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

By H. Dzinotyiweyi

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Now one easily notes that n (bWk)w;1 S bG k>n Gb c bG. and so Similarly Gb ::> bG and we have Gb = bG for all b € B. For any g € G and b € B, setting x := gb, we get g- 1 (Gx) • Ggb • Gb .. bG and so Cx • gbG = xG. e. for all A). Finally, we prove item (iii). Since B x :N is countable, by Urysohn's lemma we can find a sequence {gn} of continuous functions from AG into [o,l] such that, corresponding to each (b,m) € B x lN there is an n € lN for which Sn(b) .. J For each n fn(x) :• 2-n(x*v(Sn)) 56 for all x € AG.

If S is a-compact and Fa (S) • S then Ma(S) • u{L(S,m) : m € Ma(S)+ n ~(S)}. 4 REMARK. 2(i) does not hold if we replace Fe(S) by Fa(S) and Me(S) by Ma(S), even when S is also locally compact and commutative. 4. Now let p be a lebesgue measure with supp{p) = {(x,l) : x € [2,3]} and let H be the compact subset {(x, y) : x € [7 ,8] and y € [1, 2]}. Then IPI € Ma(S). Suppose (on the contrary) there is a measure min Ma{S) n ~q(S,H) with IPI << m. Then for some m € m taking y := 1 + ~ , we must have m(B ) = 0 where B := {(x,y) x € [7 ,8]}.

Continuous mapping of K into M(KHL). 2, 1 * K is a norm v ~. So noting that X*v*u is concentrated on KHL for all x e K, we are done. 4, we have the following result. 4 COROLLARY. a k-space. (s) and suppose that S is also Then (i) tv is ~eakly continuous; (ii) 1 is no~ continuous. (s) when S is a k-space. We exploit this observation for the rest of this section. (s) in the next Proposition. for the reader to supply. 5 PROPOSITION. (s) is a no~ closed ~ght ideal of M(S). (ii) Mn(S) is a closed subalgebra of M(S).

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