Every synaptic algebra has the monotone square root property

Abstract

A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW\(^{*}\)-algebra. In this paper we prove that a synaptic algebra A has the monotone square root property, i.e., if \(0\le a,b\in A\), then \(a\le b \Rightarrow a^{1/2}\le b^{1/2}\).