Let $A$ and $B$ be commutative Banach algebra. I have proven that if $A$ and $B$ have identity $e_A$ and $e_B$ respectivly , then $e_A\hat\otimes e_B$ is identity for $A\hat\otimes B$ (the projective tensor product of $A$ with $B$)

I want to find a proof for the converse

Any help will be appreciated.