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The order bidual of lattice ordered algebras. (English) Zbl 0549.46006

Let A be an Archimedean f-algebra with point separating order dual A’. It is shown that the space \((A')'_ n\) of all order continuous linear functionals on A’ is an Archimedean (and hence commutative!) f-algebra with respect to the Arens multiplication. Moreover, if A has a unit element, then \((A')'_ n=A''\), the whole second order dual of A. Necessary and sufficient conditions are derived for \((A')'_ n\) to be semiprime and to have a unit element respectively. It is shown that \((A')'_ n\) is semiprime if and only if the annihilator of \(\{a\in A:| a|\leq bc\) for some \(b,c\in A^+\}\) is trivial. If A is semiprime and satisfies the so-called Stone condition, then \((A')'_ {n'}\) is semiprime if and only if A has a weak approximate unit. Furthermore, \((A')'_ n\) has a unit element in this case if and only if (amongst others) every positive linear functional on A can be extended positively to its f-algebra \(Orth(A)\) of orthomorphisms. Finally, it is proved that in the latter situation Orth(A) can be embedded in \((A')'_ n\).

MSC:

46A40 Ordered topological linear spaces, vector lattices
46H05 General theory of topological algebras
06F25 Ordered rings, algebras, modules
47B60 Linear operators on ordered spaces
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[1] Arens, R., Operations induced in function classes, Monatsh. Math., 55, 1-19 (1951) · Zbl 0042.35601
[2] Arens, R., The adjoint of a bilinear operation, (Proc. Amer. Math. Soc., 2 (1951)), 839-848 · Zbl 0044.32601
[3] Beukers, F.; Huijsmans, C. B.; de Pagter, B., Unital embedding and complexification of ƒ-algebras, Math. Z., 183, 131-144 (1983) · Zbl 0494.06010
[4] Bigard, A.; Keimel, K.; Wolfenstein, S., Groupes et anneaux réticulés, (Lecture Notes in Math. No. 608 (1977), Springer: Springer Berlin) · Zbl 0384.06022
[5] Bonsall, F. F.; Duncan, J., Complete normed algebras, (Ergeb. Math. Grenzgeb. No. 80 (1973), Springer: Springer Berlin) · Zbl 0271.46039
[6] Civin, P.; Yood, B., The second conjugate space of a Banach algebra as an algebra, Pacific J. Math., 11, 847-870 (1961) · Zbl 0119.10903
[7] Day, M. M., Amenable semigroups, Illinois J. Math., 1, 509-544 (1957) · Zbl 0078.29402
[8] Duhoux, M.; Meyer, M., Extended Orthomorphisms on Archimedean Riesz Spaces, (Rapport 114 (1981), Université de Louvain) · Zbl 0526.46010
[9] Duncan, J.; Hosseiniun, S. A.R, The second dual of a Banach algebra, (Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979)), 309-325 · Zbl 0427.46028
[10] Gould, G. G.; Mahowald, M., Measures on completely regular spaces, J. London Math. Soc., 37, 103-111 (1962), (2) · Zbl 0133.07902
[11] Gulick, S. L., The bidual of a locally multiplicatively-convex algebra, Pacific J. Math., 17, 71-96 (1966) · Zbl 0137.10102
[12] Hewitt, E., Linear functionals on spaces of continuous functions, Fund. Math., 37, 161-189 (1950) · Zbl 0040.06401
[13] Huijsmans, C. B.; de Pagter, B., Ideal theory in ƒ-algebras, Trans. Amer. Math. Soc., 269, 225-245 (1982) · Zbl 0483.06009
[14] Huijsmans, C. B.; de Pagter, B., Subalgebras and Riesz subspaces of an ƒ-algebra, (Proc. London Math. Soc., 48 (1984)), 161-174, (3) · Zbl 0534.46010
[15] Luxemburg, W. A.J; Zaanen, A. C., Riesz Spaces I (1971), North-Holland: North-Holland Amsterdam · Zbl 0231.46014
[16] Maté, L., Embedding multiplier operators of a Banach algebra \(B\) into its second conjugate space
((B^{∗∗}\), Bull. Acad. Polon. Sci. Ser. Sci. Math., 13, 809-812 (1965) · Zbl 0138.38204
[17] de Pagter, B., ƒ-Algebras and Orthomorphisms, Thesis (1981), Leiden
[18] Schaefer, H. H., Banach lattices and positive operators, (Grundlehren Math. Wiss. No. 215 (1974), Springer: Springer Berlin) · Zbl 0291.46008
[19] Synnatschke, J., The operator conjugate to a regular operator and some applications to the question of complete continuity and weak complete continuity of regular operators, Vestnik Leningrad Univ. Math., 60-69 (1972) · Zbl 0234.47035
[20] Zaanen, A. C., Integration (1967), North-Holland: North-Holland Amsterdam · Zbl 0175.05002
[21] Zaanen, A. C., Riesz Spaces II (1983), North-Holland: North-Holland Amsterdam · Zbl 0519.46001
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