Maximality of Positive Operator-valued Measures

Abstract

A positive operator-valued measure is a (weak-star) countably additive set function from a σ-field Σ to the space of nonnegative bounded operators on a separable complex Hilbert space Open image in new window . Such functions can be written as M = V*E(·)V in which E is a spectral measure acting on a complex Hilbert space Open image in new window and V is a bounded operator from Open image in new window to Open image in new window such that the only closed linear subspace of Open image in new window , containing the range of V and reducing E (Σ), is Open image in new window itself. Attention is paid to an existing notion of maximality for positive operator-valued measures. The purpose of this paper is to show that M is maximal if and only if E, in the above representation of M, generates a maximal commutative von Neumann algebra.