Multipliers and modulus on Banach algebras related to locally compact groups
The modulus \(|T|\) of an operator \(T\) acting on a Banach algebra which is also a Banach lattice, in particular the moduli of multipliers on the bidual of the algebras \(L^1(G)\), \(L^\infty(G)\), \(L^1(G)^{**}\) of a locally compact group \(G\), are investigated. It turns out that the modulus of a multiplier on \(L^1(G)\) is a multiplier, but that in general this is not true for left multipliers on \(L^1(G)^{**}\). For compact left multipliers \(T\) on \(L^1(G)^{**}\) for a locally compact noncompact group \(G\) it is shown that \(\langle T(n),1\rangle =0\) \(\forall n\in L^1(G)^{**}\), thus generalizing a result previously established by the authors under additional assumptions on \(G\) [J. Funct. Anal. 132, 170-191 (1995; Zbl 0832.22007)]. A similar result holds for the Fourier algebra of a nondiscrete group. Finally the commutant of \(\text{Conv}(L^\infty(G))\) (operators commuting with convolution by an \(L^1\)-function) is shown to be the set of operators being the adjoint of convolution by a measure, whereas for a nondiscrete locally compact group, which is amenable as discrete, no such operator commutes with \(\text{Hom}L^\infty(G)\) (operators on \(L^\infty(G)\) commuting with left translation) if the corresponding measure is continuous. This extends a previous result of \textit{A. Zappa} [Colloq. Math. 39, 343-349 (1978; Zbl 0398.43005)].
scientific article; zbMATH DE number 1098593
scientific article