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International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 1, Pages 11-20
http://dx.doi.org/10.1155/S0161171282000027

The convolution-induced topology on L∞(G) and linearly dependent translates in L1(G)

G. Crombez and W. Govaerts

Seminar of Higher Analysis, State University of Ghent, Galglaan 2, GENT B-9000, Belgium

Received 8 October 1980; Revised 7 May 1981

Copyright © 1982 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given a locally compact Hausdorff group G, we consider on L∞(G) the τc-topology, i.e. the weak topology under all convolution operators induced by functions in L1(G). As a major result we characterize the trigonometric polynomials on a compact group as those functions in L1(G) whose left translates are contained in a finite-dimensional set. From this, we deduce that τc is different from the w∗-topology on L∞(G) whenever G is infinite. As another result, we show that τc coincides with the norm-topology if and only if G is discrete. The properties of τc are then studied further and we pay attention to the τc-almost periodic elements of L∞(G).