$L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}}$ 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}
}
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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