A property for inverses in a partially ordered linear algebra

Abstract:

We consider a Dedekind $\sigma$-complete partially ordered linear algebra A which has the following property: if $x \in A$ and $1 \leqslant x$, then $- u \leqslant {x^{ - 1}}$, where $u = {u^2}$. This property is used to show that A must be commutative. We also show that A is the direct sum of two algebras, each of which behaves like an algebra of real-valued functions.