arXiv subjects

Ruy Exel

Publications and source records attributed to Ruy Exel.

At least 37 records · Page 2Linked to original sources

arXiv2016

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an étale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.

math.DS↗

arXiv2015

Amenable actions of inverse semigroups

We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup $S$, the action of $S$ on its spectrum is amenable if and only if every action of $S$ is amenable.

math.OA↗

arXiv2015

K-theory for the tame C*-algebra of a separated graph

A {\it separated graph} is a pair $(E,C)$ consisting of a directed graph $E$ and a set $C=\bigsqcup_{v\in E^0}C_v$, where each $C_v$ is a partition of the set of edges whose terminal vertex is $v$. Given a separated graph $(E,C)$, such that all the sets $X\in C$ are finite, the K-theory of the graph C*-algebra $C^*(E,C)$ is known to be determined by the kernel and the cokernel of a certain map, denoted by $1_C- A_{(E,C)}$, from $\mathbb Z^{(C)}$ to $\mathbb Z^{(E^0)}$. In this paper, we compute the K-theory of the {\it tame} graph C*-algebra $\mathcal O(E,C)$ associated to $(E,C)$, which has been recently introduced by the authors. Letting $π$ denote the natural surjective homomorphism from $C^*(E,C)$ onto $\mathcal O(E,C)$, we show that $K_1(π)$ is a group isomorphism, and that $K_0(π)$ is a split monomorphism, whose cokernel is a torsion-free abelian group. We also prove that this cokernel is a free abelian group when the graph $E$ is finite, and determine its generators in terms of a sequence of separated graphs $\{(E_n, C^n)\}_{n=1}^{\infty}$ naturally attached to $(E,C)$. On the way to showing our main results, we obtain an explicit description of a connecting map arising in a six-term exact sequence computing the K-theory of an amalgamated free product, and we also exhibit an explicit isomorphism between $\mathrm{ker} (1_C - A_{(E,C)})$ and $K_1(C^*(E,C))$.

math.OA↗

arXiv2014

A remark on contracting inverse semigroups

A semi-lattice is said to be tree-like when any two of its elements are either orthogonal or comparable. Given an inverse semigroup S whose idempotent semi-lattice is tree-like, and such that all tight filters are ultra-filters, we present a necessary and sufficient condition for S to be contracting which looks closer in spirit to the notion of contracting actions than a recent condition found by the second named author and E. Pardo.

math.OA↗

arXiv2014

The tight groupoid of an inverse semigroup

In this work we present algebraic conditions on an inverse semigroup S (with zero) which imply that its associated tight groupoid G_{tight)(S) is: Hausdorff, essentially principal, minimal and contracting, respectively. In some cases these conditions are in fact necessary and sufficient.

math.OA↗

arXiv2014

Self-Similar Graph C*-Algebras and Partial Crossed Products

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 the UCT class. 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 $H$.

math.OA↗

arXiv2013

Inverse semigroup expansions and their actions on C*-algebras

In this work, we give a presentation of the prefix expansion Pr(G) of an inverse semigroup G as recently introduced by Lawson, Margolis and Steinberg which is similar to the universal inverse semigroup defined by the second named author in case G is a group. The inverse semigroup Pr(G) classifies the partial actions of G on spaces. We extend this result and prove that Fell bundles over G correspond bijectively to saturated Fell bundles over Pr(G). In particular, this shows that twisted partial actions of G (on C*-algebras) correspond to twisted (global) actions of Pr(G). Furthermore, we show that this correspondence preserves C*-algebras crossed products.

math.OA↗

arXiv2013

Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions

We attach to each finite bipartite separated graph (E,C) a partial dynamical system (Ω(E,C), F, θ), where Ω(E,C) is a zero-dimensional metrizable compact space, F is a finitely generated free group, and θ is a continuous partial action of F on Ω(E,C). The full crossed product C*-algebra O(E,C) = C(Ω(E,C)) \rtimes_θ F is shown to be a canonical quotient of the graph C*-algebra C^*(E,C) of the separated graph (E,C). Similarly, we prove that, for any *-field K, the algebraic crossed product L^{ab}_K(E,C) = C_K(Ω(E,C)) \rtimes_θ^{alg} F is a canonical quotient of the Leavitt path algebra L_K(E,C) of (E,C). The monoid V(L^{ab}_K(E,C)) of isomorphism classes of finitely generated projective modules over L^{ab}_K(E,C) is explicitly computed in terms of monoids associated to a canonical sequence of separated graphs. Using this, we are able to construct an action of a finitely generated free group F on a zero-dimensional metrizable compact space Z such that the type semigroup S(Z, F, K) is not almost unperforated, where K denotes the algebra of clopen subsets of Z. Finally we obtain a characterization of the separated graphs (E,C) such that the canonical partial action of F on Ω(E,C) is topologically free.

math.OA↗

arXiv2013

Graphs, groups and self-similarity

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

math.OA↗

arXiv2013

Representing Kirchberg algebras as inverse semigroup crossed products

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.

math.OA↗

arXiv2013

Invertibility in groupoid C*-algebras

Given a second-countable, Hausdorff, étale, amenable groupoid G with compact unit space, we show that an element a in C*(G) is invertible if and only if λ_x(a) is invertible for every x in the unit space of G, where λ_x refers to the "regular representation" of C*(G) on l_2(G_x). We also prove that, for every a in C*(G), there exists some x in G^{(0)} such that ||a|| = ||λ_x(a)||.

math.OA↗

arXiv2013

Interactions and dynamical systems of type (n,m) - A case study

We prove that the C*-algebra of the universal (n,m)-dynamical system may be obtained, up to Morita-Rieffel equivalence, as the crossed-product relative to an interaction on a commutative C*-algebra. The interaction involved is shown not to be part of an interaction group.

math.OA↗

arXiv2013

A Ruelle Operator for continuous time Markov Chains

We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t$, $t\geq 0$, taking values on S. We denote by $Ω$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($Ω, {\cal B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${\cal L}^\infty (P)$. From the probability $P$ we define a Ruelle operator ${\cal L}^t, t\geq 0$, acting on functions $f:Ω\to \mathbb{R}$ of ${\cal L}^\infty (P)$. Given $V:Ω\to \mathbb{R}$, such that is constant in sets of the form $\{X_0=c\}$, we define a modified Ruelle operator $\tilde{\cal L}_V^t, t\geq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $ν_V$ on $Ω$ associated to $\tilde{\cal L}^t_V, t\geq 0$. We also show the following property for the probability $ν_V$: for any integrable $g\in {\cal L}^\infty (P)$ and any real and positive $t$ $$ \int e^{-\int_0^t (V \circ Θ_s)(.) ds} [ (\tilde{\cal L}^t_V (g)) \circ θ_t ] d ν_V = \int g d ν_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras).

math.DS↗

arXiv2011

Dynamical systems of type (m,n) and their C*-algebras

Given positive integers n and m, we consider dynamical systems in which n copies of a topological space is homeomorphic to m copies of that same space. The universal such system is shown to arise naturally from the study of a C*-algebra we denote by O_{mn}, which in turn is obtained as a quotient of the well known Leavitt C*-algebra L_{mn}, a process meant to transform the generating set of partial isometries of L{mn} into a tame set. Describing O_{mn} as the crossed-product of the universal (m,n)-dynamical system by a partial action of the free group F_{m+n}, we show that O_{mn} is not exact when n and m are both greater than or equal to 2, but the corresponding reduced crossed-product, denoted O_{mn}^r, is shown to be exact and non-nuclear. Still under the assumption that m,n>=2, we prove that the partial action of F_{m+n} is topologically free and that O_{mn}^r satisfies property (SP) (small projections). We also show that O_{mn}^r admits no finite dimensional representations. The techniques developed to treat this system include several new results pertaining to the theory of Fell bundles over discrete groups.

math.OA↗

arXiv2011

The tiling C*-algebra viewed as a tight inverse semigroup algebra

We realize Kellendonk'?s C*-algebra of an aperiodic tiling as the tight C*-algebra of the inverse semigroup associated to the tiling, thus providing further evidence that the tight C*-algebra is a good candidate to be the natural associative algebra to go along with an inverse semigroup.

math.OA↗

arXiv2010

Partial crossed product description of the C*-algebras associated with integral domains

Recently, Cuntz and Li introduced the C^*-algebra A[R] associated to an integral domain R with finite quotients. In this paper, we show that A[R] is a partial group algebra of the group $K \rtimes K^x$ with suitable relations, where K is the field of fractions of R. We identify the spectrum of this relations and we show that it is homeomorphic to the profinite completion of R. By using partial crossed product theory, we reconstruct some results proved by Cuntz and Li. Among them, we prove that $\ar$ is simple by showing that the action is topologically free and minimal.

math.OA↗

arXiv2010

Purely infinite simple C*-algebras associated to integer dilation matrices

Given an n x n integer matrix A whose eigenvalues are strictly greater than 1 in absolute value, let σ_A be the transformation of the n-torus T^n=R^n/Z^n defined by σ_A(e^{2πix})=e^{2πiAx} for x\in R^n. We study the associated crossed-product C*-algebra, which is defined using a certain transfer operator for σ_A, proving it to be simple and purely infinite and computing its K-theory groups.

math.OA↗

arXiv2010

Twisted actions and regular Fell bundles over inverse semigroups

We introduce a new notion of twisted actions of inverse semigroups and show that they correspond bijectively to certain regular Fell bundles over inverse semigroups, yielding in this way a structure classification of such bundles. These include as special cases all the stable Fell bundles. Our definition of twisted actions properly generalizes a previous one introduced by Sieben and corresponds to Busby-Smith twisted actions in the group case. As an application we describe twisted etale groupoid C*-algebras in terms of crossed products by twisted actions of inverse semigroups and show that Sieben's twisted actions essentially correspond to twisted etale groupoids with topologically trivial twists.

math.OA↗