A proof of Weinbergâ€™s conjecture on lattice-ordered matrix algebras

Authors:
Jingjing Ma and Piotr J. Wojciechowski

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2845-2851

MSC (2000):
Primary 06F25; Secondary 15A48

DOI:
https://doi.org/10.1090/S0002-9939-02-06408-0

Published electronically:
March 15, 2002

MathSciNet review:
1908906

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Abstract | References | Similar Articles | Additional Information

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, Weinbergâ€™s conjecture is true.

- Abraham Berman and Robert J. Plemmons,
*Nonnegative matrices in the mathematical sciences*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Computer Science and Applied Mathematics. MR**544666** - G. Birkhoff and R. S. Pierce,
*Lattice-ordered rings*, An. Acad. Brasil. Cienc.**28**(1956), 41â€“69. - P. Conrad,
*Lattice-ordered groups*, Tulane Lecture Notes, Tulane University, 1970. - L. Fuchs,
*Partially ordered algebraic systems*, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR**0171864** - Jingjing Ma,
*Lattice-ordered matrix algebras with the usual lattice order*, J. Algebra**228**(2000), no. 2, 406â€“416. MR**1764570**, DOI https://doi.org/10.1006/jabr.1999.8232 - Stuart A. Steinberg,
*Finitely-valued $f$-modules*, Pacific J. Math.**40**(1972), 723â€“737. MR**306078** - Stuart A. Steinberg,
*On the scarcity of lattice-ordered matrix algebras. II*, Proc. Amer. Math. Soc.**128**(2000), no. 6, 1605â€“1612. MR**1641109**, DOI https://doi.org/10.1090/S0002-9939-99-05171-0 - E. C. Weinberg,
*On the scarcity of lattice-ordered matrix rings*, Pacific J. Math.**19**(1966), 561â€“571. MR**202775**

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Additional Information

**Jingjing Ma**

Affiliation:
Department of Mathematical Sciences, University of Houston-Clear Lake, 2700 Bay Area Boulevard, Houston, Texas 77058

Email:
ma@cl.uh.edu

**Piotr J. Wojciechowski**

Affiliation:
Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, Texas 79968

Email:
piotr@math.utep.edu

Keywords:
Lattice-ordered algebra,
matrix algebra

Received by editor(s):
March 20, 2001

Received by editor(s) in revised form:
May 16, 2001

Published electronically:
March 15, 2002

Additional Notes:
The results in this paper were presented at the conference â€śLattice-ordered groups and *f*-rings" at the University of Florida, March 2001.

Dedicated:
Dedicated to Professor Melvin Henriksen on his 75th birthday

Communicated by:
Lance W. Small

Article copyright:
© Copyright 2002
American Mathematical Society