Vector lattices in synaptic algebras

2016/05/23 by David J. Foulis, Foulis, David J., Anna Jencova +3
Mathematics · #06C15 #47B15 #91P10 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:06C15 #msc:47B15 #msc:91P10

paper · pdf · doi:10.48550/arxiv.1605.06987

24 pages, no figures

arxiv created 2016/05/23 · arxiv updated 2016/05/24

Abstract

A synaptic algebra A is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace V of A in regard to the question of when V is a vector lattice. Our main theorem states that if V contains the identity element of A and is closed under the formation of both the absolute value and the carrier of its elements, then V is a vector lattice if and only if the elements of V commute pairwise.

Related

• Symmetries in Synaptic Algebras
• Two projections in a synaptic algebra
• Spectral order on synaptic algebras
• Nest algebras in an arbitrary vector space
• Every synaptic algebra has the monotone square root property