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arXiv:1302.3201v1 [math.FA] 13 Feb 2013
Lmc−COMPACTIFICATION OF A SEMITOPOLOGICAL
SEMIGROUP AS A SPACE OF E-ULTRAFILTERS
M. AKBARI TOOTKABONI,
DEPARTMENT OF MATHEMATICS, SHAH ED UNIVERSITY
TEHRAN, IRAN
EMAIL: AKBARI@SHAH ED.AC.IR
Abstract. Let S be a semitopological semigroup and CB(S) denotes
the C
∗
-algebra of all bounded complex valued continuous functions on
S with uniform norm. A function f ∈ CB(S) is left mu ltiplicative
continuous if and only if T
µ
f ∈ CB(S) for all µ in the spectrum of CB(S),
where T
µ
f(s) = µ(L
s
f) an d L
s
f(x) = f (sx) for each s, x ∈ S. The col-
lection of all left multiplicative continuous funct ions 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 construc-
tion is applied to obtain some algebraic properties of
(ε, S
Lmc
), that S
Lmc
is the spectrum of Lmc(S), for semitopological
semigroups S. It is shown that if S is a locally compact semitopological
semigroup, then S
∗
= S
Lmc
\ ε(S) is a left ideal of S
Lmc
if and only if
for each x, y ∈ S, there exists a compact zero set A such that x ∈ A and
{t ∈ S : yt ∈ A} is a compact set.
1. Introduction
Let X be a completely regular space, C(X, R) denotes all the real
valued continuous function on X and CB(X, R) denotes all the bounded real
valued continuous function on X. The correspondences between z−filters
on X and ideals in C(X, R) that have been established in [3] are powerful
tools in the study of C(X, R). These correspondences, which also occur in
a rudimentary form in CB(X, R), are inconsequential, as many theorems
of [3] become f alse if C(X, R) is replaced by CB(X, R). However, there is
another correspondence between a certain class of z−filters on X and id eals
in CB(X, R) that leads to quite analogous theorems to those for C(X, R).
The requisite information is ou tlined in [3, 2L].
Key words and phrases. Semigroup Compactification, Lmc- Compactification, z-filter,
e-filter
2010 Mathematics Subject Classification. 22A20, 54D80.
1
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