Joseph W. Kitchen
- Associate Professor Emeritus of Mathematics
If A is a commutative C* algebra with identity, then a
well-known theorem by Gelfand states that A is
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
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
mathematics at Trinity College, Hartford, have been
studying Banach bundles and related topics. The results
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
the algebra C(S).) In this first paper,  below, they
generalize the Gelfand representation theory for
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
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
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.