The Wickstead Problem

A. E. Gutman ; A. G. Kusraev ; S. S. Kutateladze

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 293-333.

Résumé

In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.

Détail

Keywords: Band preserving operator, universally complete vector lattice, $\sigma$-distributive Boolean algebra, local Hamel basis, transcendence basis, derivation, Boolean valued representation.

Bibliographie

[1] Abramovich Yu. A., “Multiplicative representation of disjointness preserving operators”, Indag. Math. (N.S.), 45:3 (1983), 265–279 | MR | Zbl

[2] Abramovich Yu. A., Veksler A. I., and Koldunov A. V., “On disjointness preserving operators”, Dokl. Akad. Nauk SSSR, 248:5 (1979), 1033–1036 | MR

[3] Abramovich Yu. A., Veksler A. I., and Koldunov A. V., “Disjointness-preserving operators, their continuity, and multiplicative representation”, Linear Operators and Their Applications, Leningrad Ped. Inst., Leningrad, 1981 (in Russian)

[4] Abramovich Y. A. and Kitover A. K., “$d$-Independence and $d$-bases in Vector lattices”, Rev. Romaine de Math. Pures et appliquees, 44 (1999), 667–682 | MR | Zbl

[5] Abramovich Y. A. and Kitover A. K., Inverses of Disjointness Preserving Operators, Memoirs Amer. Math. Soc., 679, 2000 | Zbl

[6] Abramovich Y. A. and Kitover A. K., “$d$-Independence and $d$-bases”, Positivity, 7:1 (2003), 95–97 | DOI | MR | Zbl

[7] Abramovich Yu. A. and Wickstead A. W., “The regularity of order bounded operators into $C(K)$ II”, Quart. J. Math. Oxford, Ser. 2, 44 (1993), 57–270 | DOI | MR

[8] Aczél J. and Dhombres J., Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge etc., 1989 | MR | Zbl

[9] Albeverio S., Ayupov Sh. A., and Kudaybergenov K. K., “Derivations on the algebras of measurable operators affiliated with a type I von Neumann algebra”, Siberian Adv. Math. (to appear)

[10] Aliprantis C. D. and Burkinshaw O., Positive Operators, Acad. Press Inc., London, 1985 | MR | Zbl

[11] Ayupov Sh. A. and Kudaybergenov K. K., “Derivations and automorphisms of algebras of bounded operators on Banach–Kantorovich spaces” (to appear)

[12] Azouzi Y., Boulabiar K., and Buskes G., “The de Schipper formula and squares of Riesz spaces”, Indag. Math. (N.S.), 17:4 (2006), 265–279 | DOI | MR

[13] Bell J. L., Boolean Valued Models and Independence Proofs in Set Theory, Clarendon Press, New York etc., 1985, xx+165 pp. | MR | Zbl

[14] Ber A. F., Chilin V. I., and Sukochev F. A., “Derivations in regular commutative algebras”, Math. Notes, 75 (2004), 418–419 | DOI | MR | Zbl

[15] Berberian S. K., Baer $\ast$-Rings, Springer-Verlag, Berlin, 1972, xii+296 pp. | MR | Zbl

[16] Bernau C. B., “Orthomorphisms of Archimedean vector lattices”, Math. Proc. Cambridge Phil. Soc., 89 (1981), 119–126 | DOI | MR

[17] Beukers F., Huijsmans C. B., and de Pagter B., “Unital embedding and complexification of $f$-algebras”, Math. Z, 183 (1983), 131–144 | DOI | MR | Zbl

[18] Bigard A. and Keimel K., “Sur les endomorphismes conservant les polaires d'un groupe réticulé archimédien”, Bull. Soc. Math. France, 97 (1969), 381–398 | MR | Zbl

[19] Bigard A., Keimel K., and Wolfenstein S., Groupes et Anneaux Réticulés, Lecture Notes in Math., 608, Springer-Verlag, Berlin etc., 1977, xi+334 pp. | MR | Zbl

[20] Birkhoff G., Lattice Theory, Amer. Math. Soc. Colloq. Publ., 25, 3rd ed., Providence, 1967 | Zbl

[21] Birkhoff G. and Pierce R. S., “Lattice-ordered rings”, An. Acad. Brasil Ciênc, 28 (1956), 41–68 | MR

[22] Boulabiar K., Buskes G., and Triki A., “Recent results in lattice ordered algebras”, Contemp. Math., 328, Providence, RI, 2003, 99–133 | MR | Zbl

[23] Boulabiar K., Buskes G., and Triki A., “Results in $f$-algebras”, Positivity, 2007, 73–96 | DOI | MR | Zbl

[24] Bourbaki N., Algebra (Polynomials and Fields, Ordered Groups), Hermann, Paris, 1967 (in French)

[25] Buck R. C., “Multiplication operators”, Pacific J. Math., 11 (1961), 95–103 | MR | Zbl

[26] Bukhvalov A. V., “Order bounded operators in vector lattices and spaces of measurable functions”, Itogi Nauki i Tekhniki. Mathematical Analysis, 26, VINITI, Moscow, 1988, 3–63 (in Russian) | MR

[27] Buskes G. and Kusraev A. G., “Representation and extension of orthoregular bilinear operators”, Vladikavkaz Math. J., 9:1 (2007), 16–29 | MR

[28] Buskes G. and Redfield R. H., “Square means and geometric means in lattice-ordered groups and vector lattices” (to appear)

[29] Buskes G. and van Rooij A., “Almost $f$-algebras: commutativity and the Cauchy–Schwarz inequality”, Positivity, 4:3 (2000), 227–231 | DOI | MR | Zbl

[30] Buskes G. and van Rooij A., “Squares of Riesz spaces”, Rocky Mountain J. Math., 31:1 (2001), 45–56 | DOI | MR | Zbl

[31] Chilin V. I., “Partially ordered involutive Baer algebras”, Contemporary Problems of Mathematics. Newest Advances, 27, VINITI, Moscow, 1985, 99–128 (in Russian) | MR

[32] Conrad P. F. and Diem J. E., “The ring of polar preserving endomorphisms of an abelian lattice-ordered group”, Illinois J. Math., 15 (1971), 222–240 | MR | Zbl

[33] Duhoux M. and Meyer M., “A new proof of the lattice structure of orthomorphisms”, J. London Math. Soc., 25:2 (1982), 375–378 | DOI | MR | Zbl

[34] Goodearl K. R., Von Neumann Regular Rings, Pitman, London, 1979 | MR | Zbl

[35] Gordon E. I., “Real numbers in Boolean valued models of set theory and $K$-spaces”, Dokl. Akad. Nauk SSSR, 237:4 (1977), 773–775 | MR | Zbl

[36] Gordon E. I., “$K$-spaces in Boolean valued models of set theory”, Dokl. Akad. Nauk SSSR, 258:4 (1981), 777–780 | MR | Zbl

[37] Gordon E. I., “To the theorems of identity preservation in $K$-spaces”, Sibirsk. Mat. Zh., 23:5 (1982), 55–65 | MR | Zbl

[38] Gordon E. I., Nonstandard Methods in Commutative Harmonic Analysis, Amer. Math. Soc., Providence, RI, 1997 | MR

[39] Gutman A. E., “Banach bundles in the theory of lattice-normed spaces”, Linear Operators Compatible with Order, Sobolev Institute Press, Novosibirsk, 1995, 63–211 (in Russian) | MR | Zbl

[40] Gutman A. E., “Locally one-dimensional $K$-spaces and $\sigma$-distributive Boolean algebras”, Siberian Adv. Math., 5:2 (1995), 99–121 | MR

[41] Gutman A. E., “Disjointness preserving operators”, Vector Lattices and Integral Operators, ed. Kutateladze S. S., Kluwer, Dordrecht etc., 1996, 361–454

[42] Huijsmans C. B. and de Pagter B., “Ideal theory in $f$-algebras”, Trans. Amer. Math. Soc., 269 (1982), 225–245 | MR | Zbl

[43] Huijsmans C. B. and Wickstead A. W., “The inverse of band preserving and disjointness preserving operators”, Indag. Math., 3:2 (1992), 179–183 | DOI | MR | Zbl

[44] Jech T. J., Set Theory, The Third Millennium Edition, Springer, Berlin etc., 2002 | MR

[45] Korol' A. M. and Chilin V. I., “Measurable operators in a Boolean-valued model of set theory”, Dokl. Akad. Nauk UzSSR, 3 (1989), 7–9 (in Russian) | MR | Zbl

[46] Kusraev A. G., Dominated Operators, Kluwer, Dordrecht, 2000 | MR

[47] Kusraev A. G., “On band preserving operators”, Vladikavkaz Math. J., 6:3 (2004), 47–58 (in Russian) | MR | Zbl

[48] Kusraev A. G., “Automorphisms and derivations in the algebra of complex measurable functions”, Vladikavkaz Math. J., 7:3 (2005), 45–49 (in Russian) | MR

[49] Kusraev A. G., “On the structure of orthosymmetric bilinear operators in vector lattices”, Dokl. RAS, 408:1 (2006), 25–27 | MR

[50] Kusraev A. G., “Automorphisms and derivations in extended complex $f$-algebras”, Siberian Math. J., 47:1 (2006), 97–107 | DOI | MR | Zbl

[51] Kusraev A. G., “Analysis, algebra, and logics in operator theory”, Complex Analysis, Operator Theory, and Mathematical Modeling, eds. Korobeĭnik Yu. F. and Kusraev A. G., Vladikavkaz Scientific Center, Vladikavkaz, 2006, 171–204 (in Russian)

[52] Kusraev A. G., “When are all separately band preserving bilinear operators symmetric?”, Vladikavkaz Math. J., 9:2 (2007), 22–25 | MR

[53] Kusraev A. G. and Kutateladze S. S., Boolean Valued Analysis, Nauka, Novosibirsk; Kluwer, Dordrecht, 1999 | MR

[54] Kusraev A. G. and Kutateladze S. S., Introduction to Boolean Valued Analysis, Nauka, Moscow, 2005 (in Russian) | MR

[55] Lambek J., Lectures on Rings and Modules, Blaisdell, Toronto, 1966 | MR | Zbl

[56] Luxemburg W. A. J., Some Aspects of the Theory of Riesz Spaces, Univ. Arkansas Lect. Notes Math., 4, Fayetteville, Arkansas, 1979 | MR

[57] Luxemburg W. A. J. and Schep A., “A Radon-Nikodým type theorem for positive operators and a dual”, Indag. Math., 40 (1978), 357–375 | MR

[58] McPolin P. T. N. and Wickstead A. W., “The order boundedness of band preserving operators on uniformly complete vector lattices”, Math. Proc. Cambridge Philos. Soc., 97:3 (1985), 481–487 | DOI | MR | Zbl

[59] Meyer M., “Le stabilisateur d'un espace vectoriel réticulé”, C. R. Acad. Sci. Paris, Sér. A, 283 (1976), 249–250 | MR | Zbl

[60] Nakano H., “Teilweise geordnete algebra”, Japan J. Math., 17 (1950), 425–511 | MR

[61] Nakano H., “Product spaces of semi-ordered linear spaces”, J. Fac. Sci. Hokkaido Univ., Ser. I, 12 (1953), 163–210 | MR | Zbl

[62] de Pagter B., “A note on disjointness preserving operators”, Proc. Amer. Math. Soc., 90:4 (1984), 543–549 | DOI | MR | Zbl

[63] de Pagter B., “The space of extended orthomorphisms in Riesz space”, Pacific J. Math., 112:1 (1984), 193–210 | MR | Zbl

[64] Sakai S., $C^*$-algebras and $W^*$-algebras, Springer-Verlag, Berlin etc., 1971 | MR | Zbl

[65] Schaefer H. H., Banach Lattices and Positive Operators, Springer-Verlag, Berlin etc., 1974 | MR | Zbl

[66] Sikorski R., Boolean Algebras, Springer-Verlag, Berlin etc., 1964 | MR | Zbl

[67] New York, 1949

[68] Wickstead A. W., “Representation and duality of multiplication operators on Archimedean Riesz spaces”, Compositio Math., 35:3 (1977), 225–238 | MR | Zbl

[69] Wickstead A. W., “Extensions of orthomorphisms”, J. Austral Math. Soc., Ser A, 29 (1980), 87–98 | DOI | MR | Zbl

[70] Wickstead A. W., “The injective hull of an Archimedean $f$-algebra”, Compositio Math., 62:4 (1987), 329–342 | MR | Zbl

[71] Zaanen A. C., “Examples of orthomorphisms”, J. Approx. Theory, 13:2 (1975), 192–204 | DOI | MR | Zbl

[72] Zaanen A. C., Riesz Spaces, v. 2, North-Holland, Amsterdam etc., 1983, 720 pp. | MR | Zbl

[73] Zariski O. and Samuel P., Commutative Algebra, Springer-Verlag, Berlin, 1991