Vol. 15, No. 1, 1965

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On the invariant mean on topological semigroups and on topological groups

Edmond E. Granirer

Vol. 15 (1965), No. 1, 107–140

Let S be a topological semigroup and C(S) be the space of bounded continous functions on S. The space of translation invariant, bounded, linear functionals on C(S) and its connection with the structure of S, are investigated in this paper. For topological groups G, not necessarily locally compact, the space of bounded, linear, translation invariant functionals, on the space UC(G) of bounded uniformly continuous functions, is also investigated and its connection with the structure of G pointed out. The obtained results are applied to the study of the radical of the convolution algebra UC(G) (for locally compact groups, or for subgroups of locally convex linear topological spaces) and some results which seem to be unknown even when G is taken to be the real line are obtained.

Mathematical Subject Classification
Primary: 22.10
Received: 2 December 1963
Published: 1 March 1965
Edmond E. Granirer