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Compact orthoalgebras

Author: Alexander Wilce
Journal: Proc. Amer. Math. Soc. 133 (2005), 2911-2920
MSC (2000): Primary 06F15, 06F30; Secondary 03G12, 81P10
Published electronically: May 2, 2005
MathSciNet review: 2159769
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Abstract: We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and also the projection lattice of a Hilbert space. As the latter example illustrates, a lattice-ordered TOA need not be a topological lattice. However, we show that a compact Boolean TOA is a topological Boolean algebra. Using this, we prove that any compact regular TOA is atomistic , and has a compact center. We prove also that any compact TOA with isolated $0$ is of finite height. We then focus on stably ordered TOAs: those in which the upper set generated by an open set is open. These include both topological orthomodular lattices and interval orthoalgebras - in particular, projection lattices. We show that the topology of a compact stably-ordered TOA with isolated $0$ is determined by that of its space of atoms.

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Additional Information

Alexander Wilce
Affiliation: Department of Mathematics, Susquehanna University, Selinsgrove, Pennsylvania 17870

Keywords: Orthoalgebra, effect algebra, orthomodular lattice, topological lattice, quantum logic
Received by editor(s): August 22, 2003
Received by editor(s) in revised form: June 10, 2004
Published electronically: May 2, 2005
Communicated by: Lance W. Small
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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