Crossed Products for Interactions and Graph Algebras

We consider Exel’s interaction (V,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathcal{V, H})}$$\end{document} over a unital C*-algebra A, such that V(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{V}(A)}$$\end{document} and H(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}(A)}$$\end{document} are hereditary subalgebras of A. For the associated crossed product, we obtain a uniqueness theorem, ideal lattice description, simplicity criterion and a version of Pimsner–Voiculescu exact sequence. These results cover the case of crossed products by endomorphisms with hereditary ranges and complemented kernels. As model examples of interactions not coming from endomorphisms we introduce and study in detail interactions arising from finite graphs. The interaction (V,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathcal{V, H})}$$\end{document} associated to a graph E acts on the core FE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}_E}$$\end{document} of the graph algebra C*(E). By describing a partial homeomorphism of FE^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\mathcal{F}_E}}$$\end{document} dual to (V,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathcal{V, H})}$$\end{document} we find the fundamental structure theorems for C*(E), such as Cuntz–Krieger uniqueness theorem, as results concerning reversible noncommutative dynamics on FE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}_E}$$\end{document} . We also provide a new approach to calculation of K-theory of C*(E) using only an induced partial automorphism of K0(FE)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K_0(\mathcal{F}_E)}$$\end{document} and the six-term exact sequence.


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 such that the image of one map is in the multiplicative domain of the other. An interaction can 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 various problems, cf. [13,14,17]. However, at present there is only one significant application of an interaction (V, H) which is not a C * -dynamical system. Namely, in the recent paper [14] Exel showed that the C * -algebra O n,m introduced in [3], is Morita equivalent to 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 under consideration general structure theorems known so far concern only the case when the initial object is an injective endomorphism, cf. [7,18,35,36,43]. In particular, there are no such theorems for genuine interactions, i.e when both V and H are not multiplicative. The purpose of the present article is twofold. Firstly, we establish general tools to study the structure of C * (A, V, H) for an accessible 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. More precisely, 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 418 B. K. Kwaśniewski IEOT 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 isometry but a partial isometry. Hence the mapping F E a → sas * ∈ F E may be no longer multiplicative (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 can be viewed from many different perspectives as a model example illustrating and giving new insight, for instance, to the following objects and issues that we hope to be pursued in the future.
• Interactions with nontrivial algebras and not multiplicative dynamics. The crossed product C * (F E , V, H) is naturally isomorphic to the graph C *algebra C * (E) (Proposition 3.2). In general, (V, H) is not a C * -dynamical system and is not a part of a group interaction [13]. We precisely identify the values of n ∈ N for which (V n , H n ) is an interaction (see Proposition 3.5), and it turns out that such n's might have almost arbitrary 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). • 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 products of 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 that when E is finite and has no sinks, then C * (E) is isomorphic to Exel crossed product D E φE ,L N where D E ∼ = C(E ∞ ) is a canonical masa in F E . The spectrum of D E is identified with the space of infinite paths E ∞ , φ E is a transpose to the Markov shift on E ∞ and L is its classical Ruelle-Perron-Frobenious operator. Both φ E and L extend naturally to completely positive maps on C * (E) and the extension of φ E is called the noncommutative Markov shift, cf. e.g. [20]. However, from the point of view of the crossed 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 invariant under 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 Markov shift. Our dual and K-theoretic pictures of (V, H) (see Theorem 3.9 and Proposition 3.22, respectively) support this point of view. • Graph C * -algebras. The structure of graph algebras was originally studied via grupoids [29,30], and K-theory was calculated using a dual Pimsner-Voiculescu exact sequence and skew products of initial graphs [39,40]. The corresponding results can also 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. We show that the partial homeomorphism 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 Vol. 80 (2014) Interactions and Graph Algebras 419 uniqueness theorem [4,30,39] from our general uniqueness theorem for interactions. Similarly, we see that freeness of V is equivalent to condition (K) for E [4,30]. Thus minimality and freeness of V is equivalent to the known simplicity criteria for C * (E). Moreover, it turns out that pure infiniteness of C * (E), as defined in [29,31], is equivalent to a very strong version of topological freeness of V (see Remark 3.20), which therefore might be considered 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 direct upon the existing ones; it uses only direct limit description of the AF-core F E and the cyclic six-term exact sequence. • Topological freeness. The condition known as topological freeness was for the first time explicitly stated in [32] where the author use it to show, what we call here, uniqueness theorem. Namely, he proved that topological freeness of a homeomorphism dual to an automorphisms α of a C * -algebra A implies that any representation of A α Z whose restriction to A is injective, is automatically faithful. The converse implication (equivalence between topological freeness and the aforementioned uniqueness property) in the case 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 the assumption that A is separable. The proof is nontrivial and passes through conditions involving such notions as Connes spectrum, inner derivations, or proper outerness. Since it 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 3.19) serves as a good illustration and a starting point for further generalizations of the aforementioned notions and facts. • 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 assume that A and C * (A ∪ {s}) have a common unit. Then V(·) = s(·)s * is a completely positive map 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 is the smallest C * -algebra preserved by H and containing A. Plainly, the pair (V, H) is a corner interaction on B. Hence, potentially, our results could be applied to study the structure of Nevertheless, the dilation of V from A to B is a nontrivial procedure and in general depends on the initial representation of V via s. The core 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 above situation when A ∼ = C N is a finite dimensional commutative C *algebra, see Remark 3.10. In particular, Theorem 3.9 can be interpreted as that the partial homeomorphism dual to a dilation of the Ruelle-Perron- We begin by presenting relevant notions and statements concerning Hilbert bimodules and briefly clarifying their relationship with generalized C * -correspondences. General corner interactions are studied in Sect. 2. Section 3 is devoted to analysis of graph interactions.

Preliminaries on Hilbert Bimodules
Throughout A is a C * -algebra which (starting from Sect. 2) will always be unital. By homomorphisms, epimorphisms, etc. between C * -algebras we always mean * -preserving maps. All ideals in C * -algebras are assumed to be closed and two sided. We adhere to the convention that 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. a homeomorphism whose domain Δ and range ϕ(Δ) are open subsets of M , is topologically free if for any n > 0 the set of fixed points for ϕ n (on its natural domain) has empty interior.
If there are no nontrivial closed invariant sets, then ϕ is called minimal, and ϕ is said to be (residually) free, if it is topologically free on every closed invariant set (in the Hausdorff space case this amounts to requiring that ϕ has no periodic points).
Following [6, 1.8] and [1] by a Hilbert bimodule over A we mean X which is both a left Hilbert A-module and a right Hilbert A-module with respective inner products ·, · A and A ·, · satisfying the so-called imprimitivity condition: x · y, z A = A x, y · z, for all x, y, z ∈ X. A covariant representation of X is a pair (π A , π X ) consisting of a homomorphism π A : A → B(H) and a linear map π X : X → B(H) such that for all a ∈ A, x, y ∈ X. The crossed product A X Z is a C * -algebra generated by a copy of A and X universal with respect to covariant representations of X, see [1]. It is equipped 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 representation of A and its equivalence class in the spectrum A of A. It should not cause confusion when we consider induced representations, as for a Hilbert bimodule X over A the induced 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 representation X -Ind(π) : A → B(X ⊗ π H) where the Hilbert space X ⊗ π H is generated by simple tensors x ⊗ π h, x ∈ X, h ∈ H, satisfying Vol. 80 (2014) Interactions and Graph Algebras 421 The spaces X, X A and A X, X are ideals in A and the bimodule X implements a Morita equivalence between them. Hence X -Ind : X, X A → A X, X is a homeomorphism which we may naturally treat as a partial homeomorphism of A, see [25]. The results of [25] can be summarized as follows.
Theorem 1.1. Let X -Ind be a partial homeomorphism of A, as described above.
(i) If X -Ind is topologically free, then every faithful covariant representa- Plainly, topological freeness of (Prim (A), h X ) implies topological freeness of ( A, X -Ind), but the converse is not true and as we will see, cf. Example 3.4 below, Cuntz algebras O n provide an excellent example of this phenomenon. Remark 1.3. Schweizer [43] showed that if X is a full nondegenerate C *correspondence over a unital C * -algebra A, then the Cuntz-Pimsner algebra O X , defined as in [38], is simple 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 and topological freeness of X -Ind implies the aperiodicity of X. Moreover, the algebras O X and A X Z coincide if and only if A X, X is an essential ideal in A (which in turn is equivalent to injectivity of the left action of A on X). In particular, if the ideal A X, X is essential in A and X, X 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 with the ternary ring operation [x, y, z] := x y, z A = A x, y z, x, y, z ∈ X, 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,Proposition 7.6] as the triple (X, λ, ρ) where we consider X as a A X, X -X, X A -Hilbert bimodule and define homomorphisms λ : A → A X, X and ρ : A → X, X 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 naturally isomorphic to the covariance algebra C * (A, X), as defined in [12, 7.12], for X treated as a generalized correspondence. IEOT Proof. The Toeplitz algebra T (A, X) for the generalized correspondence X, see [12, page 57], is a universal C * -algebra generated by a copy of A and X subject to all A-A-bimodule relations plus the ternary ring relations: 3) The C * -algebra C * (A, X) is the quotient T (A, X)/(J + J r ) where J (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 ax = kx (or resp. xa = xk) for all x ∈ X. (1.4) Note that (ker λ) ⊥ = A X, X and (ker ρ) ⊥ = X, X 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 and A X Z are universal C * -algebras generated by copies of A and X subject to the same relations.
Katsura [22] obtained a version of the Pimsner-Voiculescu exact sequence for general C * -correspondences and their C * -algebras. We recall it in the case X is a Hilbert bimodule and in a form suitable for our purposes. We consider the linking algebra D X = K(X ⊕ A) in the following matrix representation where X is the dual Hilbert bimodule of X, cf. e.g. [41, pages 49, 50]. Let ι : A X, X → A, ι 11 is an isomorphism and by [22,Theorem 8.6] the following sequence is exact: where φ : A → L(X) is the homomorphism implementing the left action of A on X, and X * : K * ( A X, X ) → K * (A) is the composition of (ι 11 ) * : K * ( A X, X ) → K * (D X ) and the inverse to the isomorphism (ι 22 ) * : Vol. 80 (2014) Interactions and Graph Algebras 423

Corner Interactions and Their Crossed Products
In this section, following closely the relationship between C * -dynamical systems and interactions, we introduce corner interactions, describe the structure of the associated crossed product and establish fundamental tools for its analysis (Theorems 2.20, 2.25).

Interactions and C * -Dynamical Systems
It is instructive to consider interactions as generalization of pairs (α, L), sometimes called Exel systems [19], consisting of an endomorphism α : A → A and its transfer operator, i.e. a positive linear map L : [11]. Then L is automatically continuous, *preserving, and we also have: L(bα(a)) = L(b)a, a, b ∈ A. We say that a transfer 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 latter term is a bit unfortunate. For instance, it is used in the related context to mean a different property in [12, page 60], and also there are historical reasons to change this name, see [26]. It is important, see [23], that the range of a regular transfer operator L coincides with the annihilator (ker α) ⊥ of the kernel of α and L(1) is the unit in L(A) = (ker α) ⊥ , so in particular the latter is a complemented ideal. Definition 2.1. A pair (α, L) where L : A → A is a regular transfer operator for an endomorphism α : A → A will be called a C * -dynamical system.
A dissatisfaction concerning asymmetry in the C * -dynamical system (α, L); α is multiplicative while L is 'merely' positive linear, lead the author of [12] to the following more general notion.
3. An interaction (V, H), or even a C * -dynamical system (α, L), in general does not generate a semigroup of interactions [28] and all the more is not an element of a group interaction in the sense of [13]. This will be a generic case in our example arising from graphs, cf. Proposition 3.5 below. Accordingly, in general the facts proved in [13,28], can not be applied in our present context. Recall that the C * -algebra A has the unit 1. It follows that the algebras involved in an interaction are automatically also unital.
and H(A), respectively (in particular, they are projections).
Proof. Let us observe that for arbitrary a ∈ A. It follows that V(1) is the unit in V(A) and a similar argument works for H.
As shown in [2], in the case the conditional expectation E = α • L is given by 1) 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 cover many classic constructions, see [2].
A transfer operator for which (2.1) holds is called complete [2,23]. It is a corner retraction [26,43]. 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 [2,23,26,43]. We naturally generalize the aforementioned concepts to interactions, cf. also [28].

An interaction (V, H) is corner if and only if V(A) = V(1)AV(1) and H(A) = H(1)AH(1) are corners in A. Moreover, for a corner interaction (V, H) the following conditions are equivalent
Proof. For the first part of the assertion apply Lemma 2.5 and notice that if B is a hereditary subalgebra of A and B has a unit P , then B = P AP .
To show the second part of assertion let us suppose that Hence V is an endomorphism of A. The map H is a transfer operator for V because

H(aV(b)) = H(a)H(V(b))) = H(a)H(1)bH(1) = H(a)b.
As it is indicated by the uniqueness of the complete transfer operator, it turns out that 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 corner interaction (V, H) if and only if V(1) = 1, V(A) is a hereditary subalgebra of A and there is a projection
Moreover, in the above equivalence P and H are uniquely determined by V, and we have Proof. The necessity of the stated conditions follows from Proposition 2.8 and Lemma 2.5. For the sufficiency note that V(P ) is a unit in V(A) and therefore V( ). 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, 2, are two corner interactions and consider projections P 1 := H 1 (1) and P 2 := H 2 (1). We have and as V is injective on H i (A) = P i AP i , i = 1, 2, it follows that P 1 P 2 P 1 = P 1 and P 2 = P 2 P 1 P 2 . This implies P 1 = P 2 .

Crossed Product for Corner Interactions
From now on (V, H) will always stand for a corner interaction. We define the corresponding crossed product in universal terms.

Definition 2.10.
A covariant representation of (V, H) is a pair (π, S) consisting of a non-degenerate representation π : A → B(H) and an operator S ∈ B(H) (which is necessarily 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 Obviously, the above definition generalizes the crossed product studied in [2]. In other 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 Hilbert bimodule. To this end, we conveniently adopt Exel's construction of his generalized correspondence associated to (V, H), [12,Section 5].
Let X 0 = A A be the algebraic tensor product over the complexes, and let ·, · A and A ·, · be the A-valued sesqui-linear functions defined on We consider the linear space X 0 as an A-A-bimodule with the natural module operations:

Lemma 2.11. A quotient of X 0 becomes naturally a pre-Hilbert A-A-bimodule.
More precisely, (i) the space X 0 with a function ·, · A (respectively A ·, · ) becomes a right (respectively left) semi-inner product A-module; (ii) the corresponding semi-norms coincide on X 0 and thus the quotient space X 0 / · obtained by modding out the vectors of length zero with respect to the seminorm x := x A = A x is both a left and a right pre-Hilbert module over A; Vol. 80 (2014) Interactions and Graph Algebras 427 Proof. (i) All axioms of A-valued semi-inner products for ·, · A and A ·, · except the non-negativity are straightforward, and to show the latter one may rewrite the proof of [ where a = (a 1 , . . . , a n ) T and b = (b 1 , . . . , 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 be proved analogously. Namely, for every x, y, a, b ∈ A we have Definition 2.12. We call the completion X of the pre-Hilbert bimodule X 0 described in Lemma 2.11 a Hilbert bimodule associated to (V, H).
Remark 2.13. 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 coincide as Banach spaces, and since [12] uses to define an A-A-bimodule structure on X , when restricted respectively to AV(1)A and AH (1)A are isomorphisms. Hence we may use them to assume the identifications K V = AV(1)A and K H = AH(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 bimodule crossed product. IEOT Proposition 2.14. We have a one-to-one correspondence between covariant representations (π, S) of the interaction (V, H) and covariant representations (π, π X ) of the Hilbert bimodule X associated to (V, H). It is given by relations In particular, Proof. Let (π, S) be a covariant representation of (V, H).
defines a contractive linear mapping on X 0 / · . Clearly, it satisfies (1.1) and (1.2). Hence by continuity it extends uniquely to 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 ⊗ 1). Then for a ∈ A we have Remark 2.15. The Hilbert bimodule X is nothing but the GNS C * -correspondence determined by the completely positive map H, cf. [26]. In particular, the above proposition shows that C * (A, V, H) is isomorphic to the crossed product of A by the completely positive 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) as in [12,Section 5]. The crossed product C * (A, V, H) for the interaction (V, H) and the covariance algebra C * (A, X ) for X are naturally isomorphic.

Topological Freeness, Ideal Structure and Simplicity Criteria
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 the partial homeomorphism of A associated to X, then X -Ind = H.
Hence we see that the space and thus the mapping In general, one can deduce from Propositions 2.14, 2.19, see [25, discussion before Theorem 2.5], that open V-invariant sets in A are in a one-to-one correspondence with gauge invariant ideals in C * (A, V, H). Therefore it is useful to have a convenient description of the former.

K-Theory
We retain the notation from page 8 with the additional assumption that X is the Hilbert bimodule associated to a corner interaction (V, H). In particular,

Lemma 2.24. The following diagram commutes and the horizontal map is an isomorphism
Proof. Since V(A) is a full corner in AV(1)A it is known that the inclusion ι : V(A) → AV(1)A yields isomorphisms of K-groups, cf. e.g. [22,Proposition B.5]. We claim that the map Vol. 80 (2014) Interactions and Graph Algebras 431 where : X → X is the canonical antilinear isomorphism, is a homomorphism of C * -algebras. Plainly, it is linear, * -preserving, and the reader easily (H(A)), using the following calculations This shows our claim. The following diagram commutes (it commutes on the level of C * -algebras) However, since for any C * -algebra B the homomorphisms ι ii : B → M 2 (B), i = 1, 2, induce the same mappings on the level of K-theory, the mappings Using the above lemma we see that in sequence (1.5) we may replace K * ( A X, X ) = K * (AV(1)A) with K * (V(A)) and then X * turns into (ι 22 ) −1 * • (ι 22 • H) * = H * . Hence we get the following version of Pimsner-Voiculescu exact sequence, cf. [36,42].

Graph C * -Algebras via Interactions
In this section we introduce and study properties of graph interactions. We show that Theorem 2.20 applied to graph interactions is equivalent to the Cuntz-Krieger uniqueness theorem and its consequences. We use Theorem 2.25 to calculate K-theory for graph algebras straight from the dynamics on their AF-cores. 432 B. K. Kwaśniewski IEOT

Graph C * -Algebra C * (E) and its AF-Core
Throughout we let E = (E 0 , E 1 , r, s) to be a fixed finite directed graph. Thus E 0 is a set of vertices, E 1 is a set of edges, r, s : E 1 → E 0 are range, source maps, and we assume that both sets E 0 , E 1 are finite. We write E n , n > 0, for the set of paths μ = μ 1 . . . μ n , μ i ∈ E 1 , r(μ i ) = s(μ i+1 ), i = 1, . . . , n − 1, of length n. The maps r, s naturally extend to E n , so that (E 0 , E n , s, r) is the graph, and s extends to the set E ∞ of infinite paths μ = μ 1 μ 2 μ 3 . . . . We also put s(v) = r(v) = v for v ∈ E 0 . The elements of E 0 sinks := E 0 \ s(E 1 ) and respectively E 0 sources := E 0 \ r(E 1 ) are called sinks and sources. We also consider sets E n sinks = {μ ∈ E n : r(μ) ∈ E 0 sinks }, n ∈ N. We adhere to conventions of [4,29]. In our setting a Cuntz-Krieger Efamily compose of non-zero pair-wise orthogonal projections {P v : v ∈ E 0 } and partial isometries {S e : e ∈ E 1 } satisfying S * e S e = P r(e) and P v = Having such a family we put S μ := S μ1 S μ2 · · · S μn for μ = μ 1 . . . μ n (S μ = 0 ⇒ μ ∈ E n ) and S v := P v for v ∈ E 0 . The above Cuntz-Krieger relations extend to operators S μ , see [29, Lemma 1.1], as follows In particular, is the closure of the linear span of elements 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 universal Cuntz-Krieger E-family {s e : e ∈ E 1 }, {p v : v ∈ E 0 }. 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 0 , e ∈ E 1 , λ ∈ T. The fixed point C * -algebra for γ is called the core. It is an AF-algebra of the form F E := span {s μ s * ν : μ, ν ∈ E n , n = 0, 1, . . . } . We recall the standard Bratteli diagram for F E . For each vertex v and N ∈ N we set , N ∈ N, form an increasing family of finite-dimensional algebras, cf. e.g. [4], and : v ∈ r(E k sinks )} and each v (k) descends to v (k) with multiplicity one. We adopt the convention that if V is a subset of E 0 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 1 , and saturated, i.e. every vertex which feeds into V and only V is in V , then the subdiagram Λ(V ) of Λ(E) yields an ideal in F E which is naturally identified with F V . In general, viewing Λ(E) as an infinite directed graph the hereditary and saturated subgraphs (subdiagrams) of Λ(E) correspond to ideals in F E , see [5, 3.3].

Interactions Arising from Graphs
For each vertex v ∈ E 0 we let n v := |r −1 (v)| be the number of the edges received by v. We define an operator s in C * (E) as the sum of the partial isometries {s e : e ∈ E 1 } "averaged" on the spaces corresponding to projections {p v : v ∈ r(E 0 )} that are not sources: Since s * s = v∈r(E 1 ) p v is a projection the operator s is a partial isometry. It is an isometry iff E has no sources. We use s to define where μ ∈ E n , ν ∈ E m , n, m ∈ N, e, f ∈ E 1 , v ∈ E 0 . 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 s μ s * μ : μ ∈ E n , n ∈ N ⊂ F E . The following statement is one of the facts justifying the above definition. IEOT

} for E and faithful covariant representations (π, S) of the graph interaction (V, H). It is given by the relations
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).
e ∈ E 1 } yields a representation π of C * (E) which is well known to be faithful on F E . By the definition of (V, H) the pair (π| FE , S) where S := π(s) = e∈E1 1 √ 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 Se √ n r(e) . Indeed, for e ∈ E 1 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 1 ) we have Now note that S * S = π(H(1)) = v∈r(E 1 ) π(p v ) and thus S = e∈E 1 Sπ(p v ). Moreover, for each v ∈ r(E 1 ) we have ⎛ Hence the final space of the partial isometry Sπ(p v ) decomposes into the orthogonal sum of ranges of the projections π(s e s * e ), e ∈ r −1 (v), and consequently  A natural question to ask is when the graph interaction (V, H) is a C *dynamical system. It is somewhat surprising that this holds only if (V, H) is a part of a group interaction. We take up the rest of this subsection to clarify this issue in detail. To this end we will use a partially-stochastic matrix P = [p v,w ] arising from the adjacency matrix A E = [A E (v, w)] v,w∈E 0 of the graph E. Namely, we let where A E (v, w) = |{e ∈ E 1 : s(e) = v, r(e) = w}|. By a partially-stochastic matrix we mean 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 ≥ 1. The following conditions are equivalent: iii) operator s n is a partial isometry, (iv) nth power of the matrix P = {p v,w } v,w∈E 0 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 0 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 projection by Lemma 2.5. (iii) ⇔ (iv). Operator s n is a partial isometry iff H n (1) is a projection. Since v,w } v,w∈E 0 stands for the nth power of P . By the orthogonality of projections p w , it follows that H n (1) is a projection iff v∈E 0 p (n) v,w ∈ {0, 1} for all w ∈ E 0 , 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 transfer operator. Moreover, it is a straight forward fact that an iteration of an endomorphism and its transfer operator gives again an endomorphism and its transfer operator. Thus (φ n E , H n ) is a C * -dynamical system iff the transfer operator H n is regular, that is iff φ n E (H n (1)) = φ n E (1). However, as is not true, that is let μ ∈ E n and ν ∈ E k , k < n, be such that r(μ) = r(ν) and s(ν) ∈ E 0 sources . Notice that the that is P n is not partially-stochastic.
(v) ⇒ (iv). Suppose that v∈E 0 p (n) v,w > 0. By our assumption for each 0 < k < n the condition p v k−1 ,v k = 1 (because P is partially-stochastic). Therefore proceeding inductively for k = 1, 2, 3 . . . , n−1 we get v∈E 0 Example. It follows from Proposition 3.5 that if we consider a graph interaction (V, H) arising from the following graph H) has the property that its kth power (V k , H k ), for k > 1, is an interaction unless k = n. Hence by considering a disjoint sum of graphs of the above form one can obtain a graph interaction with an arbitrary finite distribution of powers not being interactions.

Vol. 80 (2014)
Interactions and Graph Algebras 437 In our specific situation of graph interactions we may prolong the list of equivalent conditions in Proposition 2.8 as follows.
Corollary 3.6. Let (V, H) be the interaction associated to the graph E. The following conditions are equivalent: s given by (3.2) is a power partial isometry, which hold for all μ, ν ∈ E n , n ∈ N.
A natural question to ask is when H is multiplicative. We rush to say that it is hardly the case. Proof. By Proposition 2.8 multiplicativity of H is equivalent to V(1) being a central element in F E . If r : E 1 → E 0 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 the projection V(1) = ss * = v∈r(E 1 ) 1 nv Hence r :

Dynamical Systems Dual to Graph Interactions
Let (V, H) be the interaction of the graph E. We obtain a satisfactory picture of the system dual to (V, H) using a Markov shift (Ω E , σ E ) dual to the commutative system (D E , φ E ). Namely, we put Ω E = ∞ N =0 E N sinks ∪E ∞ and let σ E : Ω E \ E 0 sinks → {μ ∈ Ω E : s(μ) / ∈ E 0 sources } be the shift defined by the formula 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 compact Hausdorff space and σ E is a local homeomorphism whose both domain and codomain are clopen. Moreover, the standard argument, cf. e.g. [20,Lemma 3.2], shows that s ν s * ν → χ Uν , ν ∈ E n , n ∈ N, establishes an isomorphism Let us consider the relation of 'eventual equality' defined on Ω E as follows: 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
form a basis for the quotient topology of Ω E / ∼. Moreover, the formula defines a partial homeomorphism of Ω E /∼with natural domain and codomain: 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 which means that U v,n is open in Ω E / ∼. We conclude that (3.7) defines a basis for the topology of Ω E / ∼ and q is an open map. Now, it is straightforward to check that (3.8) gives a well defined mapping whose domain and codomain are open sets of the form described in the assertion. The map [σ E ] is invertible as for μ ∈ Ω E such that s(μ) / ∈ E 0 sources its inverse can be described by the formula for an arbitrary edge e ∈ E 1 such that r(e) = s(μ), n+1 for v ∈ E n , n ∈ N, we see that [σ E ] is a partial homeomorphism.

Interactions and Graph Algebras 439
We show that the quotient partial reversible dynamical system (Ω E / ∼ , [σ E ]) embeds as a dense subsystem into ( F E , V). Under this embedding the relation ∼ coincides with the unitary equivalence of GNS-representations associated to pure extensions of the pure states of D E = C(Ω E ). More precisely, for any path μ ∈ Ω E the formula determines a pure state ω μ : F E → C (a pure extension of the point evaluation δ μ acting on D E = C(Ω E )). Indeed, the functional ω μ is a pure state on each F k , k ∈ N, and thus it is also a pure state on F E = k∈N F k , cf. e.g. [5, 4.16].
We denote by π μ the GNS-representation associated to ω μ and take up the rest of the subsection to prove the following Theorem 3.9. (Partial homeomorphism dual to a graph interaction) Under the above notation [μ] → π μ is a topological embedding of Ω E / ∼ as a dense subset into F E . This embedding intertwines [σ E ] and V. Accordingly, the space F E admits the following decomposition into disjoint sets is the domain of V, and V is uniquely determined by the formula V(π μ ) = π σE (μ) , μ ∈ Ω E \ E 0 sinks . In particular, π μ ∈ V(Δ), for μ ∈ E N sinks , iff there is ν ∈ E N +1 sinks such that r(μ) = r(ν), and then H(π μ ) = π ν . Similarly, π μ ∈ V(Δ), for μ ∈ E ∞ , iff there is ν ∼ μ such that s(ν) is not a source, and then for any ν 0 ∈ E 1 such that ν 0 ν 1 ν 2 . . . ∈ E ∞ we have H(π μ ) = π ν0ν1ν2,... . Remark 3.10. One may verify that if we put 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 in the process of dilations. The essential difference is that a dilation of a multiplicative map on a commutative algebra always leads a commutative C * -algebra, cf. [24,27], while a stochastic factor manifested by a lack of multiplicativity of the initial mapping inevitably leads to noncommutative objects after a dilation. Significantly, our dual picture of the graph interaction 'collapses' to the non Hausdorff quotient similar to that of Penrose tilings [8, 3.2]. IEOT We start by noting that the infinite direct sum ⊕ ∞ N =0 ⊕ w∈E 0 sinks F N (w) yields an ideal I sinks in F E generated by the projections p w , w ∈ E 0 sinks . We rewrite it in the form Plainly, F N (w) = {0} for w ∈ E 0 sinks iff there is μ ∈ E N sinks such that r(μ) = w and then (since F N (w) is a finite factor) π μ is a unique up to unitary equivalence irreducible representation of F E such that ker π μ ∩ F N (w) = {0}. Consequently, we see that establishes a homeomorphism between the corresponding discrete spaces. The complement of I sinks = ∞ N =0 G N in F E is a closed set which we identify in a usual way with the spectrum of the quotient algebra We will describe a dense subset of G ∞ exploiting the fact that states ω μ arising from μ ∈ E ∞ can be viewed as analogs of Glimm's product states for UHF-algebras, cf. e.g. [37, 6.5]. where v = r(μ N ) and since F N (v) ∼ = M n (C) there is a unitary 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 u − x < 1 2 and x ≤ 1. To get the contradiction we assume that μ k = ν k for some k > n. The element a := s μ1...μ k s * μ1...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 , then either s αf s * βf = 0 for all f ∈ E k−n and then ab = ba = 0 or b = f ∈E k−n s αf s * βf and then ab = s αμn+1...μ k s * βμn+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 (u * − x * )au = x * a(u − x) < 1/2 we get Remark 3.12. The C * -algebra G ∞ is a graph algebra arising from a graph which has no sinks. Indeed, the saturation E 0 sinks of E 0 sinks (the minimal Vol. 80 (2014) Interactions and Graph Algebras 441 saturated set containing E 0 sinks ) is the hereditary and saturated set corresponding to the ideal I sinks in F E . Hence I sinks = F E 0 sinks and Let us now treat μ ∈ E ∞ as the full subdiagram of the Bratelli diagram Λ(E) where the only vertex on the nth level is r(μ n ). Similarly, we treat μ ∈ E N sinks as the full subdiagram of Λ(E) where on the nth level for n ≤ N is r(μ n ) and for n > N is r(μ) (N ) , cf. notation in Sect. 3.1. For any μ ∈ Ω E we denote by W (μ) the full subdiagram of Λ(E) consisting of all ancestors of the base vertices of μ ⊂ Λ(E).

Lemma 3.13. For any
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 contained in ker ω μ .
Proof. We already know that [μ] → π μ is injective and restricts to homeomorphism between discrete spaces (Ω E \ E ∞ )/ ∼ and F E \ G ∞ . Hence it suffices to prove that [μ] → π μ is continuous and open when considered as a mapping from To this end, we may assume that E has no sinks, cf. Remark 3.12. Suppose then that E has no sinks.

Topological Freeness of Graph Interactions
We will now use Theorem 3.9 to identify the relevant properties of the partial homeomorphism V dual to the graph interaction (V, H). We recall that the condition (L) introduced in [29] requires that every loop in E has an exit. For convenience, by loops we will mean simple loops, that is paths μ = μ 1 . . . μ n such that s(μ 1 ) = r(μ n ) and s(μ 1 ) = r(μ k ) for k = 1, . . . , n − 1. A loop μ is said to have an exit if there is an edge e such that s(e) = s(μ i ) and e = μ i for some i = 1, . . . , n.
Since E is finite there must be a vertex v which appears as a base point of μ infinitely many times. Namely, there exists an increasing sequence {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 on two different loops. Let us consider these two cases: Vol. 80 (2014) Interactions and Graph Algebras 443 1. Suppose ν is a finite path such that v = s(ν) and w := r(ν) ∈ E 0 sinks . Consider the family of finite, and hence non-periodic, paths Plainly, all except finitely many of elements [μ (n k ) ] belong to U n,v (and they are all different). 2. Suppose ν is a finite path such that v = s(ν) and the vertex w := r(ν) is a base point for two different loops μ 0 and μ 1 . We put μ = Since there is an uncountable number of non-periodic sequences in {0, 1} N\{0} which pairwisely do not eventually coincide the paths μ corresponding to these sequences give rise to the uncountable family of non-periodic elements [μ ] in Ω E / ∼. Moreover, one readily sees that for sufficiently large n k all the equivalent classes of paths 1} N\{0} belong to U n,v . This proves our assertion.
Example. In the case C * (E) = O n is the Cuntz algebra, that is when E is the graph with a single vertex and n edges, n ≥ 2, then F E is an UHFalgebra and the states ω μ are simply Glimm's product states. In particular, it is well known that Prim (F E ) = {0} and F E is uncountable, cf. [37, 6.5.6]. Hence, on one hand the Rieffel homeomorphism given by the imprimitivity F E -bimodule X = F E sF E associated with the graph interaction (V, H) is the identity on Prim (F E ). Thereby it is not topologically free ([28, Theorem 6.5] can not be applied). On the other hand, we have just shown that F E contains uncountably many non-periodic points for X -Ind = V −1 , cf. Proposition 2.19, and hence it is topologically free.
Suppose now that μ is a loop in E. Let μ ∞ ∈ E ∞ be the path obtained by the infinite concatenation 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 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 the only representation of F E whose kernel is I μ and the singleton 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 faithful irreducible representation of F E /I μ is unitarily equivalent to the representation given by the 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 IEOT and saturated not only in W (μ ∞ ) but also in Λ(E). Therefore we let now K stand for the ideal in F E , corresponding to μ ∞ . Let P ∈ Prim (F E ). As K is simple P K implies K ∩ P = {0}. By the form of W (μ ∞ ) and hereditariness of Λ(P ), K ∩ P = {0} implies Λ(P ) ⊂ Λ(F E ) \ W (μ ∞ ) = Λ(I μ ). However, if P ⊂ I μ , we must have P = I μ because no part of Λ(I μ ) is not connected to W (μ ∞ ) (consult the form of diagrams of primitive ideals [5, 3.8]). Concluding, we get We have the following characterization of minimality of V.  Combining the above results we not only characterize freeness and topological freeness of ( F E , V) but also spot out an interesting dichotomy concerning its core subsystem ( G ∞ , V), cf. Remark 3.20 below.
forming open discrete sets in F E ; they correspond to loops without exits μ. In particular,

(I) V is topologically free if and only if every loop in E has an exit (satisfies condition (L)), (II) V is free if and only if every loop has an exit connected to this loop
(satisfies the so-called condition (K) introduced in [30], see also [4] is the source of a loop μ which has no exit, then Concluding, we deduce from our general results for corner interactions the following fundamental classic results for graph algebras, cf. [4,29,30,39].

Corollary 3.21.
Consider the graph C * -algebra C * (E) of the finite directed graph E.

K-Theory
We now turn to description of K-groups for C * (E). As K 1 groups for AFalgebras are trivial, using Pimsner-Voiculescu sequence from Theorem 2.25 applied to the graph interaction (V, H) associated to E we have where (ι * − H * ) : K 0 (V(F E )) → K 0 (F E ). Hence to calculate the K-groups for C * (E) we need to identify ker(ι * − H * ) and coker(ι * − H * ). We do it in two steps. In particular, the subgroup generated by v (N ) ∈ V , v ∈ E 0 sinks , N ∈ N, in K 0 (F E ) is free abelian. The groups K 0 (V(A)) and K 0 (H(A)) embed into K 0 (F E ) and we have The isomorphism H * : K 0 (V(A)) → K 0 (H(A)) is determined by Proof. We identify v (N ) with the K 0 -group element [s μ s * μ ] where μ ∈ E N and v = r(μ). It follows from (3.1) that the group K 0 (F E ) is the inductive limit lim Under the above isomorphisms, the bonding maps i N E : K 0 (F N ) → K 0 (F N +1 ), N ∈ N, are given on generators by the formula This immediately implies the first part of the assertion. : v ∈ E sinks } . Now (3.12) follows from (3.5). Note that (3.12) determines H * , as for v ∈ E 0 \ E 0 sinks , using only (3.11) and (3.12) we have H * (v (0) ) = H * ( s(e)=v r(e) (1) ) = s(e)=v r(e) (0) . We let Z(E 0 \ E 0 sinks ) and ZE 0 denote the free abelian groups on free generators E 0 \ E 0 sinks and E 0 , respectively. We consider the group homomorphism Δ E : Z(E 0 \ E 0 sinks ) → ZE 0 defined on generators as r(e).
The following lemma can be viewed as a counterpart of Lemmas 3.3, 3.4 in [40]. Nevertheless, it is a slightly different statement. which are given on generators by Proof. Suppose a = Accordingly, a ∈ ker Δ 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 is a general form of an element in K 0 (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 ( v v (0) where x is given by (3.13) [this is a general form of an element in K 0 (F E )]. Observe that as x − H * (x) ∈ im(ι * − H * ) the element y has the same class in coker(ι * − H * ) as A)). Applying the above argument to z and proceeding in this way N times we get that y is in the same class in coker(ι * − H * ) as Comparing coefficients in the above two formulas one can see that