Wiener pairs of measure algebras (Q748813)

Let M(G) denote the Borel regular finite measures on a locally compact group G. The present paper is concerned with whether the spectral radius norm of a subalgebra A of M(G) is equal to the norm of the Fourier transform of A. For compact non-abelian groups G this question was examined in the papers of \textit{J. B. Fountain}, \textit{R. W. Ramsay} and \textit{J. H. Williamson} [Proc. R. Ir. Acad., Sect. A 76, 235-251 (1976; Zbl 0309.43010)] and \textit{C. Karanikas} and \textit{J. H. Williamson} [Math. Proc. Camb. Philos. Soc. 95, 109-122 (1984; Zbl 0537.43003)]. It is shown that the Fourier norm of the direct sum of group algebras in M(G) coincides with the norm of the left regular representation of M(G) provided that G is an amenable group. In the last section of the paper equivalent conditions are given for the symmetry of certain subalgebras of M(G) involving the equality of the spectral radius norm with the Fourier norm.