This is an announcement for the paper “Extending representations of Banach algebras to their biduals” by Eusebio Gardellahttps://arxiv.org/find/math/1/au:+Gardella_E/0/1/0/all/0/1, Hannes Thielhttps://arxiv.org/find/math/1/au:+Thiel_H/0/1/0/all/0/1.
Abstract: We show that a representation of a Banach algebra $A$ on a Banach space $X$ can be extended to a canonical representation of $A^{**}$ on $X$ if and only if certain orbit maps $A\rightarrow X$ are weakly compact. We apply this to study when the essential space of a representation is complemented. This provides a tool to disregard the difference between degenerate and nondegenerate representations on Banach spaces. As an application we show that a $C^*$-algebra $A$ has an isometric representation on an $L^p$ -space, for $p\in [1, \infty){2}$, if and only if $A$ is commutative
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.00882