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Ideal spaces of Banach algebras. (English) Zbl 1027.46058
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)].

##### 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
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