The commutative core of a Leavitt path algebra

TL;DR

This paper characterizes the maximal commutative subalgebra within Leavitt path algebras over any unital commutative ring and extends the Cuntz-Krieger uniqueness theorem to these structures.

Contribution

It identifies the commutative core of Leavitt path algebras and generalizes the injectivity criteria for representations.

Findings

Identified the commutative core as a maximal commutative subalgebra.

Provided a characterization of injective representations.

Extended the Cuntz-Krieger uniqueness theorem.

Abstract

For any unital commutative ring $R$ and for any graph $E$, we identify the commutative core of the Leavitt path algebra of $E$ with coefficients in $R$, which is a maximal commutative subalgebra of the Leavitt path algebra. Furthermore, we are able to characterize injectivity of representations which gives a generalization of the Cuntz-Krieger uniqueness theorem.