# Extending representations of Banach algebras to their biduals

## TL;DR

This paper characterizes when representations of Banach algebras extend to their biduals, linking weak compactness of orbit maps to the structure of representations and their applications in C*-algebras and harmonic analysis.

## Contribution

It provides necessary and sufficient conditions for extending Banach algebra representations to their biduals, and explores implications for C*-algebras and $L^p$-space representations.

## Key findings

- Representation extension characterized by weakly compact orbit maps.
- Essential space of representation is complemented under certain conditions.
- C*-algebra $A$ has an isometric $L^p$-space representation iff $A$ is commutative.

## 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 $A\to X$ are weakly compact. When this is the case, we show that the essential space of the representation is complemented if $A$ has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in C*-algebras and in abstract harmonic analysis. For example, a C*-algebra $A$ has an isometric representation on an $L^p$-space, for $p\in[1,\infty)\setminus\{2\}$, if and only if $A$ is commutative. Moreover, the $L^p$-operator algebra of a locally compact group is universal with respect to arbitrary representations on $L^p$-spaces.