The order bidual of almost $f$-algebras and $d$-algebras
HTML articles powered by AMS MathViewer
- by S. J. Bernau and C. B. Huijsmans PDF
- Trans. Amer. Math. Soc. 347 (1995), 4259-4275 Request permission
Abstract:
It is shown in this paper that the second order dual $A''$ of an Archimedean (almost) $f$-algebra $A$, equipped with the Arens multiplication, is again an (almost) $f$-algebra. Also, the order continuous bidual $(Aβ)_nβ$ of an Archimedean $d$-algebra $A$ is a $d$-algebra. Moreover, if the $d$-algebra $A$ is commutative or has positive squares, then $A''$ is again a $d$-algebra.References
- Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR 809372
- Richard Arens, Operations induced in function classes, Monatsh. Math. 55 (1951), 1β19. MR 44109, DOI 10.1007/BF01300644
- Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839β848. MR 45941, DOI 10.1090/S0002-9939-1951-0045941-1
- S. J. Bernau and C. B. Huijsmans, Almost $f$-algebras and $d$-algebras, Math. Proc. Cambridge Philos. Soc. 107 (1990), no.Β 2, 287β308. MR 1027782, DOI 10.1017/S0305004100068560
- C. B. Huijsmans, The order bidual of lattice ordered algebras. II, J. Operator Theory 22 (1989), no.Β 2, 277β290. MR 1043728 β, Lattice ordered algebras and $f$-algebras: A survey, Studies in Economic Theory 2, Positive Operators, Riesz Spaces and Economics (C. D. Aliprantis, K. C. Border and W. A. J. Luxemburg, eds.), Springer, Berlin, 1991, pp. 151-169.
- C. B. Huijsmans and B. de Pagter, The order bidual of lattice ordered algebras, J. Funct. Anal. 59 (1984), no.Β 1, 41β64. MR 763776, DOI 10.1016/0022-1236(84)90052-1
- W. A. J. Luxemburg, Notes on Banach function spaces. XIVa, Nederl. Akad. Wetensch. Proc. Ser. A 68=Indag. Math. 27 (1965), 229β239. MR 0188766, DOI 10.1016/S1385-7258(65)50028-7 W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. I, North-Holland, Amsterdam, 1971.
- Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093, DOI 10.1007/978-3-642-76724-1
- Egon Scheffold, Der Bidual von $F$-Banachverbandsalgebren, Acta Sci. Math. (Szeged) 55 (1991), no.Β 1-2, 167β179 (German). MR 1124955
- Egon Scheffold, Γber den ordnungsstetigen Bidual von $FF$-Banachverbandsalgebren, Arch. Math. (Basel) 60 (1993), no.Β 5, 473β477 (German). MR 1213518, DOI 10.1007/BF01202314
- Egon Scheffold, Γber Bimorphismen und das Arens-Produkt bei kommutativen $D$-Banachverbandsalgebren, Rev. Roumaine Math. Pures Appl. 39 (1994), no.Β 3, 259β270 (German, with English summary). MR 1315492
- 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
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 4259-4275
- MSC: Primary 46H05; Secondary 46A40, 46B42
- DOI: https://doi.org/10.1090/S0002-9947-1995-1308002-0
- MathSciNet review: 1308002