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Bade’s theorem on the uniformly closed algebra generated by a Boolean algebra. (English) Zbl 0681.47020

The author gives a new proof of the by now classical result due to W. G. Bade [Trans. Am. Math. Soc. 80, 345-360 (1955; Zbl 0066.362)] stating that the uniformly closed algebra generated by a complete Boolean algebra \({\mathcal M}\) of projections on a Banach space coincides with the strongly closed algebra generated by \({\mathcal M}\). The idea is to represent the latter algebra as an \(L^ 1\)-space w.r.t. a suitable spectral measure, whose range is \({\mathcal M}\).

MSC:

47L10 Algebras of operators on Banach spaces and other topological linear spaces
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