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Oxford Quantum Talks Archive

Involutive Categories and Monoids, with a GNS-correspondence

Bart Jacobs, Radboud University Nijmegen
Quantum Physics and Logic 2010, May 2010, University of Oxford

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 arbitrary involutive symmetric monoidal categories.

[video] [streaming video]