Weakly Completely Continuous Elements of the Banach Algebra LUC(G)*
Abstract
In this paper, we study weakly
compact left multipliers on the Banach algebra
$\hbox{LUC}(G)^{*}$. We show that $G$ is compact if
and only if there exists a non-zero weakly compact left
multipliers on $\hbox{LUC}(G)^*$. We also investigate the relation
between positive left weakly completely continuous elements of the
Banach algebras $\hbox{LUC}(G)^*$ and $L^\infty(G)^*$. Finally, we
prove that $G$ is finite if and only if there exists a non-zero
multiplicative linear functional $\mu$ on $\hbox{LUC}(G)$ such
that $\mu$ is a left weakly completely continuous elements of
$\hbox{LUC}(G)^*$.
compact left multipliers on the Banach algebra
$\hbox{LUC}(G)^{*}$. We show that $G$ is compact if
and only if there exists a non-zero weakly compact left
multipliers on $\hbox{LUC}(G)^*$. We also investigate the relation
between positive left weakly completely continuous elements of the
Banach algebras $\hbox{LUC}(G)^*$ and $L^\infty(G)^*$. Finally, we
prove that $G$ is finite if and only if there exists a non-zero
multiplicative linear functional $\mu$ on $\hbox{LUC}(G)$ such
that $\mu$ is a left weakly completely continuous elements of
$\hbox{LUC}(G)^*$.
Keywords
Locally compact group, multiplier, weakly compact operator
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