Groupoid Characterization of Locally Convex Partial $^*$-Algebras

TL;DR

This paper re-characterizes locally convex partial $^*$-algebras using groupoid convolution algebras, simplifying the analysis of their complex quantum-mechanical space structures.

Contribution

It introduces a groupoid-based framework to describe locally convex partial $^*$-algebras, providing a new perspective that addresses their inherent pathologies.

Findings

Re-characterization of partial $^*$-algebras via Lie groupoids

Simplification of quantum space pathologies

Enhanced understanding of algebraic structures in quantum mechanics

Abstract

Given a locally convex space $(\mathcal{A},\tau)$ with a Hausdorff locally convex topology $\tau$ such that the following maps are continuous; $u \mapsto u^*$ for all $u \in \mathcal{A}$, $x \mapsto x\cdot y$ and $x \mapsto z\cdot x$ for every left and right multipliers of $\mathcal{A}$. In this paper we re-characterized the locally convex partial $*$-algebra $(\mathcal{A}, \Gamma,\cdot,*,\tau)$ arising from these continuous maps in terms of convolution algebra of a Lie groupoid $\Gamma \rightrightarrows \mathcal{A}$. This is advantageous because the pathologies of the underlying spaces owing to their quantum mechanical nature are easily resolved in groupoid terms.