Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I am interested in $\mathcal{B}(H)$ only as a Banach algebra (operator algebra).

Do there exist two **infinite-dimensional** Banach algebras $A, B$ such that $\mathcal{B}(H)$ is isomorphic as a Banach algebra to the **projective tensor product** $A\otimes_\gamma B$?

You may also replace the projective tensor product by any other Banach algebra tensor product which arises from a **reasonable crossnorm** (so the vNA tensor product is not good).