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

@article{Brownlowe2016L\_2mathbbZL,
title={\$L\_\{2,\mathbb\{Z\}\} \otimes L\_\{2,\mathbb\{Z\}\}\$ does not embed in \$L\_\{2,\mathbb\{Z\}\}\$},
author={Nathan Brownlowe and Adam P. W. S{\o}rensen},
journal={Journal of Algebra},
year={2016}
}
• Published 2016
• Mathematics
• Journal of Algebra
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 $L_{2,R}$.
6 Citations
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We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and $L_{2-,\mathbb{Z}}$ are not isomorphic as $*$-algebras. There are two key ingredients in the proof. One is a partial algebraicExpand
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