Leavitt $R$-algebras over countable graphs embed into $L_{2,R}$

TL;DR

This paper proves that Leavitt path algebras over a commutative ring R embed into a specific algebra L_{2,R} if and only if the underlying graph is countable, and establishes a generalized Cuntz-Krieger Uniqueness Theorem.

Contribution

It characterizes when Leavitt path algebras embed into L_{2,R} and generalizes the Cuntz-Krieger Uniqueness Theorem for these algebras.

Findings

Leavitt path algebra embeds into L_{2,R} iff the graph is countable.

A generalized Cuntz-Krieger Uniqueness Theorem is established.

Provides a criterion for embedding based on graph countability.

Abstract

For a commutative ring $R$ with unit we show that the Leavitt path algebra $L_R(E)$ of a graph $E$ embeds into $L_{2,R}$ precisely when $E$ is countable. Before proving this result we prove a generalised Cuntz-Krieger Uniqueness Theorem for Leavitt path algebras over $R$.