On Riesz homomorphisms in unital f -algebras

Elmiloud Chil

Mathematica Bohemica (2009)

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

Abstract

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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 → A a Riesz homomorphism such that T 2 ( f ) = T ( f T ( e ) ) for all f ∈ A . Then every Riesz homomorphism extension T ˜ of T from the Dedekind completion A δ of A into itself satisfies T ˜ 2 ( f ) = T ˜ ( f T ( e ) ) for all f ∈ A δ . 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.

How to cite

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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

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  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
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  15. Zaanen, A. C., Riesz Space II, North-Holland, Amsterdam (1983). (1983) MR0704021

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