Vigier's theorem for the spectral order and its applications

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 M-sa be the self-adjoint part of M and let <= be the spectral order on M-sa. We show that a decreasing net in (M-sa, <=) 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 (M-sa, <=) 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 M-sa, with respect to the spectral order. We show that it is finer than the restriction of the Mackey topology. (C) 2019 Elsevier Inc. All rights reserved.