Groupoid Characterization of Locally Convex Partial ${}^{\ast }$-Algebras

Abstract

Given a locally convex space $(\mathcal{A},\tau )$ with a Hausdorff locally convex topology $\tau$ such that the following maps are continuous; $u\mapsto u^{\ast }$ 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 $\ast$-algebra $(\mathcal{A},\Gamma ,\cdot ,\ast ,\tau )$ 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.

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