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Normed Banach Bimodule

A normed (Banach) A-B-bimodule over normed (Banach) algebras A and B over the same field R or C is a normed space (Banach space, respectively) X that is simultaneously a normed (Banach) left A-module and normed (Banach) right B-module, with a(xb)=(ax)b for all a in A, b in B, and x in X. If A=B, then X is simply called a normed (Banach) A-bimodule. A normed (Banach) A-bimodule is called symmetric or commutative if ax=xa for all a in A, x in X.

For example, if A is a Banach algebra, then its continuous dual A^* can be considered as a Banach A-bimodule with the actions

(af)(b)=f(ba),(fa)(b)=f(ab)

for f in A^* and a,b in A.

See also

Bounded Hochschild Cohomology, Derivation, Normed Banach Module

Portions of this entry contributed by Mohammad Sal Moslehian

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References

Helemskii, A. Ya. The Homology of Banach and Topological Algebras. Dordrecht, Netherlands: Kluwer, 1989.Helemskii, A. Ya. "Homology for the Algebras of Analysis." In Handbook of Algebra, Vol. 2 (Ed. M. Hazewinkel). Amsterdam, Netherlands: Elsevier, pp. 151-274, 2000.

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Normed Banach Bimodule

Cite this as:

Weisstein, Eric W., with contributions by Mohammad Sal Moslehian. "Normed Banach Bimodule." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NormedBanachBimodule.html

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