Wiener pairs of measure algebras

Abstract

This article concerns the relationship between the spectral and the norm properties of certain subalgebras of the convolution measure algebra M(G) (Borel regular finite measures) on a locally compact amenable group G. 1. Introduction. In this paper we deal with the same general problem as in [2] and [4] which is concerned with the relationship of the spectrums of a C*-enveloping algebra C*(A) of a subalgebra A of M(G). In different terminology this is the Wiener-Pitt phenomenon for A, i.e., whether the spectral radius of A is equal to the norm of the Fourier transform of A. Or in Y. Naimark's terminology whether A and C*(A) form a Wiener pair. In the above mentioned works it has been examined for the case of compact non-Abelian groups G. Also the earlier paper [7] deals with the case of Abelian groups. In the present work we examine this problem for amenable groups G for more general subalgebras of M(G) than those of [2] and [4]. Our methods here differ considerably from previous works. For example one can see that the proof of Theorem (2.6) doesn't make use of the properties of compact groups as Theorem (4.3) of [4]. In §2 we show 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 §3 we deal with the equality of these norms for more general direct sum of group algebras and we discuss the problem of symmetric subalgebras of M(G). In particular in Theorem (3.3) we give conditions for the symmetry of certain subalgebras of M(G). The conditions involve the equality of the spectral radius with the Fourier norm as well as the equality of spectrums. We begin in §2 with a brief presentation of the main definitions and symbols. For the general background, and for definitions and results not given explicit mention here, we refer to [1] and [3].