A Bitopological Representation of Pro-C^*-Algebras
by
John Mack
University of Kentucky

In [M], the following is proved.

Theorem A. Any C^*-algebra A with unit can be represented as the algebra of pairwise continuous sections s:X --> E, where X is the bitopological space of proper Glimmal ideals of A and E is a bitopological field of C^*-algebras. When A is commutative, this representation becomes the familiar Gelfand representation of A as the algebra of complex-valued continuous functions on X.

This representation can be extended to include a class of topological algebras called b^*-algebras.

Definition. A topological algebra is a Pro-C^*-algebra if it is iseomorphic to an inverse limit of C^*-algebras. An algebra A is a b^*-algebra if it is a norm-closed, advertibly complete *-subalgebra of a Pro-C^*-algebra.

Note that for any b^*-algebra A, the subalgebra A^b of norm bounded elements of A is a C^*-algebra.

Theorem B. For a b^*-algebra A with unit, let X and E be the bitopological spaces used in Theorem A to represent A^b. Then there exists a dense subspace T of X so that A is iseomorphic with an algebra of pairwise continuous sections s : T --> E. When A is commutative, this representation becomes an iseomorphism between A and the algebra of all extended complex-valued continuous functions on X that are scalar valued on T.

[M] J. Mack, A Bitopological Gelfand Theorem for C^*-Algebras, Top. Proc. 22 (1997), 285-304.

Date received: June 4, 1999