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Banach algebras, decomposable convolution operators, and a spectral mapping property. (English) Zbl 0818.46059
Function spaces, Proc. Conf., Edwardsville/IL (USA) 1990, Lect. Notes Pure Appl. Math 136, 307-323 (1992).
The paper begins with a discussion of four motivating problems concerning spectral properties of convolution operators given by measures on locally compact Abelian groups. An operator \(T\) on a complex Banach space \(X\) is called decomposable if, for every open cover \(\{U, V\}\) of the complex plane, there exist \(T\)-invariant closed subspaces \(Y\), \(Z\) of \(X\) such that \(\sigma(T| Y)\subset U\), \(\sigma(T| Z)\subset V\) and \(Y+ Z= X\); it is called superdecomposable if there exists an operator \(R\) commuting with \(T\) such that \(\sigma(T| \overline{R(X)})\subset U\), \(\sigma(T|\overline{(I-R)(X)})\subset V\). Theorems characterizing these properties are proved. The basic idea is to investigate a certain algebra of continuous functions on a compact space, not necessarily Hausdorff. Those elements of a commutative semi-simple Banach algebra are considered, whose Gelfand transform is continuous with respect to the hull-kernel topology on the maximal ideal space. This continuity property characterizes decomposability of the corresponding multiplication operator. There are interesting consequences for representations and applications to regular and non-regular Banach algebras. Finally applications to the measure algebra, and to certain natural subalgebras of it, of a locally compact Abelian group, are given.
For the entire collection see [Zbl 0746.00071].

46J05 General theory of commutative topological algebras
46H05 General theory of topological algebras
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
43A10 Measure algebras on groups, semigroups, etc.
46H40 Automatic continuity