A duality of locally compact groups which does not involve the Haar measure

TL;DR

This paper introduces a new duality framework for locally compact groups that does not rely on Haar measure, using functorial maps between categories of coinvolutive Hopf algebras and von Neumann algebras.

Contribution

It provides an explicit duality map between $C_0(G)$ and $C^*(G)$ without assuming isomorphism to their biduals, expanding the understanding of group duality beyond Haar measure dependence.

Findings

Explicit duality between $C_0(G)$ and $C^*(G)$

Description of commutative and co-commutative algebras in the duality range

Duality between enveloping von Neumann algebras of these structures

Abstract

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf \cst-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the \cst-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorphic to their bidual): these algebras have the form $C_0(G)$ or $C^*(G)$ respectively, where $G$ is a locally compact group. The von Neumann version of the functor puts into duality, in the group case, the enveloping von Neumann algebras of the algebras above: $C_0(G)^{**}$ and $C^*(G)^{**}$.