Lattice orders on matrix algebras

Abstract

We construct all the lattice orders (up to isomorphism) on a full matrix algebra over a subfield of the field of real numbers so that it becomes a lattice-ordered algebra.

Positive derivations on archimedean lattice-ordered rings

“…According to [10, Example 3.1], the only positive R-derivation on M n (R) with the usual entrywise lattice order is the zero derivation. In this section, using ideas from [10] and results from [20], we show that the only positive F -derivation on any lattice-ordered matrix algebra over a subfield F of R is the zero derivation. Note that by [4, p. 49, Lemma 1], each finite-dimensional vector lattice has a strong unit.…”

“…Section 7 uses a noncommutative generalization of the theorem of Henriksen and Smith to investigate archimedean -rings in which squares are positive. Section 8 concerns lattice-ordered matrix algebras over a subfields of R; using ideas in [10] and results in [20], we show that the only positive derivation on any such algebra is the zero derivation. We conclude in section 9 by characterizing derivations on archimedean polynomial rings that are lattice-homomorphisms or positive orthomorphisms.…”

“…By [20,Theorem 12], we may assume that M n (F ) ≥ = P A , where A = (a ij ) is a nonsingular matrix in M n (F ≥ ) and…”

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

“…Later, Steinberg studied Weinberg's conjecture over totally ordered fields (see [5]). In 2002, Ma and Wojciechowski proved Weinberg's conjecture over a totally ordered subfield of the real number field (see [4]). In 2007, Ma and Redfield proved Weinberg's conjecture over the ring of integers (see [3]).…”

Lattice orders on 2 × 2 triangular matrix algebras

“…The entrywise lattice order on M n (F ) makes it into an -algebra over F with the positive cone M n (F ≥ ), where F ≥ denotes all nonnegative real numbers in F . It was shown that entrywise lattice order is the only lattice order (under isomorphism) on M n (F ) in which the identity matrix is positive [5], and that the positive cone of each lattice order on M n (F ) is isomorphic to fM n (F ≥ ), where f is an invertible matrix in M n (F ≥ ) [6]. The above results are also true for 2 × 2 matrix algebras over any totally ordered fields [7,8].…”