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

Ricker, Werner J.

Commentat. Math. Univ. Carol. 29, No. 3, 597-600 (1988).

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 \({\mathcal M}\).

Reviewer: Dirk Werner (Berlin)

MSC:

47L10

Algebras of operators on Banach spaces and other topological linear spaces

Keywords:

complete Boolean algebra of projections on a Banach; space; strongly closed algebra; spectral measure

Citations:

Zbl 0681.47021; Zbl 0066.362