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Multipliers and convolution operators with natural spectrum on group algebras. (English) Zbl 0799.47011

Summary: These lectures are concerned with some spectral properties of convolution operators given by measures on Abelian locally compact topological groups. These properties are first presented in the more abstract framework of the theory of multipliers of commutative semisimple Banach algebras. In particular we concentrate our study on multipliers whose spectra, in a certain sense, are natural. First we consider the case of a commutative semisimple Banach algebra \(A\) with a discrete maximal ideal space \(\Delta (A)\). In this case a multiplier \(T\) has natural spectrum if and only if \(T\) is a Riesz operator, or equivalently if and only if the spectrum of \(T\) is a finite set or a sequence which converges to 0. This result applies to the case of \(A = L_ 1(G)\), where \(G\) is compact, and \(T\) is a convolution operator. It is an open problem for arbitrary Abelian locally compact group algebras to determine the measures for which the corresponding convolution operator has natural spectrum. This problem and some other related results of [K. B. Laursen and M. M. Neumann, Stud. Math. 101, No. 2 (1992)] will be discussed in the last section.

MSC:

47B38 Linear operators on function spaces (general)
46J05 General theory of commutative topological algebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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