Unitary representations of groups, duals, and characters

TL;DR

This expository work explores the structure of unitary representations of topological groups, focusing on dual spaces, their classifications, and examples, with an emphasis on the case of discrete groups.

Contribution

It systematically introduces and compares various dual spaces of unitary representations, providing detailed examples and connections to operator algebras.

Findings

Classification of dual spaces for various groups

Illustration of dual spaces with specific examples

Connection between representations and operator algebras

Abstract

This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most important notions are defined for topological groups, but a special attention is paid to the case of discrete groups. The unitary dual of a group $G$ is the space of equivalence classes of its irreducible unitary representations; it is both a topological space and a Borel space. The primitive dual is the space of weak equivalence classes of unitary irreducible representations. The normal quasi-dual is the space of quasi-equivalence classes of traceable factor representations; it is parametrized by characters, which can be finite or infinite. The theory is systematically illustrated by a series of specific examples: Heisenberg groups, affine groups of infinite fields, solvable Baumslag-Solitar groups, lamplighter groups, and general linear groups. Operator algebras play an important role in the exposition, in particular the von Neumann algebras associated to a unitary representation and C*-algebras associated to a locally compact group.