# The Analogue of the Group Algebra for Topological Semigroups by H. Dzinotyiweyi By H. Dzinotyiweyi

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6 LEMMA. Proof. Let (xa ) be a net converging to x in S. 1, and so must have a weak cluster point n (say). 2 imply that Consequently (x*~)IK = n and (x*~)IK is the only weak cluster point of ((xa*~)IK). This completes our proof. 3. 7 THEOREM. = S, then If S is such that Fe (S) Me (S) is an L-ideal of M(S) • Proof. 5, it follows that Me(S) is an L-subalgebra. We now show that Me(S) is left translation invariant. Let ~ € Me(S) be fixed v be a positive measure in Me(S) with D :• supp(v) compact. Fix x € D and let (Va) be a net of open neighbourhoods contracting to x.

For our purpose in this chapter we are interested in the first item of the proposition which says that for many stips compact neighbourhoods can be realized as closures of products of compact subgroups and countable subsets. g. 6. ated identity element 1, F be a G6-subset with 1 £ F and A be a Felatively compact subset of s. 8 PROPOSITION. A = GB (i) ; .. Zised HaaP measupe on G we have that {f £ C(GA) : x*'ll'(f) = f(x) foP all X in GA} is a (noPm) sepaPable subset of (ii) Gx = xG foP all x £ C(GA).

C. W. G. Sleijpen (,  and ). In particular, for such semigroups S, the results of sections 1 and 2 can be found in ,  and . The extension to include topological semigroups that are not necessarily locally compact is due to Dzinotyiweyi . 43 The object Me(S) was first introduced by Dzinotyiweyi subsequently studied by Dzinotyiweyi and Sleijpen 3 are  and  and  . Section taken from from are largely inspired by the results of [32) and was The results of while those of Section 4 are taken  and .