Geodesic
On Partially Ordering Operator Algebras
Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 636-643

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view. 
Demarr, Ralph. On Partially Ordering Operator Algebras. Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 636-643. doi: 10.4153/CJM-1967-057-6
@article{10_4153_CJM_1967_057_6,
     author = {Demarr, Ralph},
     title = {On {Partially} {Ordering} {Operator} {Algebras}},
     journal = {Canadian journal of mathematics},
     pages = {636--643},
     year = {1967},
     volume = {19},
     number = {1},
     doi = {10.4153/CJM-1967-057-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1967-057-6/}
}
TY  - JOUR
AU  - Demarr, Ralph
TI  - On Partially Ordering Operator Algebras
JO  - Canadian journal of mathematics
PY  - 1967
SP  - 636
EP  - 643
VL  - 19
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1967-057-6/
DO  - 10.4153/CJM-1967-057-6
ID  - 10_4153_CJM_1967_057_6
ER  - 
%0 Journal Article
%A Demarr, Ralph
%T On Partially Ordering Operator Algebras
%J Canadian journal of mathematics
%D 1967
%P 636-643
%V 19
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1967-057-6/
%R 10.4153/CJM-1967-057-6
%F 10_4153_CJM_1967_057_6

[1] 1. Birkhoff, G., Lattice theory, Amer. Math. Soc., Colloquium Publications, vol. 25, rev. ed. (1948). Google Scholar

[2] 2. DeMarr, R. E., Order convergence in linear topological spaces, Pacific J. Math., 14 (1964), 17–20. Google Scholar

[3] 3. DeMarr, R. E., Partially ordered linear spaces and locally convex linear topological spaces, Illinois J. Math., 8 (1964), 601–606. Google Scholar

[4] 4. Kadison, R. V., A representation theory for commutative topological algebras, Mem. Amer. Math. Soc., No. 7 (1951). Google Scholar

[5] 5. Kantorovich, L. V., Pinsker, A. G., and Vulikh, B. Z., Functional analysis in semi-ordered spaces (Moscow-Leningrad, 1950). Google Scholar

[6] 6. McShane, E. J., Order-preserving maps and integration processes, Ann. of Math. Studies, No. 31 (1953). Google Scholar

[7] 7. Nakano, H., Modern spectral theory (Tokyo, 1950). Google Scholar

[8] 8. Namioka, I., Partially ordered linear topological spaces, Mem. Amer. Math. Soc., No. 24 (1957). Google Scholar

[9] 9. Vulikh, B. Z., Introduction to the theory of semi-ordered spaces (in Russian; Moscow, 1961). Google Scholar

Cité par Sources :