New York Journal of Mathematics
Volume 19 (2013) 669-688


M. Akbari Tootkaboni

Lmc-compactification of a semitopological semigroup as a space of e-ultrafilters

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Published: October 20, 2013
Keywords: Semigroup Compactification, Lmc-compactification, z-filter, e-filter
Subject: 22A20, 54D80

Let S be a semitopological semigroup and CB(S) denote the C*-algebra of all bounded complex valued continuous functions on S with uniform norm. A function f∈ CB(S) is left multiplicative continuous if and only if Tμf∈ CB(S) for all μ in the spectrum of CB(S), where Tμf(s)=μ(Lsf) and Lsf(x)=f(sx) for each s,x∈ S. The collection of all the left multiplicative continuous functions on S is denoted by Lmc(S). In this paper, the Lmc-compactification of a semitopological semigroup S is reconstructed as a space of e-ultrafilters. This construction is applied to obtain some algebraic properties of (ε ,SLmc), such that SLmc is the spectrum of Lmc(S), for semitopological semigroups S. It is shown that if S is a locally compact semitopological semigroup, then S*=SLmc \ ε(S) is a left ideal of SLmc if and only if for each x,y∈ S, there exists a compact zero set A containing x such that {t∈S : yt∈A} is a compact set.

Author information

Department of Mathematics, Shahed University, Tehran, Iran