VECTOR LATTICES IN SYNAPTIC ALGEBRAS

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. (C) 2017 Mathematical Institute Slovak Academy of Sciences