Bicommutant Theorem for sigma-Complete Boolean Algebras of Projections in BanachSpaces

摘要

A classical result of J. von Neumann states that if T is a bounded selfadjointoperator in a separable Hilbert space, then the following four algebras are the same: (1) The algebra of all bounded Borel functions of T, (2) The strong (or weak) closed operator algebra generated by T and I (the identity operator), (3) The uniform closed operator algebra generated by the projections in the resolution of the identity for T and (4) The bicommutant (T)cc, of T.