A note on the centralizer of a subalgebra of the Steinberg algebra

2021 ◽  
Vol 33 (1) ◽  
pp. 179-184
Author(s):  
R. Hazrat ◽  
Huanhuan Li
Keyword(s):  
Commutative Ring ◽  
Invariant Subset ◽  
Content Type ◽  
Path Algebras ◽  
Unit Space ◽  
Leavitt Path Algebras

For an ample Hausdorff groupoid G \mathcal {G} , and the Steinberg algebra A R ( G ) A_R(\mathcal {G}) with coefficients in the commutative ring R R with unit, the centralizer is described for the subalgebra A R ( U ) A_R(U) with U U an open closed invariant subset of the unit space of  G \mathcal {G} . In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify several results in the literature, and the corresponding results for Leavitt path algebras follow.

scholarly journals Purely infinite simple Kumjian–Pask algebras

2018 ◽  
Vol 30 (1) ◽  
pp. 253-268 ◽  
Author(s):  
Hossein Larki
Keyword(s):  
Commutative Ring ◽  
Path Algebras ◽  
Higher Rank ◽  
Leavitt Path Algebras ◽  
Unital Simple

Abstract Given any finitely aligned higher-rank graph Λ and any unital commutative ring R, the Kumjian–Pask algebra {\mathrm{KP}_{R}(\Lambda)} is known as the higher-rank generalization of Leavitt path algebras. After the characterization of simple Kumjian–Pask algebras by Clark and Pangalela among others, in this article we focus on the purely infinite simple ones. Briefly, we show that if {\mathrm{KP}_{R}(\Lambda)} is simple and every vertex of Λ is reached from a generalized cycle with an entrance, then {\mathrm{KP}_{R}(\Lambda)} is purely infinite. We also prove a dichotomy for simple Kumjian–Pask algebras: If each vertex of Λ is reached only from finitely many vertices and {\mathrm{KP}_{R}(\Lambda)} is simple, then {\mathrm{KP}_{R}(\Lambda)} is either purely infinite or locally matritial. This result covers all unital simple Kumjian–Pask algebras.

scholarly journals CHAIN CONDITIONS ON ÉTALE GROUPOID ALGEBRAS WITH APPLICATIONS TO LEAVITT PATH ALGEBRAS AND INVERSE SEMIGROUP ALGEBRAS

2018 ◽  
Vol 104 (3) ◽  
pp. 403-411 ◽  
Author(s):  
BENJAMIN STEINBERG
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebras ◽  
Chain Conditions ◽  
Path Algebras ◽  
Finite Dimensional ◽  
Unit Space ◽  
Leavitt Path Algebras ◽  
Étale Groupoid ◽  
Totally Disconnected

The author has previously associated to each commutative ring with unit$R$and étale groupoid$\mathscr{G}$with locally compact, Hausdorff and totally disconnected unit space an$R$-algebra$R\,\mathscr{G}$. In this paper we characterize when$R\,\mathscr{G}$is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup 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 Reconstruction of graded groupoids from graded Steinberg algebras

2017 ◽  
Vol 29 (5) ◽  
pp. 1023-1037 ◽  
Author(s):  
Pere Ara ◽  
Joan Bosa ◽  
Roozbeh Hazrat ◽  
Aidan Sims
Keyword(s):  
Integral Domain ◽  
Ring Structure ◽  
Ring Isomorphism ◽  
Path Algebras ◽  
Unit Space ◽  
Leavitt Path Algebras

AbstractWe show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies {C^{*}}-isomorphism of {C^{*}}-algebras for graphs E and F in which every cycle has an exit.

Leavitt path algebras over arbitrary unital rings and algebras

2019 ◽  
Vol 19 (06) ◽  
pp. 2050107
Author(s):  
Hans Nordstrom ◽  
Jennifer A. Firkins Nordstrom
Keyword(s):  
Commutative Ring ◽  
General Setting ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Unital Rings ◽  
Rings And Algebras

We expand the work of Tomforde by further extending the construction of Leavitt path algebras (LPAs) over arbitrary associative, unital rings. We show that many of the results over a commutative ring hold in the more general setting, provide some useful generalizations of prior results, and give a definition for an iterated Leavitt path extension in our context.

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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