Dichotomy laws for the behaviour near the unit element of group representations

Abstract.

A well known “zero-two law" shows that if ${\left( {T{\left( t \right)}} \right)}_{{t \in \mathbb{R}}}$ is a strongly continuous one-parameter group of bounded operators on a Banach space X, and if ${\mathop {{\mathop {\lim \sup }\limits_{t \to 0^{ + } } }\parallel I - T{\left( t \right)}}\limits_{} }\parallel < 2,$ then ${\mathop {{\mathop {\lim }\limits_{t \to 0^{ + } } }\parallel I - T{\left( t \right)}}\limits_{} }\parallel = 0.$ Here we discuss analogous problems for general unital representations θ of a topological group G on a unital Banach algebra A. Let 1 be the unit of G, and I the unit element of A. We show that either ${\mathop {{\mathop {\lim \sup }\limits_{u \to 1} }\parallel \theta {\left( u \right)} - I}\limits_{} }\parallel \geqq {\sqrt 3 },$ or $\mathop{\lim}\limits_{u \to 1}\parallel \theta {(u)} -I \parallel = 0;$ if, moreover, θ admits “continuous division by any positive integer”, then, either ${\mathop {{\mathop {\lim \sup }\limits_{u \to 1} }\parallel \theta {\left( u \right)} - I}\limits_{} }\parallel \geqq 2,$ or ${\mathop {{\mathop {\lim }\limits_{u \to 1} }\parallel \theta {\left( u \right)} - I}\limits_{} }\parallel = 0.$ Our argument also gives automatic continuity results for representations of abelian Baire groups on a separable Banach algebra and representations of compact non abelian groups on a Banach algebra which are locally bounded and satisfy ${\mathop {{\mathop {\lim }\limits_{u \to 1} }\rho {\left( {\theta {\left( u \right)} - I} \right)}}} = 0.$