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Extensions of orthosymmetric lattice bimorphisms. (English) Zbl 1100.06013
For an Archimedian vector lattice \(E\) and a Dedekind complete vector lattice \(B\), it is shown that if a lattice bimorphism from \(E\times E\) to \(B\) satisfies the AF property (the image of each pair of disjoint elements is zero), then every lattice extension to the cartesian product of the Dedekind completion of \(E\) with itself also satisfies the AF property. As a consequence, the author demonstrates that the multiplication in an Archimedean d-algebra can be extended to a d-algebra multiplication in the Dedekind completion. This result was established by E. Chil [Positivity 8, 257–267 (2004; Zbl 1073.46002)] but is proved here in a brief and direct manner.

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
46A40 Ordered topological linear spaces, vector lattices
06F25 Ordered rings, algebras, modules
47B65 Positive linear operators and order-bounded operators
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[1] Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. · Zbl 0608.47039
[2] S. J. Bernau, Extension of vector lattice homomorphisms, J. London Math. Soc. (2) 33 (1986), no. 3, 516 – 524. · Zbl 0555.47023 · doi:10.1112/jlms/s2-33.3.516 · doi.org
[3] S. J. Bernau and C. B. Huijsmans, Almost \?-algebras and \?-algebras, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 287 – 308. · Zbl 0707.06009 · doi:10.1017/S0305004100068560 · doi.org
[4] G. Buskes and A. van Rooij, Almost \?-algebras: commutativity and the Cauchy-Schwarz inequality, Positivity 4 (2000), no. 3, 227 – 231. Positivity and its applications (Ankara, 1998). · Zbl 0987.46002 · doi:10.1023/A:1009826510957 · doi.org
[5] K. Boulabiar and M. A. Toumi, Lattice bimorphisms on \?-algebras, Algebra Universalis 48 (2002), no. 1, 103 – 116. · Zbl 1059.06013 · doi:10.1007/s00012-002-8206-z · doi.org
[6] Elmiloud Chil, The Dedekind completion of \?-algebras, Positivity 8 (2004), no. 3, 257 – 267. · Zbl 1073.46002 · doi:10.1007/s11117-004-1894-1 · doi.org
[7] 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. · Zbl 0728.46049
[8] C. B. Huijsmans, Lattice-ordered algebras and \?-algebras: a survey [Zbl 789:06012], Positive operators, Riesz spaces, and economics (Pasadena, CA, 1990) Springer, Berlin, 1991, pp. 151 – 169. · Zbl 0789.06012
[9] L. V. Kantorovitch, Concerning the problem of moments for finite interval, Dok. Acad. Nauk SSSR 14 (1937), 531-536.
[10] Z. Lipecki, Extension of vector-lattice homomorphisms, Proc. Amer. Math. Soc. 79 (1980), no. 2, 247 – 248. · Zbl 0441.47046
[11] Z. Lipecki, Extensions of positive operators and extreme points. III, Colloq. Math. 46 (1982), no. 2, 263 – 268. Z. Lipecki and W. Thomsen, Extensions of positive operators and extreme points. IV, Colloq. Math. 46 (1982), no. 2, 269 – 273. · Zbl 0432.47020
[12] Z. Lipecki, Extensions of positive operators and extreme points. II, Colloq. Math. 42 (1979), 285 – 289. · Zbl 0432.47019
[13] Z. Lipecki, D. Plachky, and W. Thomsen, Extensions of positive operators and extreme points. I, Colloq. Math. 42 (1979), 279 – 284. · Zbl 0432.47018
[14] Z. Lipecki, Extensions of positive operators and extreme points. III, Colloq. Math. 46 (1982), no. 2, 263 – 268. Z. Lipecki and W. Thomsen, Extensions of positive operators and extreme points. IV, Colloq. Math. 46 (1982), no. 2, 269 – 273. · Zbl 0432.47020
[15] Z. Lipecki, Extension of vector-lattice homomorphisms revisited, Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 2, 229 – 233. · Zbl 0589.46002
[16] 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. · Zbl 0425.46006
[17] Mohamed Ali Toumi, On some \?-subalgebras of a \?-algebra, Math. Rep. (Bucur.) 4(54) (2002), no. 3, 303 – 310 (2003). · Zbl 1072.06015
[18] M. A. Toumi, Structure theorem for d-algebras, submitted. · Zbl 1212.06044
[19] A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. · Zbl 0519.46001
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