Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.

A lifting of $T$ is a bounded linear map $L:A\to A \otimes_{\gamma} A$ such that $TL=Id_{A}$.

Do you know a charaterization or examples of algebras which admits a such lifting?

Any reference on this problem or explicit examples is welcome.

UPDATE: I added non unital.