Geodesic
The triadjoint of an orthosymmetric bimorphism
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 85-94 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Let $A$ and $B$ be two Archimedean vector lattices and let $( A^{\prime }) _n'$ and $( B') _n'$ be their order continuous order biduals. If $\Psi \colon A\times A\rightarrow B$ is a positive orthosymmetric bimorphism, then the triadjoint $\Psi ^{\ast \ast \ast }\colon ( A') _n'\times ( A') _n'\rightarrow ( B') _n'$ of $\Psi $ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost $f$-algebras.
Let $A$ and $B$ be two Archimedean vector lattices and let $( A^{\prime }) _n'$ and $( B') _n'$ be their order continuous order biduals. If $\Psi \colon A\times A\rightarrow B$ is a positive orthosymmetric bimorphism, then the triadjoint $\Psi ^{\ast \ast \ast }\colon ( A') _n'\times ( A') _n'\rightarrow ( B') _n'$ of $\Psi $ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost $f$-algebras.
Classification : 06F25, 47B65
Keywords: almost $f$-algebra orthosymmetric bimorphism 
@article{CMJ_2010_60_1_a5,
     author = {Toumi, Mohamed Ali},
     title = {The triadjoint of an orthosymmetric bimorphism},
     journal = {Czechoslovak Mathematical Journal},
     pages = {85--94},
     year = {2010},
     volume = {60},
     number = {1},
     mrnumber = {2595072},
     zbl = {1224.06036},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_1_a5/}
}
TY  - JOUR
AU  - Toumi, Mohamed Ali
TI  - The triadjoint of an orthosymmetric bimorphism
JO  - Czechoslovak Mathematical Journal
PY  - 2010
SP  - 85
EP  - 94
VL  - 60
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2010_60_1_a5/
LA  - en
ID  - CMJ_2010_60_1_a5
ER  - 
%0 Journal Article
%A Toumi, Mohamed Ali
%T The triadjoint of an orthosymmetric bimorphism
%J Czechoslovak Mathematical Journal
%D 2010
%P 85-94
%V 60
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2010_60_1_a5/
%G en
%F CMJ_2010_60_1_a5
Toumi, Mohamed Ali. The triadjoint of an orthosymmetric bimorphism. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 85-94. http://geodesic.mathdoc.fr/item/CMJ_2010_60_1_a5/

[1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators. Academic Press. Orlando (1985). | MR | Zbl

[2] Arens, R.: The adjoint of a bilinear operation. Proc. Amer. Math. Soc. 2 (1951), 839-848. | DOI | MR | Zbl

[3] Basly, M., Triki, A.: ${ff}$-algèbres réticulées. Preprint, Tunis (1988). | MR

[4] Bernau, S. J., Huijsmans, C. B.: The order bidual of almost $f$-algebras and $d$-algebras. Trans. Amer. Math. Soc. 347 (1995), 4259-4275. | MR | Zbl

[5] Bernau, S. J., Huijsmans, C. B.: Almost $f$-algebras and $d$-algebras. Math. Proc. Cam. Phil. Soc. 107 (1990), 287-308. | DOI | MR | Zbl

[6] Buskes, G., Rooij, A. Van: Almost $f$-algebras: Structure and the Dedekind completion. Positivity 3 (2000), 233-243. | DOI | MR

[7] Buskes, G., Rooij, A. Van: Almost $f$-algebras: commutativity and Cauchy Schwartz inequality. Positivity 3 (2000), 227-131. | DOI | MR

[8] Grobler, J. J.: Commutativity of the Arens product in lattice ordered algebras. Positivity 3 (1999), 357-364. | DOI | MR | Zbl

[9] Huijsmans, C. B.: Lattice-ordered Algebras and $f$-algebras: A survey in Studies in Economic Theory 2. Springer Verlag, Berlin-Heidelberg-New York (1990), 151-169. | MR

[10] Luxemburg, W. A. J., Zaanen, A. C.: Riesz Spaces I. North-Holland. Amsterdam (1971).

[11] Meyer-Nieberg, P.: Banach Lattices. Springer-Verlag, Berlin Heidelberg New York (1991). | MR | Zbl

[12] Schaefer, H. H.: Banach Lattices and Positive Operators. Grundlehren. Math. Wiss. Vol. 215, Springer, Berlin (1974). | MR | Zbl

[13] Scheffold, E.: ${ff}$-Banachverbandsalgebren. Math. Z. 177 (1981), 183-205. | MR | Zbl

[14] Triki, A.: On algebra homomorphism in complex $f$-algebras. Comment. Math. Univ. Carolin. 43 (2002), 23-31. | MR

[15] Zaanen, A. C.: Riesz spaces II. North-Holland, Amsterdam (1983). | MR | Zbl