On Riesz homomorphisms in unital -algebras

Elmiloud Chil

Mathematica Bohemica (2009)

  • Volume: 134, Issue: 2, page 121-131
  • ISSN: 0862-7959

Abstract

top
The main topic of the first section of this paper is the following theorem: let be an Archimedean -algebra with unit element , and a Riesz homomorphism such that for all . Then every Riesz homomorphism extension of from the Dedekind completion of into itself satisfies for all . 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 -algebras.

How to cite

top

Chil, Elmiloud. "On Riesz homomorphisms in unital $f$-algebras." Mathematica Bohemica 134.2 (2009): 121-131. <http://eudml.org/doc/38080>.

@article{Chil2009,
abstract = {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.},
author = {Chil, Elmiloud},
journal = {Mathematica Bohemica},
keywords = {vector lattice; $d$-algebra; $f$-algebra; vector lattice; -algebra; -algebra},
language = {eng},
number = {2},
pages = {121-131},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Riesz homomorphisms in unital $f$-algebras},
url = {http://eudml.org/doc/38080},
volume = {134},
year = {2009},
}

TY - JOUR
AU - Chil, Elmiloud
TI - On Riesz homomorphisms in unital $f$-algebras
JO - Mathematica Bohemica
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 134
IS - 2
SP - 121
EP - 131
AB - 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.
LA - eng
KW - vector lattice; $d$-algebra; $f$-algebra; vector lattice; -algebra; -algebra
UR - http://eudml.org/doc/38080
ER -

References

top
  1. Aliprantis, C. D., Burkinshaw, O., Positive Operators, Academic Press, Orlando (1985). (1985) Zbl0608.47039MR0809372
  2. Bernau, S. J., Huijsmans, C. B., 10.1017/S0305004100068560, Math. Proc. Camb. Philos. Soc. 107 (1990), 208-308. (1990) Zbl0707.06009MR1027782DOI10.1017/S0305004100068560
  3. Bigard, A., Keimel, K., Wolfenstein, S., Groupes et anneaux réticulés, Lect. Notes Math., 608, Springer (1977). (1977) Zbl0384.06022MR0552653
  4. Birkhoff, G., Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc., Providence, R.I. (1967). (1967) Zbl0153.02501MR0227053
  5. Birkhoff, G., Pierce, R. S., Lattice-ordered rings, Anais. Acad. Brasil. Cienc. 28 (1956), 41-69. (1956) Zbl0070.26602MR0080099
  6. Boulabiar, K., Chil, E., On the structure of Archimedean almost -algebras, Demonstr. Math. 34 (2001), 749-760. (2001) MR1869777
  7. Buskes, G., van Rooij, A., 10.1023/A:1009874426887, Positivity 4 (2000), 233-243. (2000) Zbl0967.46008MR1797126DOI10.1023/A:1009874426887
  8. Hager, A. W., Robertson, L. C., Representing and ringifying a Riesz space, Symposia Math. 21 (1977), 411-431. (1977) Zbl0382.06018MR0482728
  9. Huijsmans, C. B., de Pagter, B., 10.1016/0022-247X(86)90340-9, J. Math. Anal. Appl. 113 (1986), 163-184. (1986) Zbl0604.47024MR0826666DOI10.1016/0022-247X(86)90340-9
  10. Huijsmans, C. B., 10.1007/978-3-642-58199-1_7, Stud. Econ. Theory 2 (1991), 151-169. (1991) MR1307423DOI10.1007/978-3-642-58199-1_7
  11. Huijsmans, C. B., de Pagte, B., 10.1112/plms/s3-48.1.161, Proc. London, Math. Soc., III. Ser. 48 (1984), 161-174. (1984) MR0721777DOI10.1112/plms/s3-48.1.161
  12. Luxembourg, W. A. J., Zaanen, A. C., Riesz spaces I, North-Holland, Amsterdam (1971). (1971) 
  13. de Pagter, G., 10.2140/pjm.1984.112.193, Pacific J. Math. 112 (1984), 193-210. (1984) Zbl0541.46006MR0739146DOI10.2140/pjm.1984.112.193
  14. Triki, A., 10.1016/0022-247X(90)90227-7, J. Math. Anal. Appl. 153 (1990), 486-496. (1990) Zbl0727.47021MR1080661DOI10.1016/0022-247X(90)90227-7
  15. Zaanen, A. C., Riesz Space II, North-Holland, Amsterdam (1983). (1983) MR0704021

NotesEmbed ?

top

You must be logged in to post comments.