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

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