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Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups

  • Wolfgang Rump EMAIL logo
Published/Copyright: December 20, 2017

Abstract

It is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular lattice X, admits a natural embedding into a group G(X) with a lattice ordering so that G(X) determines X up to isomorphism. The embedding XG(X) appears to be a universal (non-commutative) group-valued measure on X, while states of X turn into real-valued group homomorphisms on G(X). The existence of completions is characterized by a generalized archimedean property which simultaneously applies to X and G(X). By an extension of Foulis’ coordinatization theorem, the negative cone of G(X) is shown to be the initial object among generalized Baer-semigroups. For finite X, the correspondence between X and G(X) provides a new class of Garside groups.


Dedicated to B. V. M.



Communicated by Manfred Droste


Acknowledgements

We thank an anonymous referee for a very careful inspection of the paper which led to a substantial and detailed report.

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Received: 2017-05-15
Revised: 2017-11-08
Published Online: 2017-12-20
Published in Print: 2018-07-01

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