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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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