Vol. 50, No. 1, 1974

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Open projections and Borel structures for Cāˆ—-algebras

Herbert Paul Halpern

Vol. 50 (1974), No. 1, 81ā€“98

In this paper the relationships existing among the Boolean σ-algebra generated by the open central projections of the enveloping von Neumann algebra of a C-algebra 𝒜, the Borel structure induced by a natural topology on the quasispectrum of 𝒜, and the type of 𝒜 are discussed. The natural topology is the hull-kernel topology. It is shown that this topology is induced by the open central projections and is the quotient topology of the factor states of 𝒜(with the relativized w-topology) under the relation of quasi-equivalence. The Borel field is shown to be Borel isomorphic with the Boolean σ-algebra multiplied by the least upper bound of all minimal central projections. Finally, it is shown that 𝒜 is GCR if and only if the Boolean σ-algebra (resp. algebra) contains all minimal projections in the center of , or equivalently, if and only if every point in the quasi-spectrum is a Eorel set.

Mathematical Subject Classification 2000
Primary: 46L05
Received: 25 September 1972
Published: 1 January 1974
Herbert Paul Halpern