Simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in Archimedean almost \(f\)-algebras. (English) Zbl 1199.06073
Summary: The paper presents simple proofs of the Cauchy-Schwartz inequality and the negative discriminant property in Archimedean almost \(f\)-algebras, based on a sequence approximation.
MSC:
| 06F25 | Ordered rings, algebras, modules |
| 47B65 | Positive linear operators and order-bounded operators |
Keywords:
Cauchy-Schwartz inequality; \(f\)-algebra; almost \(f\)-algebra; positive square \(l\)-algebraReferences:
| [1] | Aliprantis, C. D. and Burkinshaw, O., Positive Operators, Academic Press, Orlando, 1985. · Zbl 0608.47039 |
| [2] | Bernau, S. J. and Huijsmans, C. B., The Schwartz Inequality in Archimedean f-algebras, Indag. Math., 7:2(1996), 137–148. · Zbl 0861.46002 · doi:10.1016/0019-3577(96)85085-3 |
| [3] | Beukers, F., Huijsmans, C. B. and de Pagter, B., Unital Embedding and Complexification of f-algebras, Math. Z., 183(1983) 131–144. · Zbl 0494.06010 · doi:10.1007/BF01187219 |
| [4] | Boulabiar, K. and Toumi, M. A., Lattice Bimorphisms on f-algebras, Algebra Universalis, 48(2002), 103–116. · Zbl 1059.06013 · doi:10.1007/s00012-002-8206-z |
| [5] | Buskes, G. and Van Rooij, A., Almost f-algebras: Commutativity and the Cauchy-Schwarz Inequality, Positivity, 4:3, (2000), 227–231. · Zbl 0987.46002 · doi:10.1023/A:1009826510957 |
| [6] | Huijsmans, C. B. and de Pagter, B., Averaging Operators and Positive Contractive Projections, J. Math. Anal. Appl., 113(1986), 163–184. · Zbl 0604.47024 · doi:10.1016/0022-247X(86)90340-9 |
| [7] | Luxemburg, W. A. J. and Zaanen, A. C., Riesz Spaces I, North-Holland, Amsterdam, 1971. · Zbl 0231.46014 |
| [8] | de Pagter, B., f-algebras and Orthomorphisms, Thesis Leiden, University, 1981. |
| [9] | Toumi, M. A., Commutativity of Almost f-algebras, Positivity, 22:2(2007), 357–368. · Zbl 1119.06013 · doi:10.1007/s11117-005-0038-6 |
| [10] | Vismanathan, T. M., Ordered Modules of Fractions, J. Reine Angew Math., 235(1969), 78–107. · Zbl 0172.32002 · doi:10.1515/crll.1969.235.78 |
| [11] | Zaanen, A. C., Riesz spaces II, North-Holland, Amsterdam, 1983. · 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.
