On the homogeneous zero components of Leavitt algebras

TL;DR

This paper investigates the structure of the zero component of Leavitt algebras, showing it as a direct limit of certain free products and establishing the IBN property, thus advancing understanding of their algebraic properties.

Contribution

It introduces a new description of the zero component of Leavitt algebras as a direct limit of free products of Bergman algebras and proves it has the IBN property.

Findings

Zero component is a direct limit of free products of Bergman algebras.

For m=1, zero component is a direct limit of matrix algebras.

Zero component has the IBN property.

Abstract

We prove that the zero component $L(m,n)_0$ of a Leavitt algebra $L(m,n)$ with respect to the canonical grading is a direct limit $\varinjlim_{z}L(m,n)_{0,z}$, where each algebra $L(m,n)_{0,z}$ is a free product of two Bergman algebras. For the special case $m=1,n>1$, one recovers the known result that the zero component $L(1,n)_0$ is a direct limit of matrix algebras. Moreover, we show that $L(m,n)_0$ has the IBN property.