Extensions of orthosymmetric lattice bimorphisms
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- by Mohamed Ali Toumi PDF
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Abstract:
Let $E$ be an Archimedean vector lattice, let $E^{\mathfrak {d}}$ be its Dedekind completion and let $B$ be a Dedekind complete vector lattice. If $\Psi _{0}:E\times E\rightarrow B$ is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism $\Psi :E^{\mathfrak {d}}\times E^{\mathfrak {d}} \rightarrow B$ that not just extends $\Psi _{0}$ but also has to be orthosymmetric. As an application, we prove the following: Let $A$ be an Archimedean $d$-algebra. Then the multiplication in $A$ can be extended to a multiplication in $A^{\mathfrak {d}}$, the Dedekind completion of $A$, in such a fashion that $A^{\mathfrak {d}}$ is again a $d$-algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.References
- Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
- S. J. Bernau, Extension of vector lattice homomorphisms, J. London Math. Soc. (2) 33 (1986), no. 3, 516–524. MR 850967, DOI 10.1112/jlms/s2-33.3.516
- S. J. Bernau and C. B. Huijsmans, Almost $f$-algebras and $d$-algebras, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 287–308. MR 1027782, DOI 10.1017/S0305004100068560
- G. Buskes and A. van Rooij, Almost $f$-algebras: commutativity and the Cauchy-Schwarz inequality, Positivity 4 (2000), no. 3, 227–231. Positivity and its applications (Ankara, 1998). MR 1797125, DOI 10.1023/A:1009826510957
- K. Boulabiar and M. A. Toumi, Lattice bimorphisms on $f$-algebras, Algebra Universalis 48 (2002), no. 1, 103–116. MR 1930035, DOI 10.1007/s00012-002-8206-z
- Elmiloud Chil, The Dedekind completion of $d$-algebras, Positivity 8 (2004), no. 3, 257–267. MR 2120121, DOI 10.1007/s11117-004-1894-1
- J. J. Grobler and C. C. A. Labuschagne, The Riesz tensor product of Archimedean Riesz spaces, Technical Report. F. A. 39, (1986), Potchefsroom University for CHE, South Africa.
- C. B. Huijsmans, Lattice-ordered algebras and $f$-algebras: a survey [Zbl 789:06012], Positive operators, Riesz spaces, and economics (Pasadena, CA, 1990) Springer, Berlin, 1991, pp. 151–169. MR 1307423
- L. V. Kantorovitch, Concerning the problem of moments for finite interval, Dok. Acad. Nauk SSSR 14 (1937), 531-536.
- Z. Lipecki, Extension of vector-lattice homomorphisms, Proc. Amer. Math. Soc. 79 (1980), no. 2, 247–248. MR 565348, DOI 10.1090/S0002-9939-1980-0565348-6
- Z. Lipecki, Extensions of positive operators and extreme points. III, Colloq. Math. 46 (1982), no. 2, 263–268. MR 678143, DOI 10.4064/cm-46-2-263-268
- Z. Lipecki, Extensions of positive operators and extreme points. II, Colloq. Math. 42 (1979), 285–289. MR 567565, DOI 10.4064/cm-42-1-285-289
- Z. Lipecki, D. Plachky, and W. Thomsen, Extensions of positive operators and extreme points. I, Colloq. Math. 42 (1979), 279–284. MR 567564, DOI 10.4064/cm-42-1-279-284
- Z. Lipecki, Extensions of positive operators and extreme points. III, Colloq. Math. 46 (1982), no. 2, 263–268. MR 678143, DOI 10.4064/cm-46-2-263-268
- Z. Lipecki, Extension of vector-lattice homomorphisms revisited, Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 2, 229–233. MR 799083, DOI 10.1016/1385-7258(85)90010-1
- W. A. J. Luxemburg and A. R. Schep, An extension theorem for Riesz homomorphisms, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 2, 145–154. MR 535562, DOI 10.1016/S1385-7258(79)80009-8
- Mohamed Ali Toumi, On some $f$-subalgebras of a $d$-algebra, Math. Rep. (Bucur.) 4(54) (2002), no. 3, 303–310 (2003). MR 2067642
- M. A. Toumi, Structure theorem for d-algebras, submitted.
- A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021, DOI 10.1016/S0924-6509(08)70234-4
Additional Information
- Mohamed Ali Toumi
- Affiliation: Département des Mathématiques, Faculté des Sciences de Bizerte, 7021 Zarzouna, Bizerte, Tunisia
- Email: MohamedAli.Toumi@fsb.rnu.tn
- Received by editor(s): February 10, 2004
- Received by editor(s) in revised form: January 13, 2005
- Published electronically: December 5, 2005
- Additional Notes: The author thanks Professor S. J. Bernau for providing the bibliographic information of [2]
- Communicated by: Joseph A. Ball
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1615-1621
- MSC (2000): Primary 06F25, 47B65
- DOI: https://doi.org/10.1090/S0002-9939-05-08142-6
- MathSciNet review: 2204271