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Let $X$ be a compact Hausdorff space, and let $\gamma$ be an iterated function system on $X$. Kajiwara and Watatani showed that if $\gamma$ is self-similar and satisfies the open set condition and some additional technical conditions, $C(X)$ is a maximal abelian subalgebra of the Kajiwara--Watatani algebra $\O_\gamma$ associated with $\gamma$. In this paper, we extend their results by providing sufficient conditions under which $C(X)$ becomes a Cartan subalgebra of $\O_\gamma$. Additionally, we present sufficient conditions under which $C(X)$ fails to be a masa of $\O_\gamma$., Comment: 39 pages

We use the Ladder Technique to establish bijections between the ideals of related Fell bundles., Comment: 11 pages

In the setting of product systems over group-embeddable monoids, we consider nuclearity of the associated Toeplitz C*-algebra in relation to nuclearity of the coefficient algebra. Our work goes beyond the known cases of single correspondences and compactly aligned product systems over right LCM monoids. Specifically, given a product system over a submonoid of a group, we show, under technical assumptions, that the fixed-point algebra of the gauge action is nuclear iff the coefficient algebra is nuclear; when the group is amenable, we conclude that this happens iff the Toeplitz algebra itself is nuclear. Our main results imply that nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for every full product system of Hilbert bimodules over abelian monoids, over $ax+b$-monoids of integral domains and over Baumslag-Solitar monoids $BS^+(m,n)$ that admit an amenable embedding, which we provide for $m$ and $n$ relatively prime.

We introduce the notion of relative topological principality for a family $\{H_\alpha\}$ of open subgroupoids of a Hausdorff \'etale groupoid $G$. The C*-algebras $C^*_r(H_\alpha)$ of the groupoids $H_\alpha$ embed in $ C^*_r(G)$ and we show that if $G$ is topologically principal relative to $\{H_\alpha\}$ then a representation of $C^*_r(G)$ is faithful if and only if its restriction to each of the subalgebras $C^*_r(H_\alpha)$ is faithful. This variant of the ideal intersection property potentially involves several subalgebras, and gives a new method of verifying injectivity of representations of reduced groupoid C*-algebras. As applications we prove a uniqueness theorem for Toeplitz C*-algebras of left cancellative small categories that generalizes a recent result of Laca and Sehnem for Toeplitz algebras of group-embeddable monoids, and we also discuss and compare concrete examples arising from integer arithmetic., Comment: 17 pages. A few comments and references added

A long-standing open question in the theory of group actions on C*-algebras is the stable rank of the crossed product. Specifically, N. C. Phillips asked that if a finite group $G$ acts on a simple unital C*-algebra $A$ with stable rank one, does the crossed product have stable rank one? A similar question can be asked about the real rank. Most of the existing partial answers contain a reasonable restriction (mainly, a Rokhlin-type property) on the action and assumptions on $A$. We remove all extra assumptions on $A$ (for instance, stable finiteness and that the order on projections over $A$ is determined by traces) and we prove that if the action has the tracial Rokhlin property and $A$ is simple and $\sigma$-unital with stable rank one or real rank zero, then so do the crossed product and the fixed point algebra. Moreover, we show that if the Kirchberg's central sequence algebra $\mathrm{F}(A)$ has real rank zero, then the weak tracial Rokhlin property is equivalent to the tracial Rokhlin property for actions on simple unital separable C*-algebras $A$., Comment: 17 pages. arXiv admin note: text overlap with arXiv:1711.10818

We establish a generalized Rieffel correspondence for ideals in equivalent Fell bundles., Comment: 20 pages, fixed typo

Let $R$ be a rational function with degree $\geq 2$ and $X$ be its Julia set, its Fatou set, or the Riemann sphere. Suppose that $X$ is not empty. We can regard $R$ as a continuous map from $X$ onto itself. Kajiwara and Watatani showed that in the case that $X$ is the Julia set, $C_0(X)$ is a maximal abelian subalgebra of $\mathcal{O}_R(X)$, where $\mathcal{O}_R(X)$ denotes the C*-algebra associated with the dynamical system $(X,R)$ introduced by them. In this paper, we develop their result and give the equivalent condition for $C_0(X)$ to be a Cartan subalgebra of $\mathcal{O}_R(X)$., Comment: 27 pages

We consider the notion of equivariant uniform property Gamma for actions of countable discrete groups on C*-algebras that admit traces. In case the group is amenable and the C*-algebra has a compact tracial state space, we prove that this property implies a kind of tracial local-to-global principle for the C*-dynamical system, generalizing a recent such principle for C*-algebras exhibited in work of Castillejos et al. For actions on simple nuclear $\mathcal{Z}$-stable C*-algebras, we use this to prove that equivariant uniform property Gamma is equivalent to equivariant $\mathcal{Z}$-stability, generalizing a result of Gardella-Hirshberg-Vaccaro., Comment: 47 pages, various small corrections and added details. This version has been accepted for publication by APDE

We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II$_1$-factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C$^*$-dynamics. Given a countable discrete group $G$ and an amenable action $G\curvearrowright M$ on any separably acting semi-finite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing $G$-action is suitably absorbed at the level of each fibre in the direct integral decomposition of $M$, then it is tensorially absorbed by the action on $M$. As a direct application of Ocneanu's theorem, we deduce that if $M$ has the McDuff property, then every amenable $G$-action on $M$ has the equivariant McDuff property, regardless whether $M$ is assumed to be injective or not. By employing Tomita-Takesaki theory, we can extend the latter result to the general case where $M$ is not assumed to be semi-finite., Comment: 34 pages; some corrections and rewritten introduction; this version was accepted for publication in Journal of the Institute of Mathematics of Jussieu

Let $(A,\mathbb{N}^{2},\alpha)$ be a dynamical system consisting of a $C^*$-algebra $A$ and an action $\alpha$ of $\mathbb{N}^{2}$ on $A$ by automorphisms. Let $A\times_{\alpha}^{\textrm{piso}}\mathbb{N}^{2}$ be the partial-isometric crossed product of the system. We apply the fact that it is a full corner of a crossed product by the group $\mathbb{Z}^{2}$ in order to give a complete description of its primitive ideal space., Comment: This is the revised (second) version (19 pages)

We define a bicategory with \'etale, locally compact groupoids as objects and suitable correspondences, that is, spaces with two commuting actions as arrows; the 2-arrows are injective, equivariant continuous maps. We prove that the usual recipe for composition makes this a bicategory, carefully treating also non-Hausdorff groupoids and correspondences. We extend the groupoid C*-algebra construction to a homomorphism from this bicategory to that of C*-algebra correspondences. We describe the C*-algebras of self-similar groups, higher-rank graphs, and discrete Conduch\'e fibrations in our setup.

Let $\Gamma$ be a discrete group acting freely via homeomorphisms on the compact Hausdorff space $X$ and let $C(X) \rtimes_\eta \Gamma$ be the completion of the convolution algebra $C_c(\Gamma,C(X))$ with respect to a $C^*$-norm $\eta$. A non-zero ideal $J \unlhd C(X) \rtimes_\eta \Gamma$ is exotic if $J \cap C(X) = \{0\}$. We show that exotic ideals are present whenever $\Gamma$ is non-amenable and there is an invariant probability measure on $X$. This fact, along with the recent theory of exotic crossed product functors, allows us to provide answers to two questions of K. Thomsen. Using the Koopman representation and a recent theorem of Elek, we show that when $\Gamma$ is a countably-infinite group having property (T) and $X$ is the Cantor set, there exists a free and minimal action of $\Gamma$ on $X$ and a $C^*$-norm $\eta$ on $C_c(\Gamma, C(X))$ such that $C(X)\rtimes_\eta\Gamma$ contains the compact operators as an exotic ideal. We use this example to provide a positive answer to a question of A. Katavolos and V. Paulsen. The opaque and grey ideals in $C(X)\rtimes_\eta \Gamma$ have trivial intersection with $C(X)$, and a result from arXiv:1901.09683 shows they coincide when the action of $\Gamma$ is free, however the problem of whether these ideals can be non-zero was left unresolved. We present an example of a free action of $\Gamma$ on a compact Hausdorff space $X$ along with a $C^*$-norm $\eta$ for which these ideals are non-trivial, in particular, they are exotic ideals., Comment: Article is totally rewritten, reorganized, and has a new title (former title: "Exotic Ideals in Represented Free Transformation Groups") Includes some new results. 16 pages

Suppose that $\alpha$ is an action of the semigroup $\mathbb{N}^{2}$ on a $C^*$-algebra $A$ by endomorphisms. Let $A\times_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ be the associated partial-isometric crossed product. By applying an earlier result which embeds this semigroup crossed product (as a full corner) in a crossed product by the group $\mathbb{Z}^{2}$, a composition series $0\leq L_{1}\leq L_{2}\leq A\times_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ of essential ideals is obtained for which we identify the subquotients with familiar algebras., Comment: This is the revised version, 24 pages. arXiv admin note: text overlap with arXiv:1912.09682

Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterisation of injectivity for crossed products generalises a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$-algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$-algebras satisfies a number of expected properties. A notion of inner amenability for actions of locally compact groups is introduced, and a number of applications are given in the form of averaging arguments, relating approximation properties of crossed product von Neumann algebras to properties of the components of the underlying $w^*$-dynamical system. We use these results to answer a recent question of Buss-Echterhoff-Willett., Comment: Fixed a mistake in Section 6. Typo fixes

Let P be a left LCM semigroup, and $\alpha$ an action of $P$ by endomorphisms of a $C^{*}$-algebra $A$. We study a semigroup crossed product $C^{*}$-algebra in which the action $\alpha$ is implemented by partial isometries. This crossed product gives a model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules (associated with semigroup dynamical systems) studied first by Fowler, for which we provide a structure theorem as it behaves well under short exact sequences and tensor products., Comment: This is the third version (50 pages). The title has slightly changed. Some references are added

Consider the pair $(G,P)$ consisting of an abelian lattice-ordered discrete group $G$ and its positive cone $P$. Let $\alpha$ be an action of $P$ by extendible endomorphisms of a $C^*$-algebra $A$. We show that the Nica-Toeplitz algebra $\mathcal{T}_{\textrm{cov}}(A\times_{\alpha} P)$ is a full corner of a group crossed product $\mathcal{B}\rtimes_{\beta}G$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(G,A)$ generated by a collection of faithful copies of $A$, and the action $\beta$ on $\mathcal{B}$ is given by the shift on $\ell^{\infty}(G,A)$. By using this realization, we show that the ideal $\mathcal{I}$ of $\mathcal{T}_{\textrm{cov}}(A\times_{\alpha} P)$ for which the quotient algebra $\mathcal{T}_{\textrm{cov}}(A\times_{\alpha} P)/\mathcal{I}$ is the isometric crossed product $A\times_{\alpha}^{\textrm{iso}} P$ is also a full corner in an ideal $\mathcal{J}\rtimes_{\beta}G$ of $\mathcal{B}\rtimes_{\beta}G$., Comment: to appear in Houston J. Math

We provide the rigorous foundations for a categorical approach to the classification of C*-dynamics up to cocycle conjugacy. Given a locally compact group $G$, we consider a category of (twisted) $G$-C*-algebras, where morphisms between two objects are allowed to be equivariant maps or exterior equivalences, which leads to the concept of so-called cocycle morphisms. An isomorphism in this category is precisely a cocycle conjugacy in the known sense. We show that this category allows sequential inductive limits, and that some known functors on the usual category of $G$-C*-algebras extend. After observing that this setup allows a natural notion of (approximate) unitary equivalence, the main aim of the paper is to generalize the fundamental intertwining results commonly employed in the Elliott program for classifying C*-algebras. This reduces a given classification problem for C*-dynamics to the prevalence of certain uniqueness and existence theorems, and may provide a useful alternative to the Evans--Kishimoto intertwining argument in future research., Comment: 56 pages; this version has been accepted for publication in JFA. Note: some concepts introduced in this paper have been renamed

We study some relations between self-similar group actions and operator algebras. We consider KMS states on the Cuntz--Pimsner algebras constructed by Nekrashevych from self-similar actions and the GNS representations of the KMS states. The KMS states are given by the Bernoulli measure. We also consider the von Neumann algebras on the GNS spaces and show that the von Neumann algebras are type III factors., Comment: 13 pages, to appear in International Journal of Mathematics

We give a complete $K$-theoretical description of when an extension of two simple graph $C^{*}$-algebras is again a graph $C^{*}$-algebra., Comment: Accepted version, to appear in Annals of K-theory

We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. As a special case, we show that the $K$-groups of the continuous asymptotic centralizer are isomorphic to the $KK$-groups of the Kirchberg algebra., Comment: 32 pages

Toward the complete classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras, we prove several fundamental theorems that are used in the classification. In addition, as an application of them, we classify outer actions of poly-$\mathbb{Z}$ groups of Hirsch length not greater than three on unital Kirchberg algebras up to $KK$-trivial cocycle conjugacy., Comment: 60 pages

We introduce a generalization of the notion of approximately proper equivalence relations studied by Renault and with it we build an \'etale groupoid. Choosing a suitable set of continuous functions to play the role of a potential, we construct a cocycle in that groupoid and discuss the corresponding Radon-Nikodym problem.

We prove that for any countable group G there exists a free minimal continuous action of G on the Cantor set admitting an invariant Borel probability measure., Comment: to appear in Ergodic Theory and Dynamical Systems. arXiv admin note: text overlap with arXiv:1803.05461

In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly self-absorbing C*-algebras. Namely, as to be made precise in the paper, let $G$ be a well-behaved locally compact group. If $\mathcal D$ is a strongly self-absorbing C*-algebra, and $\alpha: G\curvearrowright A$ is an action on a separable, $\mathcal D$-absorbing C*-algebra that has finite Rokhlin dimension with commuting towers, then $\alpha$ tensorially absorbs every semi-strongly self-absorbing $G$-actions on $\mathcal D$. This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some non-trivial applications. Most notably it is shown that for any $k\geq 1$ and on any strongly self-absorbing Kirchberg algebra, there exists a unique $\mathbb R^k$-action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy., Comment: v2 40 pages; this version has been accepted for publication in Analysis & PDE

Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\alpha: \Gamma \curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\Theta_\alpha: X\to C$ such that $\Theta_\alpha\circ \alpha=\zeta \circ \Theta_\alpha$. We extend our theorem for (nonfree) Borel $(\Gamma,Z)$-actions, where $Z$ is a uniformly recurrent subgroup.

We prove that any countable non-amenable group G admits a free minimal amenable purely infinite action on the non-compact Cantor set. This answers a question of Kellerhals, Monod and R{\o}rdam., Comment: 16 pages

Let $\Gamma^{+}$ be the positive cone of a totally ordered abelian discrete group $\Gamma$, and $\alpha$ an action of $\Gamma^{+}$ by extendible endomorphisms of a $C^*$-algebra $A$. We prove that the partial-isometric crossed product $A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$ is a full corner of a group crossed product $\mathcal{B}\times_{\beta}\Gamma$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(\Gamma,A)$ generated by a collection of faithful copies of $A$, and the action $\beta$ on $\mathcal{B}$ is induced by shift on $\ell^{\infty}(\Gamma,A)$. We then use this realization to show that $A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$ has an essential ideal $J$, which is a full corner in an ideal $\mathcal{I}\times_{\beta}\Gamma$ of $\mathcal{B}\times_{\beta}\Gamma$.

For a number of properties of C*-algebras, including real rank zero, stable rank one, pure infiniteness, residual hereditary infiniteness, the combination of pure infiniteness and the ideal property, the property of being an AT algebra with real rank zero, and stability under tensoring with a strongly selfabsorbing C*-algebra, we prove the following. Consider an arbitrary action of a second countable compact abelian group on a separable C*-algebra. Then the fixed point algebra under the action has the given property if and only if the crossed product has the same property., Comment: AMSLaTeX; 14 pages

A characterization of relative weak mixing in W*-dynamical systems in terms of a relatively independent joining is proven., Comment: 21 pages

We introduce a graph theoretic property called Condition (N) for finitely separated graphs and prove that it is equivalent to both nuclearity and exactness of the associated universal tame graph C*-algebra., Comment: 30 pages

Let $\mathcal{G}$ be an ultragraph and let $C^*(\mathcal{G})$ be the associated $C^*$-algebra introduced by Mark Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*(\mathcal{G})$, we approach the quotient $C^*$-algebra $C^*(\mathcal{G})/I_{(H,B)}$ by the $C^*$-algebra of finite graphs and prove versions of gauge invariant and Cuntz-Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph $C^*$-algebras (in the sense of Kirchberg-R${\o}$rdam) via Fell bundles., Comment: This is modified and improved version of the paper "Quotient Ultragraph $C^*$-algebras"; [arXiv:1512.00346] some results modified, and purely infinite ultragraph $C^*$-algebras described in the sense of Kirchberg-Rordam