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

Abstract

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