Hannes Thiel: Projections onto the essential space of a representation

We consider a representation $\varphi$ of a Banach algebra \(A\) on a Banach space \(X\). The essential space of \(\varphi\) is the closed subspace of \(X\) generated by\(\varphi(A)X\). We study when this space is the range of a projection on \(X\), that is, when it is a complemented subspace of \(X\).
 
We obtain positive results assuming that \(A\) has a bounded left approximate unit and that every bounded map from \(A\) to \(X\) is weakly compact. The former is automatic for \(\ca{s}\) and for group algebras of locally compact groups. The latter is automatic when \(X\) is reflexive and in many cases when \(A\) is a \(C^*\)-algebra.
 
Our results are a tool to disregard the difference between degenerate and nondegenerate representations. They clarify and simplify the theory of representations on \(L^p\)-spaces. As a main application we show that a \(C^*\)-algebra \(A\) can be isometrically represented on an \(L^p\)-space, for \(p\in[1,\infty)\), \(p\neq 2\) if and only if \(A\) is commutative.