∗ − H ∗ ) where ( ι ∗ − H ∗ ) : K ( V ( F E )) → K ( F E ) . Hence to calculate the K -groups for C ∗ ( E ) weneed to identify ker( ι ∗ − H ∗ ) and coker( ι ∗ − H ∗ ) . We do it in two steps. Proposition 3.22 ( K -partial automorphism induced by a graph interaction) . Thegroup K ( F E ) is the universal abelian group h V i generated by the set V := { v ( N ) : v ∈ r ( E N ) , N ∈ N } of ‘endings of finite paths’, subject to relations (3.11) v ( N ) = X s ( e )= v r ( e ) ( N +1) for all v ∈ r ( E N ) \ E sinks . In particular, the subgroup generated by v ( N ) ∈ V , v ∈ E sinks , N ∈ N , in K ( F E ) is freeabelian. The groups K ( V ( A )) and K ( H ( A )) embeds into subgroups of K ( F E ) and wehave K ( V ( F E )) = h V \ { v (0) : v ∈ E sinks }i ,K ( H ( F E )) = h{ v ( N ) ∈ V : v ( N +1) ∈ V }i . The isomorphism H ∗ : K ( V ( A )) → K ( H ( A )) is determined by (3.12) H ∗ ( v ( N +1) ) = v ( N ) , N ∈ N . NTERACTIONS AND GRAPH ALGEBRAS 31
Proof.
We identify v ( N ) with the K -group element [ s µ s ∗ µ ] where µ ∈ E N and v = r ( µ ) .It follows from (3.1) that the group K ( F E ) is the inductive limit lim −→ ( K ( F N ) , i NE ) where K ( F N ) ∼ = M v ∈ r ( E N ) \ E sinks Z v ( N ) ⊕ M k =0 ,...,N M v ∈ r ( E k ) ∩ E sinks Z v ( k ) . Under the above isomorphisms, the bonding maps i NE : K ( F N ) → K ( F N +1 ) , N ∈ N ,are given on generators by the formula i NE ( v ( N ) ) = (P s ( e )= v r ( e ) ( N +1) , v / ∈ E sinks v ( N ) , v ∈ E sinks , v ∈ r ( E N ) . This immediately implies the first part of the assertion.Since H ( F E ) = H (1) F E H (1) is the closure of S N ∈ N H (1) F N H (1) and K ( H (1) F N H (1)) embeds into K ( F N ) we see by continuity pf K that K ( H ( F E )) embeds into K ( F E ) = h V i . Moreover, as H (1) = P v ∈ r ( E ) p v we get H (1) F N ( v ) H (1) = { } ⇐⇒ F N +1 ( v ) = { } ⇐⇒ v ( N +1) ∈ V, whence K ( H ( F E )) identifies with h{ v ( N ) ∈ V : v ( N +1) ∈ V }i .Similarly, since V (1) = P v ∈ r ( E ) 1 n v P e,f ∈ r − ( v ) s e s ∗ f we infer that V (1) F N ( v ) V (1) = { } forall v ∈ E and N > , and V (1) F ( v ) V (1) = { } if and only if v / ∈ E sinks . Thus we mayidentify K ( V ( F E )) with h V \ { v (0) : v ∈ E sinks }i . Now (3.12) follows from (3.5). Notethat (3.12) determines H ∗ , as for v ∈ E \ E sinks , using only (3.11) and (3.12) we have H ∗ ( v (0) ) = H ∗ ( P s ( e )= v r ( e ) (1) ) = P s ( e )= v r ( e ) (0) . (cid:3) We let Z ( E \ E sinks ) and Z E denote the free abelian groups on free generators E \ E sinks and E , respectively. We consider the group homomorphism ∆ E : Z ( E \ E sinks ) → Z E defined on generators as ∆ E ( v ) = v − X s ( e )= v r ( e ) . The following lemma can be viewed as a counterpart of Lemmas 3.3, 3.4 in [40]. Never-theless, it is a slightly different statement.
Lemma 3.23.
We have isomorphisms ker( ι ∗ − H ∗ ) ∼ = ker(∆ E ) , coker( ι ∗ − H ∗ ) ∼ = coker(∆ E ) , which are given on generators by ker ∆ E ∋ v i (0) v (0) ∈ ker( ι ∗ − H ∗ ) , Z E / im(∆ E ) ∋ [ v ] j (0) [ v (0) ] ∈ K ( F E ) / im( ι ∗ − H ∗ ) . Proof.
Suppose a = P v ∈ E \ E sinks a v v ∈ Z ( E \ E sinks ) . Then by (3.11), (3.12) we have ( ι ∗ − H ∗ )( i (0) ( a )) = X v ∈ E \ E sinks a v v (0) − X v ∈ E \ E sinks a v X s ( e )= v r ( e ) (0) = i (0) (∆ E ( a )) . Accordingly, a ∈ ker ∆ E implies i (0) ( a ) ∈ ker( ι ∗ − H ∗ ) and hence i (0) is well defined.Clearly i (0) is injective. To show that it is surjective note that(3.13) x = X v ∈ r ( E N ) \ E sinks x v v ( N ) + X k =1 ,...,N X v ∈ r ( E k ) ∩ E sinks x ( k ) v v ( k ) is a general form of an element in K ( V ( A )) and assume x is in ker( ι ∗ −H ∗ ) . The relation x = H ∗ ( x ) implies that the coefficients corresponding to sinks in the expansion (3.13)are zero. Thus x = H n ∗ ( x ) = P v ∈ E \ E sinks x v v (0) = i (0) ( a ) where a := P v ∈ E \ E sinks x v v is in ker ∆ E because i (0) ( a ) = x = H ∗ ( x ) = H ∗ ( i (0) ( a )) = i (0) (∆ E ( a )) . Hence i (0) is anisomorphism.Since i (0) intertwines ∆ E and ( ι ∗ − H ∗ ) we see that j (0) is well defined. To show that j (0) is surjective, let y = x + P v ∈ E sinks x (0) v v (0) where x is given by (3.13) (this is a generalform of an element in K ( F E ) ). Observe that as x − H ∗ ( x ) ∈ im( ι ∗ − H ∗ ) the element y has the same class in coker( ι ∗ − H ∗ ) as H ∗ ( x ) + X v ∈ E sinks x (0) v v (0) = z + X k =0 , X v ∈ r ( E k ) ∩ E sinks x ( k ) v v (0) where z = P v ∈ r ( E N ) \ E sinks x v v ( N − + P k =2 ,...,N P v ∈ r ( E k ) ∩ E sinks x ( k ) v v ( k − is in K ( V ( A )) .Applying the above argument to z and proceeding in this way N times we get that y isin the same class in coker( ι ∗ − H ∗ ) as X v ∈ r ( E N ) \ E sinks x v v (0) + X k =0 , ,...,N X v ∈ r ( E k ) ∩ E sinks x ( k ) v v (0) . Hence y = j (0) X v ∈ r ( E N ) \ E sinks x v [ v ] + X k =0 , ,...,N X v ∈ r ( E k ) ∩ E sinks x ( k ) v [ v ] . The proof of injectivity of j (0) is slightly more complicated. Let us consider a = P v ∈ E a v v ∈ Z E such that i ( a ) ∈ im( ι ∗ − H ∗ ) . Then i ( a ) = x − H ∗ ( x ) for anelement x of the form (3.13), and hence i ( a ) = X v ∈ r ( E N ) \ E sinks x v v ( N ) − X s ( e )= v r ( e ) ( N ) + X k =1 ,...,N X v ∈ r ( E k ) ∩ E sinks x ( k ) v (cid:0) v ( k ) − v ( k − (cid:1) . NTERACTIONS AND GRAPH ALGEBRAS 33
On the other hand, applying N -times relation (3.11) to i ( a ) = P v ∈ E a v v (0) we get i ( a ) = X µ ∈ E N a s ( µ ) v ( N ) r ( µ ) + X k =0 ,...,N X µ ∈ E ksinks a s ( µ ) v ( k ) r ( µ ) . Comparing coefficients in the above two formulas one can see that(3.14) a v = X r ( e )= v x s ( e ) + X k =1 ,...,N X µ ∈ E k ,r ( µ )= v a s ( µ ) for v ∈ E sinks (in particular a v = 0 for v ∈ E sinks \ r ( E ) ), and(3.15) X µ ∈ E N ,r ( µ )= v a s ( µ ) = x v − X r ( e )= v x s ( e ) , for v ∈ r ( E N ) \ E sinks . We define an element of Z ( E \ E sinks ) by b := X v ∈ r ( E N ) \ E sinks x v v + X k =0 ,...,N − X µ ∈ E k \ E ksinks a s ( µ ) r ( µ ) . Using (3.14) and (3.15), in the third equality below, we obtain ∆ E b = b − X v ∈ r ( E N ) X r ( e )= v x s ( e ) v − X µ ∈ E k ,k =1 ,...,N a s ( µ ) r ( µ )= X v ∈ r ( E N ) \ E sinks x v − X r ( e )= v x s ( e ) − X µ ∈ E N ,r ( µ )= v a s ( µ ) v + X v ∈ E \ E sinks a v − X v ∈ E sinks X r ( e )= v x s ( e ) + X k =1 ,...,N X µ ∈ E k ,r ( µ )= v a s ( µ ) v = 0 + X v ∈ E \ E sinks a v v + X v ∈ E sinks a v v = a. (cid:3) Corollary 3.24 (cf. Theorem 3.2 in [40]) . We have isomorphisms K ( C ∗ ( E )) ∼ = ker(∆ E ) , K ( C ∗ ( E )) ∼ = coker(∆ E ) . References [1] B. Abadie, S. Eilers, R. Exel,
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