On the Structure of Abstract H*-Algebras

Kevin Dunne
(University of Strathclyde)

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.

In Bob Coecke and Aleks Kissinger: Proceedings 14th International Conference on Quantum Physics and Logic (QPL 2017), Nijmegen, The Netherlands, 3-7 July 2017, Electronic Proceedings in Theoretical Computer Science 266, pp. 197–208.
Published: 27th February 2018.

ArXived at: http://dx.doi.org/10.4204/EPTCS.266.13 bibtex PDF
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