On the structure of abstract h*-algebras

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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 quantumlogic 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.

Original languageEnglish
Pages (from-to)197-208
Number of pages12
JournalElectronic Proceedings in Theoretical Computer Science, EPTCS
Volume266
DOIs
Publication statusPublished - 27 Feb 2018
Event14th International Conference on Quantum Physics and Logic and IQSA Quantum Structures Workshop - Nijmegen, Netherlands
Duration: 3 Jul 20177 Jul 2017

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Quantum theory
Algebra
Hilbert spaces

Keywords

  • generalised topos approach
  • quantum theory
  • dagger-kernel approach
  • h*-algebras

Cite this

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On the structure of abstract h*-algebras. / Dunne, Kevin.

In: Electronic Proceedings in Theoretical Computer Science, EPTCS, Vol. 266, 27.02.2018, p. 197-208.

Research output: Contribution to journalConference Contribution

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AB - 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 quantumlogic 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.

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