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On some properties of the Banach algebras \(A_ p(G)\) for locally compact groups. (English) Zbl 0599.43005

Let G be a locally compact group. For any function h on G define \(h^ v\) by \(h^ v(x)=h(x^{-1})\). For \(1\leq p\leq \infty\) let \(A_ p(G)=\{\sum^{\infty}_{1}f_ n*g^ v_ n |\) \(f_ n\in L^ q\), \(g_ n\in L^ p\), \(\sum^{\infty}_{1}\| f_ n\|_ q \| g_ n\|_ p<\infty \}\) where q is the conjugate index of p. Under pointwise operations \(A_ p(G)\) is an algebra (in the case \(p=2\) called the Fourier algebra of G). The author shows that a theorem of S. Foguel and a theorem of G. Choquet and J. Deny concerning measures on abelian groups are very special cases of theorems on multipliers of \(A_ p(G)\). He also extends two results of H. P. Rosenthal on ideals of the Fourier algebra of an abelian group and proves a theorem on existence of projections related to a theorem of J. E. Gilbert.
Reviewer: M.Leinert

MSC:

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
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