On the Structure of Abstract H*-Algebras

TL;DR

This paper explores the relationship between H*-algebras and various categorical approaches to quantum theory, establishing a structure theorem that generalizes classical results within a broader mathematical framework.

Contribution

It demonstrates that H*-algebras naturally align with states and observables in the generalized topos approach and proves a new structure theorem extending Ambrose's classical result.

Findings

H*-algebras correspond with states and observables in the topos approach

A structure theorem for H*-algebras is established

The theorem generalizes Ambrose's classical result for Hilbert spaces

Abstract

Previously we have shown that the topos approach to quantum theory of Doering and Isham can be generalised to a class of categories typically studied within the monoidal approach to quantum theory of Abramsky and Coecke. In the monoidal approach to quantum theory H*-algebras provide an axiomatisation of states and observables. Here we show that H*-algebras naturally correspond with the notions of states and observables in the generalised topos approach to quantum theory. We then combine these results with the dagger-kernel approach to quantum logic of Heunen and Jacobs, which we use to prove a structure theorem for H*-algebras. This structure theorem is a generalisation of the structure theorem of Ambrose for H*-algebras the category of Hilbert spaces.