A Groupoid Approach to Discrete Inverse Semigroup Algebras

By Ben Steinberg. 2009

216 citations recorded by the source corpus.

Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup $S$ that can be induced from associated groups as precisely those satisfying a certain "finiteness condition". This "finiteness condition" is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b0c45713e828867e80b801b0a2dd51a3c9730daa

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Citations in the corpus

ON INVERSE SEMIGROUP C ∗ -ALGEBRAS AND CROSSED PRODUCTS (2011, background): “Paterson’s universal groupoid G (S) is the groupoid of germs of the action β [4, 22, 26].”
Simplicity, primitivity and semiprimitivity of etale groupoid algebras with applications to inverse semigroup algebras (2014, background): “Algebras of ample groupoids were introduced in [24]; see also [23] for some additional results not included in [24] as well as [4–6].”
Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras (2016, methodology): “For an ample groupoid G, we write C(G) for the (full) C-algebra of G (see for example [24] or [23]) and, for a commutative ring R with 1, we write AR(G) for the Steinberg algebra of G over R (see [29] and [13]).”
The tight groupoid of an inverse semigroup (2014, methodology): “Similar characterizations of Hausdorffness for Paterson’s universal groupoid of an inverse semigroup have been obtained in [15,20,23,24].”
Recent developments around partial actions (2018, background): “hat every ideal in Aκ(G) may be obtained as the intersection of ideals induced from isotropy group algebras. Since the partial skew group ring Lc(X)⋊Gfrom [119] can be viewed as the Steinberg algebra [277] for the transformation groupoid associated to the partial action of Gon X, the algebraic versi”
Equivalent groupoids have Morita equivalent Steinberg algebras (2013, methodology): “These algebras were introduced in [22] as a model for discrete inverse semigroup algebras.”
Pseudogroups and their étale groupoids (2011, methodology): “This condition was introduced by Steinberg [31] who called an inverse semigroup satisfying the condition a weak semilattice.”
A Generalised uniqueness theorem and the graded ideal structure of Steinberg algebras (2016, background): “[19] E. Ortega, Simple Cuntz-Krieger Boolean algebras, Draft (2016).”
Uniqueness Theorems for Steinberg Algebras (2014, methodology): “Steinberg algebras were introduced in [8] in the context of discrete inverse semigroup algebras.”
A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY (2009, methodology): “More generally, Paterson [24] used ideas from functional analysis to construct topological groupoids from inverse semigroups and Renault [25] constructed inverse semigroups from topological groupoids using bisections.”
Graded Steinberg algebras and their representations (2017, methodology): “Quite recently Steinberg algebras were introduced in [Steinberg 2010; Clark et al. 2014] as an algebraisation of the groupoid C -algebras ﬁrst studied by Renault [1980].”