Means, homomorphisms, and compactifications of weighted semitopological semigroups
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Means, homomorphisms, and compactifications of weighted semitopological semigroups A A K H A D E M - M A B O U D I and M A P O U R A B D O L L A H Department of Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Islamic Republic of Iran MS received 9 November 1998; revised 16 March 1999
Abstract. We consider some almost periodic type function algebras on a weighted semitopological semigroup, and using the set of multiplicative means on each of these algebras, we define their corresponding weighted semigroup compactifications. This will constitute an effective tool for investigating the properties of the function algebras concerned. We also show that these compactifications do not retain all the nice properties of the ordinary semigroup compactifications unless we impose some restrictions on the weight functions. Keywords. Weighted semigroup; multiplicative mean; (weakly) almost periodic function; semigroup compactification.
1. Preliminaries During the past decade harmonic analysis on weighted semigroups has enjoyed considerable attention, and a good deal of results have been proved in this connection (see for example [1, 4, 5, 6]). In this article we continue our previous investigation [7], concerning function algebras on weighted semigroups, and try to use semigroup compactification techniques for characterizing these algebras. We shall also show that these techniques, although very effective for some weighted semigroups, do not carry over all the nice properties of the (nonweighted) semigroup case. Throughout this paper, S denotes a locally compact Hausdorff semitopological semigroup, unless otherwise stated. A mapping w : S ~ (0, ee) is called a weight function on S if w(st) 0, then w(x) > 1, for all x E S (see [5]). (ii) It is interesting to note that the measurable weight functions on locally compact groups are bounded away from zero and infinity, on compact sets (see Proposition 2.1 of [5]). Thus if S is a locally compact group and w is a measurable weight function on S, then we can drop the local boundedness of w and w -1. (iii) In [8] it is shown that if S is a topological semigroup with a Borel measurable weight function w such that f~ satisfies the so-called Grothendieck's double limit property, as defined in [2], and is continuous on S • S, then WAR(S, w) is a translation invariant C*subalgebra of C(S, w), and w is the identity element of W.A79(S, w). For s w) and .A79(S, w), we have the following two theorems, in which, we must note that, since a subset of C(S) is norm relatively compact if and only if it is totally bounded, a function f E C(S, w) is in .A79(S, w) if and only if for each e > 0 there exists a finite subset K of S such that min
{w w"l
}
,. w(t)w : t E K < e
forallsES.
Or equivalently
min
(w w"i
1
w(t)w : t E K < e for all s E S.
Theorem 1.3. Let w be a weight function on S, such that fl~-I is bounded for all s E S, and the map sf~-l is norm continuous, then l~lgC(S, w) is a translation invariant C*subalgebra of C(S, w) which contains w as the identity e
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