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Normed Banach Bimodule

Suppose that A and B are two normed (Banach) algebras. A vector space X is called an A-B-bimodule whenever it is simultaneously a normed (Banach) left A-module, a normed (Banach) right B-module, and a(mb)=(am)b. If A=B, then M is simply said 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 dual A^* can be considered as a Banach A-bimodule with the actions

(af)(b)=f(ba),(fa)(b)=f(ab):f in A^*,a,b in A.

See also

Normed Banach Module

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. "The Homology in Algebra of Analysis." In Handbook of Algebra, Vol. 2. Amsterdam, Netherlands: Elsevier, 1997.

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Normed Banach Bimodule

Cite this as:

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

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