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On basic concepts of non-commutative topology. (English) Zbl 0596.46064
Summary: In the present paper we give an intrinsic axiomatic definition of a general non-commutative topology in terms of the lattice of all projections in an arbitrary atomic \(W^*\)-algebra B. The system of axioms connects the properties of non-commutative topology with order, Jordan and \(C^*\)-structures on B. For all that two key ideas are pursued: firstly, to generalize to non-abelian case the description of any topology by means of the bounded lower semicontinuous functions cone and, secondly, to provide that the set of ”continuous” elements in B be a \(C^*\)-algebra. Moreover, we give an effective characterization of compactness and show that a non-commutative topology is locally compact iff it is the Akemann-Giles topology associated with a certain \(C^*\)- algebra.
MSC:
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
54A99 Generalities in topology
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References:
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