Ideal spaces of Banach algebras. (English) Zbl 1027.46058

Somerset, D. W. B.

Proc. Lond. Math. Soc., III. Ser. 78, No. 2, 369-400 (1999).

This paper is concerned with the problem of representing a Banach algebra \(A\) as a subdirect product of more tractable quotient algebras, using the topology of \(A\) to topologize the bundle in some useful way. The author’s main interest is to determine the appropriate topologies to be used on the base space. He considers a number of topologies on the lattice \(\text{Id}(A)\) of all closed two-sided ideals of \(A\) and studies their basic properties. The topologies have their background in the theory of bitopological spaces. To a considerable extent the paper continues the work of F. Beckhoff in [Stud. Math. 115, 189-205 (1995; Zbl 0836.46038) and ibid. 118, 63-75 (1996; Zbl 0854.46045)].

Reviewer: A.R.Villena (Granada)

MSC:

46H10	Ideals and subalgebras
46J20	Ideals, maximal ideals, boundaries
46L05	General theory of \(C^*\)-algebras
46M20	Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)

Keywords:

bundle representations; \(C^*\)-bundles; Banach bundles

Citations:

Zbl 0836.46038; Zbl 0854.46045