L2,Z⊗L2,ZL2,Z⊗L2,Z does not embed in L2,ZL2,Z

For a commutative ring R   with unit we investigate the embedding of tensor product algebras into the Leavitt algebra L2,RL2,R. We show that the tensor product L2,Z⊗L2,ZL2,Z⊗L2,Z does not embed in L2,ZL2,Z (as a unital ⁎-algebra). We also prove a partial non-embedding result for the more general L2,R⊗L2,RL2,R⊗L2,R. Our techniques rely on realising Thompson's group V   as a subgroup of the unitary group of L2,RL2,R.