scholarly journals Topological full groups of ample groupoids with applications to graph algebras

2019 ◽  
Vol 30 (04) ◽  
pp. 1950018 ◽  
Author(s):  
Petter Nyland ◽  
Eduard Ortega
Keyword(s):  
Embedding Theorem ◽  
Full Group ◽  
Graph Algebras ◽  
Graph Groupoids ◽  
Locally Compact ◽  
Abstract Group ◽  
Path Algebras ◽  
Compact Case ◽  
Leavitt Path Algebras ◽  
Definition Of

We study the topological full group of ample groupoids over locally compact spaces. We extend Matui’s definition of the topological full group from the compact to the locally compact case. We provide two general classes of étale groupoids for which the topological full group, as an abstract group, is a complete isomorphism invariant, hereby extending Matui’s Isomorphism Theorem. As an application, we study graph groupoids and their topological full groups, and obtain sharper results for this class. The machinery developed in this process is used to prove an embedding theorem for ample groupoids, akin to Kirchberg’s Embedding Theorem for [Formula: see text]-algebras. Consequences for graph [Formula: see text]-algebras and Leavitt path algebras are also spelled out. In particular, we improve on a recent embedding theorem of Brownlowe and Sørensen for Leavitt path algebras.

scholarly journals What is typical?

2011 ◽  
Vol 48 (A) ◽  
pp. 379-389 ◽  
Author(s):  
Günter Last ◽  
Hermann Thorisson
Keyword(s):  
Topological Group ◽  
Random Element ◽  
Random Measure ◽  
Measurable Space ◽  
Locally Compact ◽  
Recent Developments ◽  
Compact Case ◽  
Definition Of

Let ξ be a random measure on a locally compact second countable topological group, and letXbe a random element in a measurable space on which the group acts. In the compact case we give a natural definition of the concept that the origin is a typical location forXin the mass of ξ, and prove that when this holds, the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.

Graph algebras and the Gelfand–Kirillov dimension

2018 ◽  
Vol 17 (05) ◽  
pp. 1850095 ◽  
Author(s):  
José M. Moreno-Fernández ◽  
Mercedes Siles Molina
Keyword(s):  
Graph Algebras ◽  
Full Generality ◽  
Path Algebras ◽  
Direct Limits ◽  
Leavitt Path Algebras ◽  
Gk Dimension ◽  
Kirillov Dimension

We study some properties of the Gelfand–Kirillov dimension in a non-necessarily unital context, in particular, its Morita invariance when the algebras have local units, and its commutativity with direct limits. We then give some applications in the context of graph algebras, which embraces, among some others, path algebras and Cohn and Leavitt path algebras. In particular, we determine the GK-dimension of these algebras in full generality, so extending the main result in A. Alahmadi, H. Alsulami, S. K. Jain and E. Zelmanov, Leavitt Path algebras of finite Gelfand–Kirillov dimension, J. Algebra Appl. 11(6) (2012) 1250225–1250231.

scholarly journals GCR and CCR Steinberg Algebras

2019 ◽  
Vol 72 (6) ◽  
pp. 1581-1606 ◽  
Author(s):  
Lisa O. Clark ◽  
Benjamin Steinberg ◽  
Daniel W. van Wyk
Keyword(s):  
Representation Theory ◽  
Locally Compact ◽  
Path Algebras ◽  
Leavitt Path Algebras

AbstractKaplansky introduced the notions of CCR and GCR $C^{\ast }$-algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^{\ast }$-algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.

scholarly journals MODULES OVER ÉTALE GROUPOID ALGEBRAS AS SHEAVES

2014 ◽  
Vol 97 (3) ◽  
pp. 418-429 ◽  
Author(s):  
BENJAMIN STEINBERG
Keyword(s):  
Inverse Semigroup ◽  
Group Algebras ◽  
Semigroup Algebras ◽  
Locally Compact ◽  
Morita Equivalent ◽  
Path Algebras ◽  
Unit Space ◽  
Leavitt Path Algebras ◽  
Étale Groupoid ◽  
Totally Disconnected

AbstractThe author has previously associated to each commutative ring with unit $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Bbbk $ and étale groupoid $\mathscr{G}$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk $-algebra $\Bbbk \, \mathscr{G}$. The algebra $\Bbbk \, \mathscr{G}$ need not be unital, but it always has local units. The class of groupoid algebras includes group algebras, inverse semigroup algebras and Leavitt path algebras. In this paper we show that the category of unitary$\Bbbk \, \mathscr{G}$-modules is equivalent to the category of sheaves of $\Bbbk $-modules over $\mathscr{G}$. As a consequence, we obtain a new proof of a recent result that Morita equivalent groupoids have Morita equivalent algebras.

scholarly journals A class of corners of a Leavitt path algebra

2019 ◽  
Vol 2 (4) ◽  
pp. 75-81
Author(s):  
Deo Thanh Trinh
Keyword(s):  
Directed Graph ◽  
Finite Subset ◽  
Path Algebra ◽  
Graph Algebras ◽  
Leavitt Path Algebra ◽  
C Algebras ◽  
Path Algebras ◽  
Leavitt Path Algebras

Let E be a directed graph, K a field and LK(E) the Leavitt path algebra of E over K. The goal of this paper is to describe the structure of a class of corners of Leavitt path algebras LK(E). The motivation of this work comes from the paper “Corners of Graph Algebras” of Tyrone Crisp in which such corners of graph C*-algebras were investigated completely. Using the same ideas of Tyrone Crisp, we will show that for any finite subset X of vertices in a directed graph E such that the hereditary subset HE(X) generated by X is finite, the corner ( ) ( )( )     K v X v X v L E v is isomorphic to the Leavitt path algebra LK(EX) of some graph EX. We also provide a way how to construct this graph EX.

scholarly journals What is typical?

2011 ◽  
Vol 48 (A) ◽  
pp. 379-389 ◽  
Author(s):  
Günter Last ◽  
Hermann Thorisson
Keyword(s):  
Topological Group ◽  
Random Element ◽  
Random Measure ◽  
Measurable Space ◽  
Locally Compact ◽  
Recent Developments ◽  
Compact Case ◽  
Definition Of

Let ξ be a random measure on a locally compact second countable topological group, and let X be a random element in a measurable space on which the group acts. In the compact case we give a natural definition of the concept that the origin is a typical location for X in the mass of ξ, and prove that when this holds, the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.

scholarly journals On the representations of Leavitt path algebras

2011 ◽  
Vol 333 (1) ◽  
pp. 258-272 ◽  
Author(s):  
Daniel Gonçalves ◽  
Danilo Royer
Keyword(s):  
Path Algebras ◽  
Leavitt Path Algebras

scholarly journals FP-injective semirings, semigroup rings and Leavitt path algebras

2016 ◽  
Vol 45 (5) ◽  
pp. 1893-1906 ◽  
Author(s):  
Marianne Johnson ◽  
Tran Giang Nam
Keyword(s):  
Semigroup Rings ◽  
Path Algebras ◽  
Leavitt Path Algebras
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