New C*-completions of discrete groups and related spaces

TL;DR

This paper introduces a unified framework for C*-completions of discrete groups based on ideals in ℓ∞(Γ), leading to new characterizations of properties like a-T-menability and property (T), and discovering exotic quantum groups.

Contribution

It develops a general framework for C*-completions associated with ideals in ℓ∞(Γ), providing new characterizations of classical properties and examples of exotic quantum groups.

Findings

First C*-algebraic characterization of a-T-menability

New characterization of property (T)

Examples of exotic quantum groups

Abstract

Let $\Gamma$ be a discrete group. To every ideal in $\ell^{\infty}(\G)$ we associate a C$^*$-algebra completion of the group ring that encapsulates the unitary representations with matrix coefficients belonging to the ideal. The general framework we develop unifies some classical results and leads to new insights. For example, we give the first C$^*$-algebraic characterization of a-T-menability; a new characterization of property (T); new examples of "exotic" quantum groups; and, after extending our construction to transformation groupoids, we improve and simplify a recent result of Douglas and Nowak.