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Riesz multipliers on commutative semisimple Banach algebras. (English) Zbl 0682.46035
A linear continuous mapping T: $$A\to A$$, A a Banach algebra, is said to be a multiplier of A if $$(Tx)y=x(Ty)$$ holds for each $$x,y\in A.$$
A result of J. K. Wang [Pac. J. Math. 11, 1131-1149 (1961; Zbl 0127.333)] shows that, if A is a complex commutative semisimple Banach algebra, T a multiplier of A, then there exists a unique bounded continuous function $$\phi_ T$$ defined in the regular maximal ideal space $$\Delta$$ (A) of A, such that $(Tx){\hat{\;}}(m)=\phi_ T(m)x{\hat{\;}}(m),$ for all $$x\in A$$, $$m\in \Delta (A)$$ (x{\^ } denotes the Gelfand transform of x). By using the Wang function we investigate the spectral properties of a multiplier defined on a commutative semisimple Banach algebra A. In particular we describe the class of all Riesz operators which are multipliers of a commutative semisimple Banach algebra having discrete regular maximal ideal space.
In section 4 we assume that in A the set of all isometric multipliers of A onto A is a compact set in the strong operator topology, which separates points of $$\Delta$$ (A) (this situation occurs when $$A=L_ 1(G)$$, the group algebra for a compact abelian group). We characterize all the Riesz multipliers on such a class of Banach algebras, among the set $$M_ 0(A)$$ of all multipliers such that $$\phi_ T$$ vanishes at infinity of $$\Delta$$ (A).
Reviewer: P.Aiena

##### MSC:
 46J05 General theory of commutative topological algebras 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 46J20 Ideals, maximal ideals, boundaries
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