$L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$

TL;DR

This paper proves that the tensor product of two Leavitt algebras over integers cannot be embedded into a single Leavitt algebra, using Thompson's group V to establish non-embedding results.

Contribution

It demonstrates the non-embedding of tensor products of Leavitt algebras into a single algebra, extending understanding of their algebraic structure and embeddings.

Findings

Tensor product $L_{2,bZ} ensor L_{2,bZ}$ does not embed into $L_{2,bZ}$.

Partial non-embedding results for $L_{2,R} ensor L_{2,R}$.

Use of Thompson's group V to analyze unitary groups in Leavitt algebras.

Abstract

For a commutative ring $R$ with unit we investigate the embedding of tensor product algebras into the Leavitt algebra $L_{2,R}$. We show that the tensor product $L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$ (as a unital $*$-algebra). We also prove a partial non-embedding result for the more general $L_{2,R} \otimes L_{2,R}$. Our techniques rely on realising Thompson's group $V$ as a subgroup of the unitary group of $L_{2,R}$.