Projections on weak$^*$-closed subspaces
Abstract
Let $G$ be a locally compact group and $S$ a weak $^*$-closed translation
invariant subspace of $L^\infty(G)$. M.E.B. Bekka proved that $S$ is the range of a projection on $L^\infty(G)$
which commutes with translation if and only if $S$ is the range of a projection on $L^\infty(G)$ which commutes with convolution. Our
first purpose in this paper is to generalize Bekka’s results for a certain class
of left Banach $G$-module. This result is used to show that $G$ is amenable if
and only if whenever $X$ is a left Banach $G$-module and $S$ is a
weak$^*$-closed right invariant subspace of $X^*$ which is complemented in $X^*$,
then $S$ is the range of a projection on $X^*$ which commutes with convolution.\par
Finally, we explore the link between the projections properties and amenability of group algebras
invariant subspace of $L^\infty(G)$. M.E.B. Bekka proved that $S$ is the range of a projection on $L^\infty(G)$
which commutes with translation if and only if $S$ is the range of a projection on $L^\infty(G)$ which commutes with convolution. Our
first purpose in this paper is to generalize Bekka’s results for a certain class
of left Banach $G$-module. This result is used to show that $G$ is amenable if
and only if whenever $X$ is a left Banach $G$-module and $S$ is a
weak$^*$-closed right invariant subspace of $X^*$ which is complemented in $X^*$,
then $S$ is the range of a projection on $X^*$ which commutes with convolution.\par
Finally, we explore the link between the projections properties and amenability of group algebras
Refbacks
- There are currently no refbacks.