The convolution algebra of an absolutely locally compact topos

TL;DR

This paper introduces a new convolution algebra associated with absolutely locally compact toposes and admissible sheaves of rings, generalizing known algebraic constructions and providing various completions into Banach and C*-algebras.

Contribution

It defines a convolution algebra for a new class of toposes and constructs its Banach and C*-completions, extending classical algebraic frameworks.

Findings

Construction of the convolution algebra $\\mathcal{C}_c(\mathcal{T},A)$ for absolutely locally compact toposes.

Development of norms leading to Banach and C*-algebra completions.

Examples linking the construction to known algebraic structures like groupoid convolution algebras.

Abstract

We introduce a class of toposes called "absolutely locally compact" toposes and of "admissible" sheaf of rings over such toposes. To any such ringed topos $(\mathcal{T},A)$ we attach an involutive convolution algebra $\mathcal{C}_c(\mathcal{T},A)$ which is well defined up to Morita equivalence and characterized by the fact that the category of non-degenerate modules over $\mathcal{C}_c(\mathcal{T},A)$ is equivalent to the category of sheaf of $A$-module over $\mathcal{T}$. In the case where $A$ is the sheaf of real or complex Dedekind numbers, we construct several norms on this involutive algebra that allows to complete it in various Banach and $C^*$-algebras: $L^1(\mathcal{T},A)$, $C^*_{red}(\mathcal{T},A)$ and $C^*_{max}(\mathcal{T},A)$. We also give some examples where this construction corresponds to well known constructions of involutive algebras, like groupoids convolution algebra and Leavitt path algebras.