Let $A$ and $B$ be two Banach algebras such that $B$ is a Banach $A$-bimodue and $T:A\rightarrow B$ a linear map satisfying $T(aa')=aT(a')+T(a)a'+T(a)T(a')$ for all $a,a'\in A$.
If the algerba multiplication on $B$ is the trivial action, i.e., $bb'=0$ for all $b,b'\in B$, then $T$ is said to be a derivation. If the module actions are as $ab=ba=0$, then it is an algerba homomorphism.
Is there a special name for $T$ in general? And what are some properties of such mappings?
Thanks in advance.