Vigier's theorem for the spectral order and its applications

Abstract

The paper mainly deals with suprema and infima of self-adjoint operators in a von Neumann algebra M with respect to the spectral order. Let Msa be the self-adjoint part of M and let ⪯ be the spectral order on Msa. We show that a decreasing net in (Msa,⪯) with a lower bound has the infimum equal to the strong operator limit. The similar statement is proved for an increasing net bounded above in (Msa,⪯). This version of Vigier's theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self-adjoint operators in terms of the strong operator limit and operator means. As an application of our results on suprema and infima, we study the order topology on Msa with respect to the spectral order. We show that it is finer than the restriction of the Mackey topology.