Title: Leavitt Path Algebras having Bases Consisting Solely of Strongly Regular Elements
Speaker: Dan Bossaller (Ohio University)
Abstract: Leavitt path algebras were introduced independently by Abrams and Aranda-Pino in 2005 and Ara, Moreno, and Pardo in 2007 as purely algebraic analogues of the graph $C^*$ algebras introduced by Kumjian, Pask, Raeburn, and Renault in 1997 and 1998. Various recent papers deal with the family of so called ``invertible algebras," those algebras over arbitrary (not necessarily commutative) unital rings which have bases that consist solely of invertible elements. Many familiar algebras satisfy this property, including all finite dimension algebras over fields other than $\mathbb{F}_2$ and all $n \times n$ matrix algebras over unital rings. López-Permouth and Pilewski gave a complete characterization of precisely which Leavitt path algebras of finite graphs are invertible. In this talk I will introduce the concept of a locally invertible algebra, that is, an algebra $A$ having basis $\mathcal{B}$ such that for every $b \in \mathcal{B}$, there exists some idempotent $e$ for which $b$ is invertible in the corner algebra $eAe$. We will show that this definition is equivalent to the algebra having a basis consisting solely of strongly regular elements, and then we will give various examples and non-examples of locally invertible algebras. Afterwards we will examine local invertibility in the context of Leavitt path algebras, in particular giving a complete characterization of strongly regular monomials and, as a corollary, showing that all von Neumann regular and all directly finite Leavitt path algebras are locally invertible. If time permits, we will show that the algebraic analogues of many algebras from operator theory are locally invertible.
