Isomorphisms of spectral lattices

TL;DR

This paper characterizes spectral order isomorphisms between spectral sublattices of direct sums of AW*-factors, providing a complete description especially for atomic AW*-algebras and matrix algebras.

Contribution

It proves that spectral order isomorphisms decompose into isomorphisms of individual summands, extending understanding of spectral lattice structures in AW*-algebras.

Findings

Spectral order isomorphisms decompose into summand isomorphisms.

Complete description for atomic AW*-algebras including matrix algebras.

General form of spectral order orthoisomorphisms established.

Abstract

The paper deals with spectral order isomorphisms between certain spectral sublattices of direct sums of AW*-factors. We prove that these maps consist of spectral order isomorphisms between spectral sublattices of individual direct summands. Consequently, we obtain a complete description of spectral order isomorphisms in the case of atomic AW*-algebras. This includes the setting of matrix algebras. Moreover, we also exhibit the general form of spectral order orthoisomorphisms between various spectral sublattices of direct sums of AW*-factors.