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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?

Thank you.

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  • $\begingroup$ I don't think there is a name for this kind of map. "What are some properties" is not a good question. $\endgroup$
    – Nik Weaver
    Commented Aug 23, 2019 at 13:46
  • $\begingroup$ @Nik Weaver Thank you. By "some properties" I mean how they bahave on specific algebras like semisimple Banach algerbas or others. $\endgroup$
    – Fermat
    Commented Aug 24, 2019 at 5:44

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