A Proof of Weinberg’s Conjecture on Lattice-ordered Matrix Algebras

Let F be a subfield of the field of real numbers and let Fn (n ≥ 2) be the n× n matrix algebra over F. It is shown that if Fn is a lattice-ordered algebra over F in which the identity matrix 1 is positive, then Fn is isomorphic to the lattice-ordered algebra Fn with the usual lattice order. In particular, Weinberg’s conjecture is true. Let L be a totally ordered field, and let Ln (n ≥ 2) be the n× n matrix algebra over L. Then Ln may be lattice-ordered by requiring that a matrix in Ln is positive exactly when each of its entries is positive, that is, the positive cone is (L)n. This lattice order is called the usual lattice order of Ln. Let Q be the field of rational numbers. In 1966, Weinberg conjectured that (Q)n is the only lattice order of Qn (up to an isomorphism) such that Qn is a lattice-ordered algebra (`-algebra) over Q in which 1 is positive, and he proved his conjecture for n = 2 [8]. Recently some conditions have been obtained to ensure an `-algebra Ln, in which 1 is positive, is isomorphic to the `-algebra Ln with the usual lattice order [5], [7]. In this paper, we show that Weinberg’s conjecture is true for a matrix `-algebra over any subfield of real numbers. More precisely, suppose that F is a subfield of the field of real numbers; it is shown that an `-algebra Fn over F in which 1 is positive is isomorphic to the `-algebra Fn with the usual lattice order. We begin by collecting some definitions and results we will use later. The reader is referred to Birkhoff & Pierce [2] and Fuchs [4] for the general theory of latticeordered rings (`-rings). A partially ordered ring (po-ring) R is an (associative) ring which is partially ordered, and in which i) a ≥ b implies a + c ≥ b + c, for any c ∈ R, and ii) a ≥ 0 and b ≥ 0 imply ab ≥ 0. Let R be a po-ring. The set R = {a ∈ R : a ≥ 0} is called the positive cone of R. Clearly R is closed under addition and multiplication, and R ∩ −R = {0}. Conversely, if P is a subset of a ring R which is closed under addition and multiplication, and satisfies P ∩ −P = {0}, then the partial order ≥ defined by a ≥ b if and only if a − b ∈ P makes R into a po-ring with the positive cone equal to P . We will refer to such a Received by the editors March 20, 2001 and, in revised form, May 16, 2001. 2000 Mathematics Subject Classification. Primary 06F25; Secondary 15A48.