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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].

##### MSC:
 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