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On the scarcity
of lattice-ordered matrix algebras II

Author: Stuart A. Steinberg
Journal: Proc. Amer. Math. Soc. 128 (2000), 1605-1612
MSC (1991): Primary 06F25; Secondary 15A48
Published electronically: September 23, 1999
MathSciNet review: 1641109
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Abstract: We correct and complete Weinberg's classification of the lattice-orders of the matrix ring ${\Bbb Q}_2$ and show that this classification holds for the matrix algebra $F_2$ where $F$ is any totally ordered field. In particular, the lattice-order of $F_2$ obtained by stipulating that a matrix is positive precisely when each of its entries is positive is, up to isomorphism, the only lattice-order of $F_2$ with $1>0$. It is also shown, assuming a certain maximum condition, that $(F^+)_n$ is essentially the only lattice-order of the algebra $F_n$ in which the identity element is positive.

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Additional Information

Stuart A. Steinberg
Affiliation: Department of Mathematics, The University of Toledo, Toledo, Ohio 43606-3390

Keywords: Lattice-ordered algebra, matrix algebra
Received by editor(s): March 27, 1998
Received by editor(s) in revised form: July 17, 1998
Published electronically: September 23, 1999
Communicated by: Ken Goodearl
Article copyright: © Copyright 2000 American Mathematical Society

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