# Joseph W. Kitchen

- Associate Professor Emeritus of Mathematics

**External address:**PO Box 90320, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2800

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, [1] 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 [3]. These papers contain a number of

constructions for Banach bundles and laid the

foundation for later

work. In [2] 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 [3] also

contains extensions and refinements of these results.

Tensor products also figure significantly in [7].

The papers which followed [1],[2], and [3] pursue more

specialized topics. Paper [4], 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 [12],

the most recent paper.