×

Representations of non-commutative Banach algebras by continuous functions. (English) Zbl 0766.46036

The method of localization of a (non-commutative) unital Banach algebra \({\mathcal A}\) over a central subalgebra \({\mathcal B}\) leads to a representation of \({\mathcal A}\) as a space of bounded vector-valued functions. More precisely, let \({\mathcal M}({\mathcal B})\) be the maximal ideal space of \({\mathcal B}\) and define \(I_ x\) for \(x\in{\mathcal M}({\mathcal B})\) to be the smallest closed two-sided ideal in \({\mathcal A}\) containing \(x\). Let \(\Phi_ x:{\mathcal A}\to{\mathcal A}/I_ x\) denote the canonical quotient map. Then a representation \(\Phi:{\mathcal A}\to V({\mathcal M}({\mathcal B})\), \(({\mathcal A}(I_ x))_ x)\) of \({\mathcal A}\) as a space of bounded vector-valued functions on \({\mathcal M}(B)\) can be defined by \(\Phi(a)(x)=\Phi_ x(a)\). If \(B\) is sufficiently rich, then the map \(\Phi\) is even an isomorphism, but the question consists in the complete characterization of \(\Phi({\mathcal A})\). It is the main object of the paper, to characterize \(\Phi({\mathcal A})\) as space of all continuous functions on \({\mathcal M}(B)\) using suitable continuity concepts.
Three different methods are introduced by the authors. The first one characterizes \(\Phi({\mathcal A})\) using a certain concept of upper semi- continuity, the second method is based on continuous cross sections on vector bundles and the third one uses pointwise limits. The methods are applied to study the Calkin algebra of algebras of singular integral operators on composed curves and of algebras of multiplication and convolution operators as well as algebras of approximation methods.
Reviewer: H.Junek (Potsdam)

MSC:

46H15 Representations of topological algebras
46H30 Functional calculus in topological algebras
46H35 Topological algebras of operators
47A60 Functional calculus for linear operators
PDFBibTeX XMLCite