  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 $L_{2,R}$.
