## 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
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.