About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Chil, Elmiloud
On Riesz homomorphisms in unital $f$-algebras. (English). Mathematica Bohemica, vol. 134 (2009), issue 2, pp. 121-131
MSC: 06F25, 46A40 | MR 2535141 | Zbl 1212.06043 | DOI: 10.21136/MB.2009.140648
Keywords:
vector lattice; $d$-algebra; $f$-algebra
Summary:
The main topic of the first section of this paper is the following theorem: let $A$ be an Archimedean $f$-algebra with unit element $e$, and $T\: A\rightarrow A$ a Riesz homomorphism such that $T^2(f)=T(fT(e))$ for all $f\in A$. Then every Riesz homomorphism extension $\widetilde T$ of $T$ from the Dedekind completion $A^{\delta }$ of $A$ into itself satisfies $\widetilde T^2(f)=\widetilde T(fT(e))$ for all $f\in A^{\delta }$. In the second section this result is applied in several directions.\ As a first application it is applied to show a result about extensions of positive projections to the Dedekind completion. A second application of the above result is a new approach to the Dedekind completion of commutative $d$-algebras.
References:
[1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators. Academic Press, Orlando (1985). MR 0809372 | Zbl 0608.47039
[2] Bernau, S. J., Huijsmans, C. B.: Almost $f$-algebras and $d$-algebras. Math. Proc. Camb. Philos. Soc. 107 (1990), 208-308. DOI 10.1017/S0305004100068560 | MR 1027782 | Zbl 0707.06009
[3] Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et anneaux réticulés. Lect. Notes Math., 608, Springer (1977). MR 0552653 | Zbl 0384.06022
[4] Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc., Providence, R.I. (1967). MR 0227053 | Zbl 0153.02501
[5] Birkhoff, G., Pierce, R. S.: Lattice-ordered rings. Anais. Acad. Brasil. Cienc. 28 (1956), 41-69. MR 0080099 | Zbl 0070.26602
[6] Boulabiar, K., Chil, E.: On the structure of Archimedean almost $f$-algebras. Demonstr. Math. 34 (2001), 749-760. MR 1869777
[7] Buskes, G., van Rooij, A.: Almost $f$-algebras: Structure and the Dedekind completion. Positivity 4 (2000), 233-243. DOI 10.1023/A:1009874426887 | MR 1797126 | Zbl 0967.46008
[8] Hager, A. W., Robertson, L. C.: Representing and ringifying a Riesz space. Symposia Math. 21 (1977), 411-431. MR 0482728 | Zbl 0382.06018
[9] Huijsmans, C. B., de Pagter, B.: Averaging operators and positive contractive projections. J. Math. Anal. Appl. 113 (1986), 163-184. DOI 10.1016/0022-247X(86)90340-9 | MR 0826666 | Zbl 0604.47024
[10] Huijsmans, C. B.: Lattice-ordered algebras and $f$-algebras: A survey. Positive operators, Riesz Spaces and Economics. Stud. Econ. Theory 2 (1991), 151-169. DOI 10.1007/978-3-642-58199-1_7 | MR 1307423
[11] Huijsmans, C. B., de Pagte, B.: Subalgebras and Riesz subspaces of an $f$-algebra. Proc. London, Math. Soc., III. Ser. 48 (1984), 161-174. DOI 10.1112/plms/s3-48.1.161 | MR 0721777
[12] Luxembourg, W. A. J., Zaanen, A. C.: Riesz spaces I. North-Holland, Amsterdam (1971).
[13] de Pagter, G.: The space of extended orthomorphisms in a Riesz space. Pacific J. Math. 112 (1984), 193-210. DOI 10.2140/pjm.1984.112.193 | MR 0739146 | Zbl 0541.46006
[14] Triki, A.: Extensions of positive projections and averaging operators. J. Math. Anal. Appl. 153 (1990), 486-496. DOI 10.1016/0022-247X(90)90227-7 | MR 1080661 | Zbl 0727.47021
[15] Zaanen, A. C.: Riesz Space II. North-Holland, Amsterdam (1983). MR 0704021
Partner of
EuDML logo