zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
46A40Ordered topological linear spaces, vector lattices
46H05General theory of topological algebras
06F25Ordered algebraic structures
47B60Operators on ordered spaces
WorldCat.org
Full Text: DOI
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