- Associate Professor Emeritus of Mathematics
Joseph W. Kitchen
If A is a commutative C* algebra with identity, then a well-known theorem by Gelfand states that A is isomorphic to the algebra C(S) of all continuous complex-valued functions on a compact Hausdorff space S. If A is a non-commutative C* algebra with identity, then the Dauns-Hofmann theorem states that A is also isomorphic to the space of all continuous functions on a space S, except that the functions are no longer complex-valued; instead, they are Banach-space-valued, and to complicate matters still more, the Banach spaces can vary from point to point of S. In other words, A is isomorphic to the space of all sections of a sheaf-like object E --> S in which the stalk above each point s of the base space S is a Banach space E. These sheaf-like objects are called . Since the early 80's Dr. Kitchen and a former Ph.D. student of his, Dr. David A. Robbins, who is now a professor of mathematics at Trinity College, Hartford, have been studying Banach bundles and related topics. The results of their collaboration are the six papers listed below. They began by studying the relationships between Banach bundles and Banach modules. (For starters, if E --> S is a Banach bundle, then the section space is a Banach module over the algebra C(S).) In this first paper,  below, they generalize the Gelfand representation theory for commutative Banach algebras to Banach modules over such algebras. Many of the results of this paper are generalized and refined in . These papers contain a number of constructions for Banach bundles and laid the foundation for later work. In  the construction of tensor products of Banach bundles (projective and inductive) is undertaken, and these tensor products are related to tensor products of Banach modules and their representations. Paper  also contains extensions and refinements of these results. Tensor products also figure significantly in . The papers which followed ,, and  pursue more specialized topics. Paper , for instance concerns duality for Banach bundles. If E --> S is a bundle of Banach spaces is there a bundle E* --> S which can appropriately be called its dual, and how do its sections relate to the sections of the given bundle? The answers to these questions involve not only Banach bundles, but, more generally, bundles of locally convex topological vector spaces. These more general bundles also appear in , the most recent paper.