Related papers: Renault's Equivalence Theorem for Reduced Groupoid…

It is well known that a measured groupoid G defines a von Neumann algebra W*(G), and that a Lie groupoid G canonically defines both a C*-algebra C*(G) and a Poisson manifold A*(G). We show that the maps G -> W*(G), G -> C*(G) and G -> A*(G)…

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…

We introduce a notion of ideal-related K-theory for rings, and use it to prove that if two complex Leavitt path algebras are Morita equivalent (respectively, isomorphic), then the ideal-related K-theories (respectively, the unital…

In this paper we describe an operation on directed graphs which produces a graph with fewer vertices, such that the C*-algebra of the new graph is Morita equivalent to that of the original graph. We unify and generalize several related…

In this paper, we will prove that if $A$ is a $C^*$-algebra with an effective coaction $\epsilon$ by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an…

In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced $\mathrm{C}^*$-algebra $C_r^*(G)$ has real rank zero and strict comparison even though $C_r^*(G)$ may not be nuclear in general. Moreover, if we further…

We calculate the $K$-theory of the reduced $C^*$-algebra $C^*_r(G)$ of a reductive $p$-adic group $G$. To do so, we show that each direct summand in Plymen's Plancherel decomposition of $C^*_r(G)$ is Morita equivalent to a twisted crossed…

We show that semigroup C*-algebras are groupoid C*-algebras.

We study the groupoid C*-algebras associated to the equivalence relation induced by a quotient map on a locally compact Hausdorff space. This C*-algebra is always a Fell algebra, and if the quotient space is Hausdorff, it is a…

We unify the Kumjian-Renault Weyl groupoid construction with the Lawson-Lenz version of Exel's tight groupoid construction. We do this by utilising only a weak algebraic fragment of the C*-algebra structure, namely its *-semigroup reduct.…

We show that important structural properties of C*-algebras and the multiplicity numbers of representations are preserved under Morita equivalence.

In this paper we generalize the notion of Cuntz-Pimsner algebras of $C^*$-correspondences to the setting of subproduct systems. The construction is justified in several ways, including the Morita equivalence of the operator algebras under…

We provide an exposition and proof of Renault's equivalence theorem for crossed products by locally Hausdorff, locally compact groupoids. Our approach stresses the bundle approach, concrete imprimitivity bimodules and is a preamble to a…

We provide a complete classification of the class of unital graph $C^*$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is determined by ordered, filtered…

Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of $^*$-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$.…

We show that if $(A,G,\alpha)$ is a groupoid dynamical system with $A$ continuous trace, then the crossed product $A\rtimes_{\alpha}G$ is Morita equivalent to the C*-algebra $C*(\underline G,\underline E)$ of a twist $\underline E$ over a…

A theory of Galois co-objects for von Neumann bialgebras is introduced. This concept is closely related to the notion of comonoidal W*-Morita equivalence between von Neumann bialgebras, which is a Morita equivalence taking the…

We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group $G$ and a given normal subgroup $H$. We also give conditions for the existence of a…

We study $C^*$-algebras arising from $C^*$-correspondences, which was 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…

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if…