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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 $(A')'\sb n$ 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 $(A')'\sb n=A''$, the whole second order dual of A. Necessary and sufficient conditions are derived for $(A')'\sb n$ to be semiprime and to have a unit element respectively. It is shown that $(A')'\sb n$ is semiprime if and only if the annihilator of $\{a\in A:\vert a\vert\le bc$ for some $b,c\in A\sp+\}$ is trivial. If A is semiprime and satisfies the so-called Stone condition, then $(A')'\sb {n'}$ is semiprime if and only if A has a weak approximate unit. Furthermore, $(A')'\sb n$ 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 $(A')'\sb n$.

##### MSC:
 46A40 Ordered topological linear spaces, vector lattices 46H05 General theory of topological algebras 06F25 Ordered algebraic structures 47B60 Operators on ordered spaces
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##### 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, 839-848 (1951) · Zbl 0044.32601 [3] Beukers, F.; Huijsmans, C. B.; De Pagter, B.: Unital embedding and complexification of f-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) · Zbl 0384.06022 [5] Bonsall, F. F.; Duncan, J.: Complete normed algebras. Ergeb. math. Grenzgeb. no. 80 (1973) · 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) · Zbl 0526.46010 [9] Duncan, J.; Hosseiniun, S. A. R: The second dual of a Banach algebra. Proc. roy. Soc. Edinburgh sect. A 84, 309-325 (1979) · Zbl 0427.46028 [10] Gould, G. G.; Mahowald, M.: Measures on completely regular spaces. J. London math. Soc. 37, 103-111 (1962) · 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 f-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 f-algebra. Proc. London math. Soc. 48, 161-174 (1984) · Zbl 0534.46010 [15] Luxemburg, W. A. J; Zaanen, A. C.: Riesz spaces I. (1971) · Zbl 0231.46014 [16] Maté, L.: Embedding multiplier operators of a Banach algebra B into its second conjugate space b\ast\ast. Bull. acad. Polon. sci. Ser. sci. Math. 13, 809-812 (1965) · Zbl 0138.38204 [17] De Pagter, B.: F-algebras and orthomorphisms. Thesis (1981) [18] Schaefer, H. H.: Banach lattices and positive operators. Grundlehren math. Wiss. no. 215 (1974) · Zbl 0296.47023 [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) [20] Zaanen, A. C.: Integration. (1967) · Zbl 0175.05002 [21] Zaanen, A. C.: Riesz spaces II. (1983) · Zbl 0519.46001