Compact Topological Semigroups Associated With Oids

A semigroup S is called an absolute coretract if for any continuous homomorphism f from a compact Hausdorff right topological semigroup T onto a compact Hausdorff right topological semigroup containing S algebraically there exists a homomorphism g \colon S→ T such that f\circ g=idS. The semigroup β\ben contains isomorphic copies of any countable absolute coretract. In this article we define a class C of semigroups of idempotents each of which is a decreasing chain of rectangular semigroups. It is proved that every semigroup from C is an absolute coretract and every finite semigroup of idempotents, which is an absolute coretract, belongs to C .

Introduction. Limit properties of the convolution sequence of a regular measure on a compact topological semigroup are examined in this paper. Similar questions, as they arise in the case of a compact group, were examined byKAWADA & ITO [4], Recently BELLMAN [1] and GRENANDER [3] considered special limit theorems for products of independent identically distributed random operators. Such problems are closely related to those in this paper. It should be noted that similar questions arise when considering the structure of stationary stochastic processes [7]. Various results on compact semigroups are used in characterizing the class of limit measures [5], [6], [8].

In this paper we present a number of sufficient conditions on a sequence of probability measuresµ n on a locally compact (second countable) Hausdorff topological semigroupS that guarantee the weak convergence of the sequence of convolution productsµ k,n ≡µ k + 1 *···*µ n (k<n), asn→∞, for allk≥0.

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $$*$$ -semigroup S when it is infinite non-discrete cancellative, $$M_a(S)^{**}$$ does not admit an involution, and $$M_a(S)^{**}$$ has a trivolution with range $$M_a(S)$$ if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only if G is finite. However, $$A(G)^{**}$$ always admits trivolution.

In the study of compact semigroups the constructive method rather than the representational method is usually the better plan of attack. As it was pointed out by Hofmann and Mostert in the introduction to their book [10] this method is more productive than searching for a representation theory. Hofmann and Mostert described a constructive method called the Hormos and showed that any irreducible compact semigroup is obtained by the Hormos construction. Many of the important examples of irreducible semigroups which motivated their work were obtained by Hunter [11; 12; 13; 14].In this paper, we apply the constructive method of ideal extensions [5] in algebraic semigroups to topological semigroups which are not necessarily compact. Many of Hunter's examples and examples of the Hormos technique can also be obtained by our method of topological ideal extensions. The topological ideal extension method, however, is, in general, a different type of construction technique.

Algebraic and topological equivalences in the Stone-Čech compactification of a discrete semigroup

were discussed in [5]. Let Z(v) be the support (or spectrum) of V. One can just as well take S as the closure of Un (,(V))n = S(V) since all the convolutions v(i) are concentrated on S(v). It was shown that limn zv, =IL exists in the sense that limn,f ffdVn =ffd,u for every continuous f on S. Further, ,u is regular with support the kernel K (minimal two-sided ideal) of S(v) and is idempotent, that is,

Countably compact sequential topological semigroup with two-sides cancellations is a topological group

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.

We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group.

Let S be a topological semigroup, ω be a weight function on S, and M a (S, ω) be the weighted semigroup algebra of S. Let (S; M a (S, ω)) be the C *-algebra of all M a (S, ω)-measurable functions g on S such that g/ω vanishes at infinity. Here, we consider (S; M a (S, ω))* as a closed subalgebra of M a (S, ω)** equipped with the first Arens product, and study the relation between these two algebras. In particular, we show that (S; M a (S, ω))* is a closed ideal of M a (S, ω)** only when S is compact.

Let $S$ be a topological semigroup such that the almost periodic functions on $S$ have a left invariant mean (this is the case, for example, when $S$ has finite intersection property for closed right ideals). Then whenever $S$ acts equicontinuously on a compact Hausdorff space $X$, there exists a compact group $G$ of homeomorphisms acting equicontinuously on a retract $Y$ of $X$ such that $S$ has a common fixed point in $X$ if and only if $G$ has a common fixed point in $Y$. This result generalises some recent work of T. Mitchell. As an application, we show that whenever $S$ acts equicontinuously on the closed unit interval $I$, then $I$ contains a common fixed point for $S$.

Let S S be a topological semigroup such that the almost periodic functions on S S have a left invariant mean (this is the case, for example, when S S has finite intersection property for closed right ideals). Then whenever S S acts equicontinuously on a compact Hausdorff space X X , there exists a compact group G G of homeomorphisms acting equicontinuously on a retract Y Y of X X such that S S has a common fixed point in X X if and only if G G has a common fixed point in Y Y . This result generalises some recent work of T. Mitchell. As an application, we show that whenever S S acts equicontinuously on the closed unit interval I I , then I I contains a common fixed point for S S .

Let LUC(S) be the space of bounded left uniformly continuous functions and WAPfS) the space of bounded weakly almost periodic functions on a topological semigroup S .For a large class of topological semigroups called stips that are not relatively neocompact, we show that the quotient space LUC(5')/ WAP(S) contains an isometric linear copy of l°° .The result is particularly interesting on noting that the underlying topological semigroup is not assumed to support an absolutely continuous measure-a theme studied by the author in a previous paper.Let S be a locally compact Hausdorff topological semigroup, C(S) the space of all bounded real-valued continuous functions on 5", LUC(»S) := {/ € C(S) : the map x -► xf of S into C(S) is norm continuous} the space of left uniformly continuous functions on S and WAP(S) := {/ e C(S) : the set {xf: x e S} is relatively weakly compact} the space of weakly almost periodic functions on 5. (Here the function xf is given by xf(y) := f(xy) for all / e C(S) and x, y e S.) The space of right uniformly continuous functions, KUC(S), is the right-handed counterpart of LUC(S) and the space of uniformly continuous functions on S is UC(5) := LUC(S) n RUC(S).If we use the weak topology in place of norm topology on C(S), we obtain the weakly uniformly continuous functions WUC(S) in place of US(5).For any subsets A and B of 5 and points x in S, we write AB := {ab : a e A and b e B}, A~XB := {y e S :