The subject of this mathematical research project is the theory of Banach algebras, which may be thought of as linear algebra in infinitely many dimensions. One can do arithmetic (add and multiply) in this setting, and one can also take limits. A fundamental theme of research in this area is to understand how the arithmetic and the limit-taking influence one another. For instance, does a map between Banach algebras satisfying prescribed arithmetic identities automatically preserve limits? The principal investigators will work on questions of this nature. More specifically, they will continue their investi- gation of the continuity and algebraic structure of derivations from Banach algebras to modules. Necessary and sufficient conditions on the algebra and/or the module will be sought which will insure that each derivation is continuous, each continuous derivation into a dual bimodule is inner and each continuous derivation into the dual space of the algebra is inner. Particular atention will be paid to Banach algebras of integrable functions and other Banach algebras arising naturally in analysis and applications.
