A proof of Weinberg’s conjecture on lattice-ordered matrix algebras

Abstract:

Let $\mathbf {F}$ be a subfield of the field of real numbers and let $\mathbf {F}_{n}$ ($n \geq 2$) be the $n \times n$ matrix algebra over $\mathbf {F}$. It is shown that if $\mathbf {F}_{n}$ is a lattice-ordered algebra over $\mathbf {F}$ in which the identity matrix 1 is positive, then $\mathbf {F}_{n}$ is isomorphic to the lattice-ordered algebra $\mathbf {F}_{n}$ with the usual lattice order. In particular, Weinberg’s conjecture is true.