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Let $S$ be a semitopological semigroup and $\mathcal{CB}(S)$ denotes the$C^*$-algebra of all bounded complex valued continuous functions on $S$ withuniform norm. A function $f\in \mathcal{CB}(S)$ is left multiplicative\linebreak continuous if and only if $\mathbf{T}_{\mu}f\in \mathcal{CB}(S)$ forall $\mu$ in the spectrum of $\mathcal{CB}(S)$, where$\mathbf{T}_{\mu}f(s)=\mu(L_sf)$ and $L_sf(x)=f(sx)$ for each $s,x\in S$. Thecollection of all left multiplicative continuous functions on $S$ is denoted by$Lmc(S)$. In this paper, the $Lmc-$compactification of a semitopologicalsemigroup S is reconstructed as a space of $e-$ultrafilters. This constructionis applied to obtain some algebraic properties of \linebreak $(\varepsilon,\s)$, that $ \s $ is the spectrum of $Lmc(S)$, for semitopological semigroups$S$. It is shown that if S is a locally compact semitopological semigroup, then$S^*=\s \setminus \varepsilon(S)$ is a left ideal of $\s$ if and only if foreach $x,y\in S$, there exists a compact zero set $A$ such that $x\in A$ and$\{t\in S:yt\in A\}$ is a compact set.

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