An amenability property of algebras of functions on semidirect products of semigroups

Bao-Ting Lerner

doi:10.1155/s0161171283000253

Abstract

Let S1 and S2 be semitopological semigroups, S1 τ S2 a semidirect product. An amenability property is established for algebras of functions on S1 τ S2. This result is used to decompose the kernel of the weakly almost periodic compactification of S1 τ S2 into a semidirect product.

Highlights

S algebras of functions on S I 2 which generalizes a result of Junghenn (5) and provides conditions under which the kernel of the weakly almost periodic (WAP)-compactification of S I S 2 can be decomposed into a semidirect product
We shall denote by ql SI / SI$2 and q2 $2, SI$2 the injection mappings (ql(Sl) (Sl,l), q2(s2) (l,s2), for sI e SI, s2 e $2)
S2 and S 2 respectively, X is a compact topological group which is a homomorphic image of the canonical strongly almost periodic (SAP)-compactification of S I, and equality denotes canonical isomorphism

Summary

SI ABSTRACT

The decomposition may be written symbolically as (S I$2 )F SIG $2H (i.i) where G {f(.,l) f F} and H {f(l,’): f F} and equality denotes canonical isomorphism 0 being another semidirect product) Applications of this decomposition were made to the almost periodic (AP), strongly almost periodic (SAP) and left-uniformly continuous (LUC) cases. If S I S 2 is any commutative topological semigroup with identity for SIS which WAP(SI) AP(SI), (i.i) fails even if 2 is taken to be the special case of a direct product (Junghenn (4)). S algebras of functions on S I 2 which generalizes a result of Junghenn (5) and provides conditions under which the kernel of the WAP-compactification of S I S 2 can be decomposed into a semidirect product.

Ls define operators

Hausdorff topological space and a semigroup such that for each y

SIS compactification of

By the definition of

Observe further that since