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

By H. Dzinotyiweyi

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Extra info for Analogue of the Group Algebra for Topological Semigroups (Chapman & Hall/CRC Research Notes in Mathematics Series)

Example text

Cs). Proof. (S) be fixed. Let C be any compact subset of S and e (x) be a net converging to x ins. l(ii), the mapping a s + luI (s -lC) of S into 1R is continuous i f 27 lim I JJ I (x:1c) ~ I JJ I (x-lC) • a Now let £ > 0 be given and choose an open neighbourhood W of C such that IJJI(s-1 (W\C)) <£for all sinS. l(i) £. (s). (s) and D, F be any compact subsets of S. 4. (s). € The preceding Theorem suggests that before studying the space Me(S) in detail it is essential to know whether Me(S) does not always coincide with Ma(S).

For many more examples of stips we refer the interested reader to Sleijpen [92]. The following exercises can easily be solved now by routine arguments. ] (ii) If S and T are stips then S x T is a stip. (iii) If S is a stip and T is a locally compact quotient semigroup of S with identity element, then T is a stip. 50 (iv) Let G1 := {x E S : there exists y E S with xy an algebraic subgroup of Sand G1 = s- 11 n ls-1 • If 1 E s 1 , then G1 is an open subgroup of s. If 1 E s 1 and Sis connected, then Sis a group.

Hence v*x ~ v*y and we can find hE M(S)A with h(v*x) I h(v*y). In particular h(v) I 0 and . A h(v*s> def1n1ng X E s by x(s) := h(v) we get x(x) I x(y) and item (iii) follows. Finally we assume item (iii) and prove item (i). Let p be any non-zero measure in Ma(S) and H(S) be the linear span of S in C(S). Then H(S) separates points of S and contains the constant semicharacter 1. 4, we have that H(SA) is (norm) dense in L1 (S,Ipl). Since f fdp defines a non-zero functional on L1 (S,Ipl), there exists g E H(SA) such that +I I gdp ~ o.

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