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 $\mathcal{M}$ with respect to the spectral order. Let $\mathcal{M}_{sa}$ be the self-adjoint part of $\mathcal{M}$ and let $\preceq$ be the spectral order on $\mathcal{M}_{sa}$. We show that a decreasing net in $(\mathcal{M}_{sa},\preceq)$ with a lower bound has the infimum equal to the strong operator limit. The similar statement is proved for increasing net bounded above in $(\mathcal{M}_{sa},\preceq)$. 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 $\mathcal{M}_{sa}$ with respect to the spectral order. We show that it is finer than the restriction of the Mackey topology.