Involutive Categories and Monoids, with a GNS-correspondence

TL;DR

This paper develops the theory of involutive categories, demonstrating their role in describing involutive monoids and establishing a GNS-like correspondence between states and inner products within this framework.

Contribution

It introduces the foundational theory of involutive categories and extends the GNS correspondence to arbitrary involutive categories.

Findings

Involutive categories naturally describe involutive monoids.

Categories of Eilenberg-Moore algebras of involutive monads are involutive.

A GNS-like bijective correspondence is established in involutive categories.

Abstract

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive categories.