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

HTML articles powered by AMS MathViewer

- by Jingjing Ma and Piotr J. Wojciechowski PDF
- Proc. Amer. Math. Soc.
**130**(2002), 2845-2851 Request permission

## 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, Weinberg’s conjecture is true.## References

- Abraham Berman and Robert J. Plemmons,
*Nonnegative matrices in the mathematical sciences*, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**544666** - Cahit Arf,
*Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper*, J. Reine Angew. Math.**181**(1939), 1–44 (German). MR**18**, DOI 10.1515/crll.1940.181.1 - 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, 406–416. MR**1764570**, DOI 10.1006/jabr.1999.8232 - Stuart A. Steinberg,
*Finitely-valued $f$-modules*, Pacific J. Math.**40**(1972), 723–737. MR**306078**, DOI 10.2140/pjm.1972.40.723 - Stuart A. Steinberg,
*On the scarcity of lattice-ordered matrix algebras. II*, Proc. Amer. Math. Soc.**128**(2000), no. 6, 1605–1612. MR**1641109**, DOI 10.1090/S0002-9939-99-05171-0 - E. C. Weinberg,
*On the scarcity of lattice-ordered matrix rings*, Pacific J. Math.**19**(1966), 561–571. MR**202775**, DOI 10.2140/pjm.1966.19.561

## 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
- 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 “Lattice-ordered groups and
*f*-rings" at the University of Florida, March 2001. - Communicated by: Lance W. Small
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1908906

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