Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph $C^*$-algebras

TL;DR

This paper demonstrates that corners of Leavitt path algebras and related structures can be represented as Steinberg algebras, establishing connections to graph $C^*$-algebras and groupoid $C^*$-algebras.

Contribution

It shows that all nonzero corners of Leavitt path algebras are isomorphic to Steinberg algebras, linking algebraic and topological structures in graph algebras.

Findings

Corners of Leavitt path algebras are Steinberg algebras.

Morita equivalent algebras to Leavitt path algebras are Steinberg algebras.

Corners of graph $C^*$-algebras are isomorphic to graph $C^*$-algebras of groupoids.

Abstract

We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra $L_K(E)$ of an arbitrary graph $E$ with coefficients in a field $K$ is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every $K$-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a $C^*$-algebra of a countable graph is isomorphic to the $C^*$-algebra of an ample groupoid.