Vol. 55, No. 2, 1974

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On some group algebra modules related to Wiener’s algebra M1

Teng-Sun Liu, Arnoud C. M. van Rooij and Ju-Kwei Wang

Vol. 55 (1974), No. 2, 507–520

Along with his study of the general Tauberian theorem in L1, N. Wiener introduced the algebra M1 which consists of all those continuous functions f on the real line R for which

xm∈a[nx,n+1]|f(x)| < ∞.
n= −∞

He proved that many features of L1, including the general Tauberian theorem, are shared by M1. In this paper to generalize M1 to an arbitrary locally compact group G. While doing this, a host of L1(G)-modules mutually related by conjugation and the operation of forming multiplier modules. 1(G) is among them. In case G is abelian, 1(G) is a Segal algebra, so that it has the same ideal-theoretical structure as L1(G). If further G = R, 1(G) reduces to the Wiener algebra M1 with an equivalent norm.

Mathematical Subject Classification 2000
Primary: 43A15
Received: 29 November 1971
Revised: 31 January 1974
Published: 1 December 1974
Teng-Sun Liu
Arnoud C. M. van Rooij
Ju-Kwei Wang