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We prove that every action of a finite group all of whose Sylow subgroups are cyclic on the K-theory of a Kirchberg algebra can be lifted to an action on the Kirchberg algebra. The proof uses a construction of Kirchberg algebras generalizing the one of Cuntz-Krieger algebras, and a result on modules over finite groups. As a corollary, every automorphism of the K-theory of a Kirchberg algebra can be lifted to an automorphism of the Kirchberg algebra with same order.

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... In [8] Katsura defined a nice class of C * -algebras that exhausts all the Kirchberg algebras in the UCT class. The construction of these C * -algebras has two layers: the first is the graph skeleton, that gives to the C * -algebra most of the desired structural properties, and the second layer that consists of partial unitaries associated to every vertex, that provide the necessary richness in K-theory. ...

... This Lemma will be the crucial technical tool for the computation of the homology of the Katsura-Exel-Pardo groupoid. In section 2 we introduce the Katsura-Exel-Pardo groupoid, that is, a groupoid associated to a selfsimilar graph introduced by Exel and Pardo in [5] that realizes the C * -algebra defined by Katsura [8]. After a quick overview of the basics properties of this groupoid found in [5,Section 18], we move to the computation of the homology. ...

... In [5] and later in [6] it was shown that C * (G A,B ) is isomorphic to the C * -algebra O A,B constructed in [8]. We define the following condition on the pair of matrices A and B: Given groups G 1 , G 2 , G 3 , . . . ...

We compute the homology of the groupoid associated to the Katsura algebras, and show that they capture the $K$-theory of the $C^*$-algebras, and hence satisfying the (HK) conjecture posted by Matui. Moreover, we show that several classifiable simple $C^*$-algebras are groupoid $C^*$-algebras of this class.

... Self-similar k-graph C*-algebras were initially studied by the authors in [21]. These algebras embrace many known important C*-algebras as special classes, such as k-graph C*-algebras introduced by Kumjian-Pask [17], Exel-Pardo algebras [12], unital Katsura algebras [16], and Nekrashevych algebras [23]. Roughly speaking, a self-similar k-graph C*-algebra O G,Λ is a universal C*-algebra generated by a unitary representation u of G and a Cuntz-Krieger representation s of a k-graph Λ, which are compatible with the ambient self-similar action of G on Λ: u g s μ = s g·μ u g| μ for all g ∈ G and μ ∈ Λ. ...

... Remark 7.6. Katsura in [16] constructed a unital C*-algebra O T e 1 ,B which is the universal C*-algebra generated by a family of mutually orthogonal projections {q v } v∈Λ 0 , a family of partial unitaries {r v } v∈Λ 0 with r * v r v = r v r * v = q v , and a family of partial isometries {t v,w,z : T e 1 (v, w) = 0, z ∈ Z} satisfying the following properties: ...

... Remark 7.7. Katsura in [16], and Exel-Pardo in [12] both showed that O Z,Λ ( ∼ = O T e 1 ,B ) is simple, separable, amenable, purely infinite, and satisfies the UCT. Here we would like to comment that this can also be obtained by invoking our results in [21]. ...

Let $G$ be a countable discrete amenable group, and $\Lambda$ be a strongly connected finite $k$-graph. If $(G,\Lambda)$ is a pseudo free and locally faithful self-similar action which satisfies the finite-state condition, then the structure of the KMS simplex of the C*-algebra $\O_{G,\Lambda}$ associated to $(G,\Lambda)$ is described: it is either empty or affinely isomorphic to the tracial state space of the C*-algebra of the periodicity group $\Per_{G,\Lambda}$ of $(G,\Lambda)$, depending on whether the Perron-Frobenius eigenvector of $\Lambda$ preserves the $G$-action. As applications of our main results, we also exhibit several classes of important examples.

... Our results are motivated by a construction of Exel and Pardo [8], who studied a family of self-similar actions of groups on path spaces. Their main motivation was to provide a unified theory that accommodates both Nekrashevych's Cuntz-Pimsner algebras and a family of "Katsura algebras" [13] that includes all Kirchberg algebras. We seek a common setting for the analyses of KMS states on self-similar groups in [17] and on the Toeplitz-Cuntz-Krieger algebras of graphs [7,12,10,11]. ...

... At this point, we discuss an important class of examples that motivated the previous work of Exel and Pardo [8]. The Cuntz-Pimsner algebras of the following self-similar groupoids are a family of algebras constructed by Katsura [13] to provide models of Kirchberg algebras. These examples also fit the theory of [8], and in particular have the property s(g · e) = s(e) = g · s(e). ...

... We have used the calligraphic A to distinguish the automaton from the matrix A in the definition of the Katsura action, thereby avoiding a clash with the established notation in[13] and[8]. ...

We consider self-similar actions of groupoids on the path spaces of finite directed graphs, and construct examples of such self-similar actions using a suitable notion of graph automaton. Self-similar groupoid actions have a Cuntz-Pimsner algebra and a Toeplitz algebra, both of which carry natural dynamics lifted from the gauge actions. We study the equilibrium states (the KMS states) on the resulting dynamical systems. Above a critical inverse temperature, the KMS states on the Toeplitz algebra are parametrised by the traces on the full $C^*$-algebra of the groupoid, and we describe a program for finding such traces. The critical inverse temperature is the logarithm of the spectral radius of the incidence matrix of the graph, and at the critical temperature the KMS states on the Toeplitz algebra factor through states of the Cuntz-Pimsner algebra. Under a verifiable hypothesis on the self-similar action, there is a unique KMS state on the Cuntz-Pimsner algebra. We discuss an explicit method of computing the values of this KMS state, and illustrate with examples.

... The purpose of this paper is to give a unified treatment to two classes of C*-algebras which have been studied in the past few years from rather different points of view, namely Katsura algebras [18], and certain algebras constructed by Nekrashevych [24], [26] from self-similar groups. ...

... The realization that these classes are indeed closely related, as well as the fact that they could be given a unified treatment, came to our mind as a result of our attempt to understand Katsura's algebras O A,B from the point of view of inverse semigroups. The fact, proven by Katsura in [18], that all Kirchberg algebras in the UCT class may be described in terms of his O A,B was, in turn, a strong motivation for that endeavor. ...

... Example. As in [18], let us assume we are given two N × N matrices A and B with integer entries, and such that A i,j ≥ 0, for all i and j. We may then consider the graph E with vertex set E 0 = {1, 2, . . . ...

Given a graph E, an action of a group G on E, and a G-valued cocycle φ on the edges of E, we define a C*-algebra denoted , which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup built naturally from the triple . As a tight C*-algebra, is also isomorphic to the full C*-algebra of a naturally occurring groupoid . We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our , and many of their known properties are shown to follow from our general theory.

... Finally, Katsura [15] constructed a class of algebras O A,B (where A ∈ M n (Z + ), B ∈ M n (Z)) of combinatorial nature, generalizing Cuntz-Krieger algebras. Moreover, by using the Kirchberg-Phillips Theorem, he showed that any Kirchberg algebra can be represented (up to isomorphism) by an algebra of this family. ...

... In this section we will quickly recall the definition and basic properties of Katsura algebras that will be needed in the sequel. All the contents of this section are borrowed from [15]. ...

... (1) The C * -algebra O A,B is separable, nuclear and in the UCT class [15,Proposition 2.9]. ...

In this note we show that a combinatorial model of Kirchberg algebras in the
UCT, namely the Katsura algebras O_{AB}, can be expressed both as groupoid
C*-algebras and as inverse semigroup crossed products. We use this picture to
obtain results about simplicity, pure infiniteness and nuclearity of O_{AB}
using different methods than those used by Katsura.

... It is built upon the works of [31] on self-similar groups, originated from the study of Grigorchuk groups in geometric group theory. It turns out that the natural construction in [17] produces a class of interesting and imporant operator algebras, which embraces Katsura and Nekrashevych C*-algebras ( [21,31,32]). This has been extensively generalized to other contexts recently. ...

... Self-similar graphs. Motivated by the constructions of Katsura [21] and Nekrashevych [32], the notion of self-similar graphs was first introduced in [17]. It encodes a two-way action between a group and a directed graph. ...

We study the Wold decomposition for representations of a self-similar semigroup $P$ action on a directed graph $E$. We then apply this decomposition to the case where $P=\bN$ to study the C*-envelope of the associated universal non-selfadjoint operator algebra $\A_{\bN, E}$ by carefully constructing explicit non-trivial dilations for non-boundary representations. In particular, it is shown that the C*-envelope of $\A_{\bN, E}$ coincides with the self-similar C*-algebra $\O_{\bZ, E}$.

... For finite cyclic groups acting on AF algebras, this question is already open (see [10,Exercise 10.11.3]). State-of-the-art results for Kirchberg algebras can be found in [155] -in particular, Katsura shows that actions of Z/n on KT u (A) lift to actions on A when A is a unital Kirchberg algebra satisfying the UCT. Combining Corollary 9.10 with a result of Barlak and Szabó ([7,Corollary 2.7]), an interesting case of this problem in the stably finite case can be obtained as follows. ...

... Similarly, concatenating the right columns gives an exact sequence, using the Puppe exact sequence in KK( · , SM n ⊗ D) applied to the diagonal inclusion C → M n (see [11,Theorem 19.4.3]). 155 By the five lemma it follows that Ω A is an isomorphism. ...

We classify the unital embeddings of a unital separable nuclear $C^*$-algebra satisfying the universal coefficient theorem into a unital simple separable nuclear $C^*$-algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear $C^*$-algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of $K$-theory and traces.

... Recently in [15] Exel and Pardo introduced C * -algebras O G,E giving a unified treatment of two classes of C * -algebras which have attracted significant recent attention, namely Katsura C * -algebras [17] and Nekrashevych's self-similar group C * -algebras [21,22]. Katsura C * -algebras are important as they provide concrete models for all UCT Kirchberg algebras [18]; self-similar C * -algebras have provided the first known example of a groupoid whose C * -algebra is simple and whose Steinberg algebra over some fields is non-simple [9]. ...

... The vertex-trivial case. In [17] Katsura introduced C * -algebras O A,B which we call Katsura C * -algebras. Here we consider an algebraic analogue, denoted O alg A,B (R), and prove that all such * -algebras are Exel-Pardo * -algebras using the translation of the matrices A, B into an action of Z on a graph discovered by Exel and Pardo in [14] (see Definition 1.12). ...

We introduce an algebraic version of the Katsura C∗-algebra of a pair A,B of integer matrices and an algebraic version of the Exel–Pardo C∗-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura C∗-algebras are all isomorphic to Steinberg algebras.

... Combining [EP17], [BKQ17] and [BKQS18], one deduces that O(K A,B ) is isomorphic to the Katsura algebra O A,B introduced in [Kat08]. Hence the set-up described above enables one to define G-valued cocycles of the type η f on K A,B , hence coactions of G on O A,B , for any group G. ...

... As the set E 1 i,j of edges in E from j from i, which consists of a i,j elements, is left invariant by the action of Z, we may for example pick an element g i,j ∈ G for every (i, j) ∈ N × N satisfying that a i,j = 0 and define f : E 1 → G by f (e) = g i,j whenever e ∈ E 1 i,j . We note that, more generally, we could have handled in a similar way any countable row-finite graph E having no sources, as in [Kat08]. ...

Given a group cocycle on a finitely aligned left cancellative small category (LCSC) we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSC's, and to construct coactions of groups on Exel-Pardo algebras. Finally we discuss the universal group of a small category and connectedness of skew product categories.

... Roughly speaking, O G,Λ is generated by a universal pair {u, s} of representations, where u is a unitary representation of G in O G,Λ and s is a Cuntz-Krieger representation of Λ in O G,Λ such that u and s are compatible with respect to the underlying self-similar action of (G, Λ). It turns out that those C*-algebras are so broad that they include many known important C*-algebras, such as unital k-graph C*algebras introduced by Kumjian-Pask [18], Exel-Pardo algebras [9], unital Katsura algebras [17], and Nekrashevych algebras [23]. In [20], we proved that O G,Λ is always nuclear and satisfies the UCT, completely characterized the simplicity of O G,Λ , and further showed that if O G,Λ is simple then O G,Λ is either stably finite or purely infinite. ...

... Therefore, π is injective. Proposition 3. 17. Let (G, Λ) be a self-similar P -graph with Property (FV). ...

Let $(G, \Lambda)$ be a self-similar $k$-graph with a possibly infinite vertex set $\Lambda^0$. We associate a universal C*-algebra $\mathcal{O}_{G,\Lambda}$ to $(G,\Lambda)$. The main purpose of this paper is to investigate the ideal structures of $\mathcal{O}_{G,\Lambda}$. We prove that there exists a one-to-one correspondence between the set of all $G$-hereditary and $G$-saturated subsets of $\Lambda^0$ and the set of all gauge-invariant and diagonal-invariant ideals of $\mathcal{O}_{G,\Lambda}$. Under some conditions, we characterize all primitive ideas of $\mathcal{O}_{G,\Lambda}$. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$-graph C*-algebras in depth.

... This note we fill the gap, by reducing the problem row-finite graphs with no sources, and then showing how characterizations stated in [8] works correctly under this restrictions. The results in this note are essential to extend the scope of the results obtained in [8] for Katsura algebras over finite matrices to the general case; this guarantees, by [9], that every Kirchberg algebra in the UCT is the full groupoid C * -algebra of a second countable amenable ample groupoid. ...

... This is not a good idea for giving an intrinsic definition of the object, but it is very helpful to deal with the details in the definition. Nevertheless, the model we will follow is that of Katsura algebras [9], where the unitary associated to an element of Z is written in terms of partial unitaries associated to the projections p x for x ∈ E 0 . ...

In this note we extend the construction of a C *-algebra associated to a self-similar graph to the case of arbitrary countable graphs. We reduce the problem to the row-finite case with no sources, by using a desingularization process. Finally, we characterize simplicity in this case.

... This note we fill the gap, by reducing the problem row-finite graphs with no sources, and then showing how characterizations stated in [8] works correctly under this restrictions. The results in this note are essential to extend the scope of the results obtained in [8] for Katsura algebras over finite matrices to the general case; this guarantees, by [9], that every Kirchberg algebra in the UCT is the full groupoid C * -algebra of a second countable amenable ample groupoid. ...

... This is not a good idea for giving an intrinsic definition of the object, but it is very helpful to deal with the details in the definition. Nevertheless, the model we will follow is that of Katsura algebras [9], where the unitary associated to an element of Z is written in terms of partial unitaries associated to the projections p x for x ∈ E 0 . ...

In this note we extend the construction of a $C^*$-algebra associated to a self-similar graph to the case of arbitrary countable graphs. We reduce the problem to the row-finite case with no sources, by using a desingularization process. Finally, we characterize simplicity in this case.

... The main object they were interested in is O G,E , which is a unital universal C*-algebra generated by a unitary representation of G and a Cuntz-Krieger representation of E, such that these two representations are compatible with the given self-similar action (G, E). Surprisingly, this construction embraces many important and interesting C*algebras, such as unital Katsura algebras [15,16] and certain C*-algebras of self-similar groups constructed by Nekrashevych [25]. To study O G,E , Exel-Pardo in [10] associated the self-similar action (G, E) an inverse semigroup S G,E . ...

... Suppose that Λ is a 1-graph. Then O G,Λ is the C*-algebra studied by Exel-Pardo in [10], which includes Katsura algebras ( [16]) and C*-algebras of self-similar groups constructed by Nekrashevych ( [23,25]). 4. Suppose that Λ is a single-vertex k-graph. ...

In this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $\Lambda$, and associate it a universal C*-algebra $\O_{G,\Lambda}$. We prove that $\O_{G,\Lambda}$ can be realized as the Cuntz-Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then $\O_{G,\Lambda}$ is shown to be isomorphic to a "path-like" groupoid C*-algebra. This facilitates studying the properties of $\O_{G,\Lambda}$. We show that $\O_{G,\Lambda}$ is always nuclear and satisfies the Universal Coefficient Theorem; we characterize the simplicity of $\O_{G,\Lambda}$ in terms of the underlying action; and we prove that, whenever $\O_{G,\Lambda}$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $\Lambda$ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.

... We then apply the construction in [44], which builds on [43]. This approach has also been mentioned in [20,Remark 3.4]. In contrast to the construction in [20], we obtain groupoids with totally disconnected unit spaces in this way. ...

... This approach has also been mentioned in [20,Remark 3.4]. In contrast to the construction in [20], we obtain groupoids with totally disconnected unit spaces in this way. ...

We show that outer approximately represenbtable actions of a finite cyclic group on UCT Kirchberg algebras satisfy a certain quasi-freeness type property if the corresponding crossed products satisfy the UCT and absorb a suitable UHF algebra tensorially. More concretely, we prove that for such an action there exists an inverse semigroup of homogeneous partial isometries that generates the ambient C*-algebra and whose idempotent semilattice generates a Cartan subalgebra. We prove a similar result for actions of finite cyclic groups with the Rokhlin property on UCT Kirchberg algebras absorbing a suitable UHF algebra. These results rely on a new construction of Cartan subalgebras in certain inductive limits of Cartan pairs. We also provide a characterisation of the UCT problem in terms of finite order automorphisms, Cartan subalgebras and inverse semigroups of partial isometries of the Cuntz algebra $\mathcal{O}_2$. This generalizes earlier work of the authors.

... The second class generalized in [EP13] are the C*-algebras constructed by Katsura [Kat08] from pairs of integer matrices A and B, denoted O A,B . The pair (A, B) gives rise to a self-similar graph action (Z, E A , ϕ), where E A is the graph whose incidence matrix is A and the action of Z and the cocycle ϕ are determined by the entries of A and B. It is shown in [EP13], Example 3.4, that O A,B ∼ = O Z,E A . ...

... The pair (A, B) gives rise to a self-similar graph action (Z, E A , ϕ), where E A is the graph whose incidence matrix is A and the action of Z and the cocycle ϕ are determined by the entries of A and B. It is shown in [EP13], Example 3.4, that O A,B ∼ = O Z,E A . From [Kat08], it is a fact that every Kirchberg algebra arises as O A,B for some A and B, so the algebras we consider here constitute a large class. ...

In a recent paper, Pardo and the first named author introduced a class of
C*-algebras which which are constructed from an action of a group on a graph.
This class was shown to include many C*-algebras of interest, including all
Kirchberg algebras. In this paper, we study the conditions under which these
algebras can be realized as partial crossed products of commutative C*-algebras
by groups. In addition, for any $n\geq 2$ we present a large class of groups
such that for any group $H$ in this class, the Cuntz algebra $\mathcal{O}_n$ is
isomorphic to a partial crossed product of a commutative C*-algebra by $H$.

... Example. As in [7], given a positive integer N , let A ∈ M N ( + ) without zero rows and let B ∈ M N ( ) be such that ...

... It may be proved without much difficulty that O G,E is isomorphic to Katsura's [7] When N = 1, the relevant graph for Katsura's algebras is the same as the one we used above in the description of Nekrashevych's example. However the former is not a special case of the latter because, contrary to what is required in [11], the group action might not be faithful. ...

We study a family of C*-algebras generalizing both Katsura algebras and
certain algebras introduced by Nekrashevych in terms of self-similar groups.

... Zhang 440 bras A in the Cuntz standard form [2]. Later, this result was extended by Spielberg who showed this question has an affirmative answer for G D Z p , where p is a prime number, and for an arbitrary unital UCT Kirchberg algebra [38]. Finally, in [27], this was further extended by Katsura to actions of finite groups whose Sylow subgroups are cyclic. ...

... To give a unified framework like graph C * -algebras for the Katsura's [10] and Nekrashevyche's algebras [16,17], Exel and Pardo introduced self-similar graphs and their C * -algebras in [6]. They then associated an inverse semigroup and groupoid model to this class of C * -algebras and studied structural features by underlying self-similar graphs. ...

We introduce the Exel-Pardo $*$-algebra $\mathrm{EP}_R(G,\Lambda)$ associated to a self-similar $k$-graph $(G,\Lambda,\varphi)$. We prove the $\mathbb{Z}^k$-graded and Cuntz-Krieger uniqueness theorems for such algebras and investigate their ideal structure. In particular, we modify the graded uniqueness theorem for self-similar 1-graphs, and then apply it to present $\mathrm{EP}_R(G,\Lambda)$ as a Steinberg algebra and to study the ideal structure.

... In [16] Katsura introduced C * -algebras O A,B which we call Katsura C * -algebras. Here we consider an algebraic analogue, denoted O alg A,B (R), and prove that all such * -algebras are Exel-Pardo * -algebras using the translation of the matrices A, B into an action of Z on a graph discovered by Exel and Pardo in [13] (see Definition 1.2). ...

We introduce an algebraic version of the Katsura C∗-algebra of a pair A,B of integer matrices and an algebraic version of the Exel–Pardo C∗-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura C∗-algebras are all isomorphic to Steinberg algebras.

... Roughly speaking, O G, is generated by a universal pair {u, s} of representations, where u is a unitary representation of G in O G, and s is a Cuntz-Krieger representation of in O G, such that u and s are compatible with respect to the underlying self-similar action of (G, ). It turns out that those C*-algebras are so broad that they include many known important C*-algebras, such as unital k-graph C*-algebras introduced by Kumjian-Pask [18], Exel-Pardo algebras [9], unital Katsura algebras [17], and Nekrashevych algebras [23]. In [21], we proved that O G, is always nuclear and satisfies the Universal 2 H. Li and D. Yang Coefficient Theorem (UCT), completely characterized the simplicity of O G, , and further showed that if O G, is simple, then O G, is either stably finite or purely infinite. ...

Let $(G,\unicode[STIX]{x1D6EC})$ be a self-similar $k$ -graph with a possibly infinite vertex set $\unicode[STIX]{x1D6EC}^{0}$ . We associate a universal C*-algebra ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ to $(G,\unicode[STIX]{x1D6EC})$ . The main purpose of this paper is to investigate the ideal structures of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . We prove that there exists a one-to-one correspondence between the set of all $G$ -hereditary and $G$ -saturated subsets of $\unicode[STIX]{x1D6EC}^{0}$ and the set of all gauge-invariant and diagonal-invariant ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . Under some conditions, we characterize all primitive ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$ -graph C*-algebras in depth.

... In [7], Exel and Pardo introduced self-similar graph C * -algebras O G,E to give a unified framework like graph C * -algebras for the Katsura's [10] and Nekrashevych's algebras [18,19]. These C * -algebras were initially considered in [7] only for countable discrete groups G acting on finite graphs E with no sources, and then generalized in [2,8] for larger classes. ...

We investigate the pure infiniteness and stable finiteness of the Exel-Pardo $C^*$-algebras $\mathcal{O}_{G,E}$ for countable self-similar graphs $(G,E,\varphi)$. In particular, we associate a specific ordinary graph $\widetilde{E}$ to $(G,E,\varphi)$ such that some properties such as simpleness, stable finiteness or pure infiniteness of the graph $C^*$-algebra $C^*(\widetilde{E})$ imply that of $\mathcal{O}_{G,E}$. Among others, this follows a dichotomy for simple $\mathcal{O}_{G,E}$: if $(G,E,\varphi)$ contains no $G$-circuits, then $\mathcal{O}_{G,E}$ is stably finite; otherwise, $\mathcal{O}_{G,E}$ is purely infinite. Furthermore, Li and Yang recently introduced self-similar $k$-graph $C^*$-algebras $\mathcal{O}_{G,\Lambda}$. We also show that when $|\Lambda^0|<\infty$ and $\mathcal{O}_{G,\Lambda}$ is simple, then it is purely infinite.

... For k = 1, the K-theory of the C*-algebra O Z,Λ was studied in numerous papers such as [6,9,12,14]. Now we assume that k ≥ 2. It is easier to discuss the K-theory of O Z,Λ⋆Z k ⋊ γ −1 Z k and our discussion is based on the combination of previous profound work ([1, Corollary 2.5], [10, 6.10], [22,Theorem 2], [23]) give rise to a cohomology spectral sequence (see [24]) {E p,q r } r≥1,p,q∈Z , where for p, q ∈ Z, we have E p,q ...

We pose a conjecture on the K-theory of the self-similar $k$-graph C*-algebra of a standard product of odometers. We generalize the C*-algebra $\mathcal{Q}_S$ to any subset of $\mathbb{N}^\times \setminus \{1\}$ and then realize it as the self-similar $k$-graph C*-algebra of a standard product of odometers.

... The work in [24] was revisited by the last two named authors, along with Laca and Raeburn, in [18], where they also studied the Toeplitz extension. Motivated by operator algebraic considerations, Exel and Pardo introduced the notion of a self-similar action of a group on the path space of a directed graph [6], and used this new notion of self-similarity to produce a class of C * -algebras which unified Nekrashevych's Cuntz-Pimsner algebras from [24] and the Katsura algebras from [14]. Working in parallel to [6], the authors of [18] produced a follow-up [19] in which they defined a self-similar groupoid action on the path space of a directed graph, and associated Toeplitz and Cuntz-Pimsner algebras to these new self-similar actions. ...

We introduce the notion of a self-similar action of a groupoid G on a finite higher-rank graph. To these actions we associate a compactly aligned product system of Hilbert bimodules, and we show that the corresponding Nica-Toeplitz and Cuntz-Pimsner algebras are universal for generators and relations. We consider natural actions of the real numbers on both algebras and study the KMS states of the associated dynamics. For large inverse temperatures, we describe the simplex of KMS states on the Nica-Toeplitz algebra in terms of traces on the full C*-algebra of G. To study the KMS structure of the Cuntz-Pimsner algebra, we restrict to strongly-connected finite higher-rank graphs and a preferred dynamics. We generalise the G-periodicity group of Li and Yang, which is built to encode the periodicity of the underlying graph in the presence of the action of G. We prove that the KMS states of the Cuntz-Pimsner algebra are parametrised by states on the C*-algebra of the G-periodicity group, and we show that if the graph is G-aperiodic and the action satisfies a finite-state condition, then there is at most one KMS state on the Cuntz-Pimsner algebra. We illustrate our results by introducing the notion of a coloured-graph automaton, which we use to construct examples of self-similar actions. We compute the unique KMS states of the Cuntz-Pimsner algebra for some concrete examples.

... In [18], Katsura associates a C * -algebra to a pair of square integer matrices A, B and studies their properties. These were recast as C * -algebras of self-similar graphs in [13]. ...

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C ∗ C^{*} -algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that the C ∗ C^{*} -algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in Z 2 \mathbb {Z}_2 is not simple.

... A simple Katsura algebra with non-Hausdorff groupoid. In [18], Katsura associates a C * -algebra to a pair of square integer matrices A, B and studies their properties. These were recast as C * -algebras of self-similar graphs in [13]. ...

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C *-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that C *-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in Z 2 is not simple.

... A simple Katsura algebra with non-Hausdorff groupoid. In [18], Katsura associates a C * -algebra to a pair of square integer matrices A, B and studies their properties. These were recast as C * -algebras of self-similar graphs in [13]. ...

We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C*-algebra associated to non-Hausdorff \'etale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that C*-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coefficients in $\mathbb{Z}_2$ is not simple.

... Then the completion of this pre-correspondence is the C * -correspondence X(E) over c 0 (E 0 ). A Toeplitz representation (or just a representation; see [17]) of X(E) in a C * -algebra B is a pair (ψ, π), where ψ : X(E) → B is a linear map and π : c 0 (E 0 ) is a homomorphism such that for ξ, η ∈ X(E) and a ∈ c 0 (E 0 ), ψ(ξa) = ψ(ξ)π(a) ψ(aξ) = π(a)ψ(ξ) π( ξ, η c 0 (E 0 ) ) = ψ(ξ) * ψ(η). ...

We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated with finitely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Sz\'ep product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing $C^*$-algebras from left cancellative small categories and discuss their relationship.

... Moreover, its isomorphism class depends only on the K-theory of A and the class of the unit [1 A ]. In [1] the authors used this type of construction to prove that any order two automorphism of the K-theory of a unital Kirchberg algebra A satisfying UCT with [1 A ] = 0 in K 0 (A) lifts to an order two automorphism of A. For more about lifting automorphisms of K-groups to Kirchberg algebras, see [8] and [15]. ...

Given a locally compact group $G$ and a unitary representation $\rho:G\to U({\mathcal H})$ on a Hilbert space ${\mathcal H}$, we construct a $C^*$-correspondence ${\mathcal E}(\rho)={\mathcal H}\otimes_{\mathbb C} C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal O}_{{\mathcal E}(\rho)}$. We prove that for $G$ compact, ${\mathcal O}_{{\mathcal E}(\rho)}$ is strong Morita equivalent to a graph $C^*$-algebra. If $\lambda$ is the left regular representation of an infinite, discrete and amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(\lambda)}$ is simple and purely infinite, with the same $K$-theory as $C^*(G)$. If $G$ is compact abelian, any representation decomposes into characters and determines a skew product graph. We illustrate with several examples and we compare ${\mathcal E}(\rho)$ with the crossed product $C^*$-correspondence.

... Are there any other obstructions? For example, can we realize all of Katsura's algebras O A,B [Kat08] (also see [EP,Example 3.4 ...

For a closed subgroup of a locally compact group the Rieffel induction process gives rise to a $C^*$-correspondence over the $C^*$-algebra of the subgroup. We study the associated Cuntz-Pimsner algebra and show that, by varying the subgroup to be open, compact, or discrete, there are connections with the Exel-Pardo correspondence arising from a cocycle, and also with graph algebras.

... This C * -algebra is defined as the Cuntz-Pimsner algebra of a C * -correspondence M over C(E 0 ) ⋊ G and contains a copy of the graph algebra C * (E). In particular for G = Z acting on a graph E with N × N incidence matrix A by fixing the vertices and permuting the edges in a way determined by another N × N integer matrix B (which also determines the cocycle), they prove that O G,E is isomorphic to Katsura's algebra O A,B , see [21], used to model all Kirchberg algebras. Note that in this case C(E 0 ) ⋊ G is isomorphic to the direct sum of N copies of C(T) ∼ = C * (Z). ...

If $G$ acts on a $C^*$-correspondence ${\mathcal H}$, then by the universal
property $G$ acts on the Cuntz-Pimsner algebra ${\mathcal O}_{\mathcal H}$ and
we study the crossed product ${\mathcal O}_{\mathcal H}\rtimes G$ and the fixed
point algebra ${\mathcal O}_{\mathcal H}^G$. Using intertwiners, we define the
Doplicher-Roberts algebra ${\mathcal O}_\rho$ of a representation $\rho$ of a
compact group $G$ on ${\mathcal H}$ and prove that ${\mathcal O}_{\mathcal
H}^G$ is isomorphic to ${\mathcal O}_\rho$. When the action of $G$ commutes
with the gauge action on ${\mathcal O}_{{\mathcal H}}$, then $G$ acts also on
the core algebras ${\mathcal O}_{\mathcal H}^{\mathbb T}$, where $\mathbb T$
denotes the unit circle. We give applications for the action of a group $G$ on
the $C^*$-correspondence ${\mathcal H}_E$ associated to a directed graph $E$.
If $G$ is finite and $E$ is discrete and locally finite, we prove that the
crossed product $C^*(E)\rtimes G$ is isomorphic to the $C^*$-algebra of a graph
of $C^*$-correspondences and stably isomorphic to a locally finite graph
algebra. If $C^*(E)$ is simple and purely infinite and the action of $G$ is
outer, then $C^*(E)^G$ and $C^*(E)\rtimes G$ are also simple and purely
infinite with the same $K$-theory groups. We illustrate with several examples.

... To mention an example of interest to us, in [8,9] a unified treatment was given to a certain class of C*-algebras studied by Katsura in [11], alongside Nekrashevych's C*-algebras introduced and discussed in [17][18][19]. The unifying principle is the notion of self similar graphs introduced in [9], which gives rise to C*-algebras that can be effectively studied via the already mentioned theory of tight representations of inverse semigroups. ...

In this work we present algebraic conditions on an inverse semigroup S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}$$\end{document} (with zero) which imply that its associated tight groupoid Gtight(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_\mathrm{tight}(\mathcal {S})$$\end{document} is: Hausdorff, essentially principal, minimal and contracting, respectively. In some cases these conditions are in fact necessary and sufficient.

... This was extended to finite cyclic group actions of prime order in [34], where also the assumption of the Cuntz standard form could be dropped. Finally in [13], this was further extended to actions of finite groups whose Sylow subgroups are cyclic. ...

This paper serves as a source of examples of Rokhlin actions or approximately
representable actions of finite groups on C*-algebras satisfying a certain
UHF-absorption condition. We show that given any finite group $G$ and a
separable, unital C*-algebra $A$ that absorbs $M_{|G|^\infty}$ tensorially, one
can lift any group homomorphism $G\to\operatorname{Aut}(A)/_{\approx_u}$ to an
honest Rokhlin action $\gamma$ of $G$ on $A$. Unitality may be dropped in
favour of stable rank one. If $A$ belongs to a certain class of C*-algebras
that is classifiable by a suitable invariant (e.g. the Elliott invariant), then
in fact every $G$-action on the invariant lifts to a Rokhlin action of $G$ on
$A$. For the crossed product C*-algebra $A\rtimes_\gamma G$ of a Rokhlin action
on a UHF-absorbing C*-algebra, an inductive limit decomposition is obtained in
terms of $A$ and $\gamma$. If $G$ is assumed to be abelian, then the dual
action $\hat{\gamma}$ is locally representable in a very strong sense. If
additionally $A$ is nuclear, simple and has a unique trace, then $\hat{\gamma}$
has finite Rokhlin dimension. We then show how some well-known constructions of
finite group actions with certain predescribed properties can be recovered and
extended by the main results of this paper, when paired with known
classification theorems. Among these is a range result of Izumi of
approximately representable cyclic group actions on $\mathcal{O}_2$ and
Blackadar's famous construction of symmetries on the CAR algebra whose fixed
point algebras have non-trivial $K_1$-groups.

We introduce the Exel-Pardo ∗-algebra EPR(G,Λ) associated to a self-similar k-graph (G,Λ,φ). We also prove the Zk-graded and Cuntz–Krieger uniqueness theorems for such algebras and investigate their ideal structure. In particular, we modify the graded uniqueness theorem for self-similar 1-graphs and then apply it to present EPR(G,Λ) as a Steinberg algebra and to study the ideal structure.

We extend Nekrashevych's $KK$-duality for $C^*$-algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid $(G,E)$ acting faithfully on a finite directed graph $E$, we associate two $C^*$-algebras, $\mathcal{O}(G,E)$ and $\widehat{\mathcal{O}}(G,E)$, to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in $KK$-theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.

We prove a sandwiching lemma for inner-exact locally compact Hausdorff \'etale groupoids. Our lemma says that every ideal of the reduced $C^*$-algebra of such a groupoid is sandwiched between the ideals associated to two uniquely defined open invariant subsets of the unit space. We obtain a bijection between ideals of the reduced $C^*$-algebra, and triples consisting of two nested open invariant sets and an ideal in the $C^*$-algebra of the subquotient they determine that has trivial intersection with the diagonal subalgebra and full support. We then introduce a generalisation to groupoids of Ara and Lolk's relative strong topological freeness condition for partial actions, and prove that the reduced $C^*$-algebras of inner-exact locally compact Hausdorff \'etale groupoids satisfying this condition admit an obstruction ideal in Ara and Lolk's sense.

Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K -groups of the (reduced) groupoid C $^*$ -algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum–Connes conjecture. The construction is based on the triangulated category approach to the Baum–Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.

We pose a conjecture on the K-theory of the self-similar k-graph C*-algebra of a standard product of odometers. We generalize the C*-algebra \({\mathcal {Q}}_S\) to any subset of \({\mathbb {N}}^\times \setminus \{1\}\) and then realize it as the self-similar k-graph C*-algebra of a standard product of odometers.

We provide a Cuntz-Pimsner model for graph of groups C⁎-algebras. This allows us to compute the K-theory of a range of examples and show that graph of groups C⁎-algebras can be realised as Exel-Pardo algebras. We also make a preliminary investigation of whether the crossed product algebra of Baumslag-Solitar groups acting on the boundary of certain trees satisfies Poincaré duality in KK-theory. By constructing a K-theory duality class we compute the K-homology of these crossed products.

We investigate the pure infiniteness and stable finiteness of the Exel-Pardo C⁎-algebras OG,E for countable self-similar graphs (G,E,φ). In particular, we associate a specific ordinary graph E˜ to (G,E,φ) such that some properties such as simpleness, stable finiteness or pure infiniteness of the graph C⁎-algebra C⁎(E˜) imply that of OG,E. Among others, this follows a dichotomy for simple OG,E: if (G,E,φ) contains no G-circuits, then OG,E is stably finite; otherwise, OG,E is purely infinite. Furthermore, Li and Yang recently introduced self-similar k-graph C⁎-algebras OG,Λ. We also show that when |Λ0|<∞ and OG,Λ is simple, then it is purely infinite.

In this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $\Lambda $ and associate it a universal C$^\ast $-algebra ${{\mathcal{O}}}_{G,\Lambda }$. We prove that ${{\mathcal{O}}}_{G,\Lambda }$ can be realized as the Cuntz–Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then ${{\mathcal{O}}}_{G,\Lambda }$ is shown to be isomorphic to a “path-like” groupoid C$^\ast $-algebra. This facilitates studying the properties of ${{\mathcal{O}}}_{G,\Lambda }$. We show that ${{\mathcal{O}}}_{G,\Lambda }$ is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of ${{\mathcal{O}}}_{G,\Lambda }$ in terms of the underlying action, and we prove that, whenever ${{\mathcal{O}}}_{G,\Lambda }$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $\Lambda $ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.

It is proven that Matui's AH~conjecture is true for Katsura--Exel--Pardo groupoids $\mathcal{G}_{A,B}$ associated to integral matrices $A$ and $B$. This conjecture relates the topological full group of an ample groupoid with the homology groups of the groupoid. We also give a criterion under which the topological full group $[[\mathcal{G}_{A,B}]]$ is finitely generated.

We provide a Cuntz-Pimsner model for graph of groups $C^*$-algebras. This allows us to compute the $K$-theory of a range of examples and show that graph of groups $C^*$-algebras can be realised as Exel-Pardo algebras. We also make a preliminary investigation of whether the crossed product algebra of Baumslag-Solitar groups acting on the boundary of certain trees satisfies Poincar\'e duality in $KK$-theory. By constructing a $K$-theory duality class we compute the $K$-homology of these crossed products.

Let G be a countable discrete amenable group, and Λ be a strongly connected finite k-graph. If (G,Λ) is a pseudo free and locally faithful self-similar action which satisfies the finite-state condition, then the structure of the KMS simplex of the C*-algebra O G,Λ associated to (G,Λ) is described: it is either empty or affinely isomorphic to the tracial state space of the C*-algebra of the periodicity group Per G,Λ of (G,Λ), depending on whether the Perron-Frobenius eigenvector of Λ is G-equivariant. As applications of our main results, we also exhibit several classes of important examples.

We study amenable minimal Cantor systems of free groups arising from the
diagonal actions of the boundary actions and certain Cantor systems. It turns
out that for any virtually free group, (explicitly given) continuum many
Kirchberg algebras are realized as a crossed product of its amenable minimal
Cantor system. In particular this shows that there exist continuum many
Kirchberg algebras each of which is decomposed to the crossed product of an
amenable minimal Cantor systems of any virtually free group. The techniques
developed in our study also enable us to compute the K-theory of certain
amenable minimal Cantor systems. As an application we compute the K-theory for
the diagonal actions of the boundary actions and the odometer transformations.
The results of computations classify them in terms of the topological full
groups, continuous orbit equivalence, strong orbit equivalence, and the crossed
products.

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.

This is the fourth and final part in the series of papers [Trans. Am. Math. Soc. 356, No. 11, 4287–4322 (2004; Zbl 1049.46039); Int. J. Math. 17, No. 7, 791–833 (2006; Zbl 1107.46040); Ergodic Theory Dyn. Syst. 26, No. 6, 1805–1854 (2006; Zbl 1136.46041)] considering C * -algebras associated with topological graphs. In the present paper, we get a sufficient condition on topological graphs so that the associated C * -algebras are simple and purely infinite. Using this result, we give a method to construct all Kirchberg algebras as C * -algebras associated with topological graphs.

Let M be a coherent D-module (e.g., an overdetermined system of partial dierential equations) on the complexification of a real analytic manifold M. Assume that the characteristic variety ofM is hyperbolic with respect to a submanifold N M. Then, it is well-known that the Cauchy problem forM with data on N is well posed in the space of hyperfunctions. In this paper, under the additional assumption thatM has regular singu- larities along a regular involutive submanifold of real type, we prove that the Cauchy problem is well posed in the space of distributions. When M is induced by a single dierential operator (or by a normal square system) with characteristics of constant multiplicities, our hypotheses correspond to Levi conditions, and we recover a classical result.

We first prove that every σ-unital, purely infinite, simple C*-algebra is either unital or stable. Consequently, purely infinite simple C*-algebras have real rank zero. In particular, the Cuntz algebras On (2 ≤ n ≤ +∞) and the Cuntz-Krieger algebras OA, where A can be any irreducible matrix, contain abundant projections. This includes an answer for a question raised by B. Blackadar in [5, 2.10]. We then prove that the corona and multiplier algebras associated with many interesting C*-algebras have real rank zero. As special cases, we consider the multiplier and corona algebras associated with certain simple AF algebras, the stabilizations of type II1 and type III factors, the Cuntz algebras and certain Cuntz-Krieger algebras, the Bunce-Deddens algebras and some irrational rotation algebras. A recent result of L. G. Brown and G. K. Pedersen in [12, 3.21] is included as a special case. In particular, K1(A) = 0, where A is a σ-unital, purely infinite simple C*-algebra, if and only if the generalized Weylvon Neumann theorem holds in M(A ⊗ A).

We associate to each row-finite directed graph E a universal Cuntz-Krieger C - algebra C (E), and study how the distribution of loops in E affects the structure of C (E). We prove that C (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the Cuntz-Krieger uniqueness theorem and give a characterisation of when C (E) is purely infinite. If the graph E satisfies (L) and is cofinal, then we have a dichotomy: if E has no loops, then C (E) is AF; if E has a loop, then C (E) is purely infinite. If A is an n Theta n f0; 1g-matrix, a Cuntz-Krieger A-family consists of n partial isometries S i on Hilbert space satisfying S i S i = n X j=1 A(i; j)S j S j : (1) Cuntz and Krieger proved that, provided A satisfies a fullness condition (I) and the partial isometries are all nonzero, two such families generate isomorphic C -algebras; thus the Cuntz-Krieger algebra O A can be well-define...

Let H be a compact quantum group with faithful Haar measure and bounded counit. If H acts on a C*-algebra A, we show that A is nuclear if and only if its fixed-point subalgebra is nuclear. As a consequence H is a nuclear C*-algebra. Comment: 12 pages, LateX 2e

The possibility that nuclear (or amenable) C*-algebras should be classified up to isomorphism by their K-theory and related invariants was raised in an article by Elliott [48] (written in 1989) in which he showed that a certain class of inductive limit algebras (AT-algebras of real rank zero) admits such a classification. Elliott made the inspired suggestion that his classification theorem perhaps covers all separable, nuclear C*-algebras of real rank zero, stable rank one, and with torsion free K
0 and K
1-groups. This was the first formulation of the Elliott conjecture. Evidence in favor of the conjection was shortly after provided by Elliott and Evans ([51]) who showed that all irrational rotation C*-algebras belong to the class covered by Elliott’s classification theorem. The Elliott conjecture was later modified so as to encompass all nuclear, separable, simple C*-algebras (not necessarily of stable rank one and real rank zero, and without the restrictions on the K-theory). Ed Effros writes about Elliott’s conjecture: “This was regarded as ridiculous by many (including myself), and we waited for the counter-examples to appear. We are still waiting.”

Basic properties of finite group actions with the Rohlin property
on unital C*-algebras are investigated. A characterization of
finite group actions with the Rohlin property on the Cuntz algebra
$\mathcal{O}_2$
is given in terms of central sequences, which may
be considered as an equivariant version of E. Kirchberg and N. C.
Phillips's characterization of $\mathcal{O}_2$ . A large class of
symmetries on $\mathcal{O}_2$ are classified in terms of the fixed-point algebras for
conjugacy and the crossed products for cocycle conjugacy. Model
actions of symmetries of $\mathcal{O}_2$
are constructed for given K-theoretical invariants.

I. Introduction To K-Theory.- 1. Survey of topological K-theory.- 2. Overview of operator K-theory.- II. Preliminaries.- 3. Local Banach algebras and inductive limits.- 4. Idempotents and equivalence.- III. K0-Theory and Order.- 5. Basi K0-theory.- 6. Order structure on K0.- 7. Theory of AF algebras.- IV. K1-Theory and Bott Periodicity.- 8. Higher K-groups.- 9. Bott Periodicity.- V. K-Theory of Crossed Products.- 10. The Pimsner-Voiculescu exact sequence and Connes' Thorn isomorphism.- 11. Equivariant K-theory.- VI. More Preliminaries.- 12. Multiplier algebras.- 13. Hilbert modules.- 14. Graded C*-algebras.- VII. Theory of Extensions.- 15. Basic theory of extensions.- 16. Brown-Douglas-Fillmore theory and other applications.- VIII. Kasparov's KK-Theory.- 17. Basic theory.- 18. Intersection product.- 19. Further structure in KK-theory.- 20. Equivariant KK-theory.- IX. Further Topics.- 21. Homology and cohomology theories on C*-algebras.- 22. Axiomatic K-theory.- 23. Universal coefficient theorems and Kunneth theorems.- 24. Survey of applications to geometry and topology.

To a special embedding Φ of circle algebras having the same spectrum, we associate an r-discrete, locally compact groupoid, similar to the Cuntz-Krieger groupoid. Its C * -algebra, denoted O Φ , is a continuous version of the Cuntz-Krieger algebras O A . The algebra O Φ is generated by an AT-algebra and a nonunitary isometry. We compute its K-theory under the assumption that the AT-algebra is simple.

We classify finite group actions on some classes of C∗-algebras with the Rohlin property in terms of the K-groups. A group cohomological obstruction is obtained for the K-groups of a C∗-algebra allowing such an action, for example, the Tate cohomology groups with the K-groups as the coefficient modules must vanish. Model actions with the Rohlin property are constructed for given K-theoretical invariants. Quasi-free actions of finite cyclic groups on the Cuntz algebras and their dual actions are investigated from the view point of the Rohlin property.

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.

This is the fourth and final part in the series of papers [Trans. Am. Math. Soc. 356, No. 11, 4287–4322 (2004; Zbl 1049.46039); Int. J. Math. 17, No. 7, 791–833 (2006; Zbl 1107.46040); Ergodic Theory Dyn. Syst. 26, No. 6, 1805–1854 (2006; Zbl 1136.46041)] considering C * -algebras associated with topological graphs. In the present paper, we get a sufficient condition on topological graphs so that the associated C * -algebras are simple and purely infinite. Using this result, we give a method to construct all Kirchberg algebras as C * -algebras associated with topological graphs.

We study C∗-algebras arising from C∗-correspondences, which were introduced by the author. We prove the gauge-invariant uniqueness theorem, and obtain conditions for our C∗-algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6-term exact sequence of K-groups involving the K-groups of our C∗-algebras.

We compute the K-theory of the Cuntz-Krieger C^*-algebras associated to infinite matrices.

We prove the following theorem: let $A$ be a UCT Kirchberg algebra, and let $\alpha$ be a prime-order automorphism of $K_*(A)$, with $\alpha([1_A])=[1_A]$ in case $A$ is unital. Then $\alpha$ is induced from an automorphism of $A$ having the same order as $\alpha$. This result is extended to certain instances of an equivariant inclusion of Kirchberg algebras. As a crucial ingredient we prove the following result in representation theory: every module over the integral group ring of a cyclic group of prime order has a natural presentation by generalized lattices with no cyclotomic summands.

We introduce a method to define $C^*$-algebras from $C^*$-correspondences. Our construction generalizes Cuntz-Pimsner algebras, crossed products by Hilbert $C^*$-modules, and graph algebras.

Let G 0 and G 1 be countable abelian groups. Let γ i be an automorphism of G i of order two. Then there exists a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with [1 A ] = 0 in K 0 (A), and an automorphism α ∈ Aut(A) of order two, such that K 0 (A) ≅ G 0 , such that K 1 (A) ≅ G 1 , and such that α * : K i (A) → K i (A) is γ i . As a consequence, we prove that every -graded countable module over the representation ring R( ) of is isomorphic to the equivariant K-theory K (A) for some action of on a unital Kirchberg algebra A .
Along the way, we prove that every not necessarily finitely generated [ ]-module which is free as a -module has a direct sum decomposition with only three kinds of summands, namely [ ] itself and on which the nontrivial element of acts either trivially or by multiplication by −1.

We introduce a new class of C^*-algebras, which is a generalization of both graph algebras and homeomorphism C^*-algebras. This class is very large and also very tractable. We prove the so-called gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem, and compute the K-groups of our algebras.

Given an arbitrary infinite 0--1 matrix A having no identically zero rows, we define an algebra OA as the universal C*-algebra generated by partial isometries subject to conditions that generalize, to the infinite case, those introduced by Cuntz and Krieger for finite matrices. We realize OA as the crossed product algebra for a partial dynamical system and, based on this description, we extend to the infinite case some of the main results known to hold in the finite case, namely the uniqueness theorem, the classification of ideals, and the simplicity criteria. OA is always nuclear and we obtain conditions for it to be unital and purely infinite.

Starting from Kirchberg's theorems announced in 1994, namely O_2 tensor A is isomorphic to O_2 for separable unital nuclear simple A and O_infinity tensor A is isomorphic to A if in addition A is purely infinite, we prove that KK-equivalence implies isomorphism for nonunital separable nuclear purely infinite simple C*-algebras. It follows that if A and B are unital separable nuclear purely infinite simple C*-algebras which satisfy the Universal Coefficient Theorem, and if there is a graded isomorphism from K_* (A) to K_* (B) which preserves the class of the identity, then A is isomorphic to B. Our main technical results are, we believe, of independent interest. We say that two asymptotic morphisms t ---> \phi_t and t ---> \psi_t from A to B are asymptotically unitarily equivalent if there exists a continuous unitary path t ---> u_t in the unitization B^+ such that || u_t \phi_t (a) u_t^* - \psi_t (a) || ---> 0 for all a in A. Let A be separable, nuclear, unital, and simple, and let D be unital. We prove that any asymptotic morphism from A to K tensor O_infinity tensor D is asymptotically unitarily equivalent to a homomorphism, and two homotopic homomorphisms from A to K tensor O_infinity tensor D are asymptotically unitarily equivalent.

Graph algebras Published for the Conference Board of the Mathematical Sciences

- I Raeburn

Raeburn, I. Graph algebras. CBMS Regional Conference Series in Mathematics, 103. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005.

Actions on Kirchberg algebras Brought to you by | Columbia University Library The Burke Library New York Authenticated | 128

- Katsura

Katsura, Actions on Kirchberg algebras Brought to you by | Columbia University Library The Burke Library New York Authenticated | 128.59.62.83 Download Date | 2/18/13 12:15 AM

The classification of purely infinite C Ã -algebras using Kasparov's theory, preprint 1994 Cuntz-Krieger algebras of directed graphs

- E Kirchberg
- Kpr
- A Kumjian
- D Pask
- I Raeburn

Kirchberg, E., The classification of purely infinite C Ã -algebras using Kasparov's theory, preprint 1994. [KPR] Kumjian, A., Pask, D., Raeburn, I., Cuntz-Krieger algebras of directed graphs, Pac. J. Math. 184 (1998), no. 1, 161–174.

[Sp] Spielberg, J. Non-cyclotomic Presentations of Modules and Prime-order Automorphisms of Kirchberg Algebras

- W R Scott
- Group
- N J Englewood Cliffs

[Sc]
Scott, W. R. Group theory. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.
[Sp]
Spielberg, J. Non-cyclotomic Presentations of Modules and Prime-order Automorphisms of
Kirchberg Algebras. Preprint 2005, math.OA/0504287.
[Z]

The classification of purely infinite C * -algebras using Kasparov's theory

- E Kirchberg

Kirchberg, E. The classification of purely infinite C * -algebras using Kasparov's theory. Preprint
1994.

Classification of nuclear C * -algebras. Encyclopaedia of Mathematical Sciences, 126. Operator Algebras and Non-commutative Geometry

- M Rørdam

Rørdam, M. Classification of nuclear C * -algebras. Encyclopaedia of Mathematical Sciences, 126.
Operator Algebras and Non-commutative Geometry, 7. Springer-Verlag, Berlin, 2002, pages 1-145.