Projections onto the essential space of a representation

Classification of operator algebras: complexity, rigidity, and dynamics

01 March 14:00 - 15:00

Hannes Thiel - Westfälische Wilhelms-Universität Münster

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.
Marius Dadarlat
Purdue University
Søren Eilers
University of Copenhagen
Asger Törnquist
University of Copenhagen


Søren Eilers


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