Groupoids and Steinberg Algebras

A groupoid is a generalisation of group in which composition is only partially defined. In first half of this talk, I will give an overview of groupoid theory and show how groupoids provide a unifying model for a number of seemingly unrelated mathematical structures. In the second half of the talk, I will give an overview of the theory of Steinberg algebras. A Steinberg algebra is constructed from an `ample' topological groupoid. Once again, these algebras can be used to model a number of seemingly unrelated algebraic constructions.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

The lattice of ideals in the Steinberg algebra of a strongly effective groupoid

A topological groupoid is generalization of topological group where the binary operation is only partially defined. Ample groupoids are a special class of topological groupoid that have an especially well behaved topology. To each ample groupoid G and commutative ring R (with 1), we consider the Steinberg R-algebra of G, which has become an important object of study in both ring theory and functional analysis. In this talk, I will present results about the ideal structure of a certain class of Steinberg algebras. In particular, we will take a close look at the lattice of ideals and an innovative approach to studying the join and meet operations.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Groupoids in analysis and algebra.

Groupoid algebras are studied in both analytic and algebraic contexts. In analysis, groupoid C*-algebras play a fundamental role in the theory and include many important subclasses. Steinberg algebras are their purely algebraic analogue. They were introduced in 2010 (by B. Steinberg) and have proven themselves to be useful in unexpected ways. In this talk, I demonstrate how groupoids and their associated algebras serve as a powerful tool for bridging the gap between abstract algebra and analysis. We will first explore this interplay in the context of graph algebras, and then broaden our scope to more general classes of groupoid algebras.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Strongly graded groupoids and strongly graded Steinberg algebras

We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $\kappa: \mathcal G \to \Gamma$, to a discrete group $\Gamma$, such that $\kappa^{-1}(\gamma)\kappa^{-1}(\delta) = \kappa^{-1}(\gamma \delta)$, for all $\gamma, \delta \in \Gamma$. We introduce the category of graded $\mathcal G$-sheaves, and prove an analogue of Dade's Theorem: $\mathcal G$ is strongly graded if and only if every graded $\mathcal G$-sheaf is induced by a $\mathcal G_{\epsilon}$-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. Applying this result, we obtain a complete graphical characterisation of strongly graded Leavitt path and Kumjian-Pask algebras