Local maps and the representation theory of operator algebras

TL;DR

This paper employs representation theory to demonstrate reflexivity of certain derivation and multiplier spaces over operator algebras, revealing new structural insights and applications in automorphic semicrossed products and tensor algebras.

Contribution

It introduces novel results on the reflexivity of derivation and multiplier spaces and shows that finite dimensional nest representations distinguish points in tensor algebras.

Findings

Reflexivity of derivation and multiplier spaces established

Bounded local derivations on automorphic semicrossed products are actual derivations

Finite dimensional nest representations separate points in tensor algebras

Abstract

Using representation theory techniques we prove that various spaces of derivations or one-sided multipliers over certain operator algebras are reflexive. A sample result: any bounded local derivation (local left multiplier) on an automorphic semicrossed product is a derivation (resp. left multiplier). In the process we obtain various results of independent interest. In particular, the finite dimensional nest representations of the tensor algebra of a topological graph separate points.