Spectral Lattices

Abstract

Spectral orthomorphisms between the spectral lattices of JBW algebras which preserve the scales extend to Jordan homomorphisms for a large class of algebras. Spectral lattice homomorphism is automatically a σ -lattice homomorphism. The range projection map is, up to a Jordan homomorphism, the only natural map from the spectral lattice onto the projection lattice. Continuity of the range projection determines finiteness of the algebra in Murray-von Neumann comparison theory.

Automorphisms of spectral lattices of unbounded positive operators

“…(For a treatment on operator algebraic approach to quantum theory we refer the reader to [5].) Spectral order, first mentioned probably by Arveson [2], has been studied in the context of bounded operators and matrices so far [1,4,6,7,9,10,11]. However, unlike the standard operator order, the spectral order can be naturally and directly extended to unbounded operators as well.…”