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The order topology on the projection lattice of a Hilbert space. (English) Zbl 1257.06013

Working on the complete lattice of projections on a Hilbert space, the authors consider the norm and the strong operator topologies as well as the order topology. These topologies are compared and a conjecture raised by V. Palko in [Proc. Am. Math. Soc. 123, No. 3, 715–721 (1995; Zbl 0813.06014)], namely, “the order topology is the intersection of the norm and the strong operator topologies”, is solved in the negative, so that those topologies are different in general. The paper contributes to a further understanding of the order topology and its relationship with both the norm and the strong operator topologies. Some particular cases in which Palko’s conjecture is true, related to separability of the given space, are also shown.

MSC:

06F30 Ordered topological structures
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
54H99 Connections of general topology with other structures, applications

Citations:

Zbl 0813.06014
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References:

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