$$f$$ f -Representation of a function algebra

$$f$$ f -Representation of a function algebra We investigate representations $$\Phi :A\longrightarrow \mathcal {L}_{b}(X)$$ Φ : A ⟶ L b ( X ) , where $$A$$ A is a unital function algebra and $$\mathcal {L}_{b}(X)$$ L b ( X ) is the space of all order bounded operators on a vector lattice $$X$$ X . Given an element $$x\in X$$ x ∈ X , the orbit space $$\Phi \left[ x\right] $$ Φ x generated by $$\Phi $$ Φ at $$x$$ x is the subspace $$\begin{aligned} \Phi \left[ x\right] =\left\{ \Phi (a)(x):a\in A\right\} . \end{aligned}$$ Φ x = Φ ( a ) ( x ) : a ∈ A . In this paper we make a detailed study of the orbite space $$\Phi \left[ x\right] , x\in X$$ Φ x , x ∈ X . It turn out that they are vector lattices with a weak order unit. Moreover, It is proved that for any representation $$\Phi :A\longrightarrow \mathcal {L}_{b}(X)$$ Φ : A ⟶ L b ( X ) can be extended to a representation of the order bidual $$A^{\sim }{}^{\sim }$$ A ∼ ∼ of $$A$$ A . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

$$f$$ f -Representation of a function algebra

Positivity , Volume 19 (4) – Jan 29, 2015

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Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0325-9
Publisher site
See Article on Publisher Site

Abstract

We investigate representations $$\Phi :A\longrightarrow \mathcal {L}_{b}(X)$$ Φ : A ⟶ L b ( X ) , where $$A$$ A is a unital function algebra and $$\mathcal {L}_{b}(X)$$ L b ( X ) is the space of all order bounded operators on a vector lattice $$X$$ X . Given an element $$x\in X$$ x ∈ X , the orbit space $$\Phi \left[ x\right] $$ Φ x generated by $$\Phi $$ Φ at $$x$$ x is the subspace $$\begin{aligned} \Phi \left[ x\right] =\left\{ \Phi (a)(x):a\in A\right\} . \end{aligned}$$ Φ x = Φ ( a ) ( x ) : a ∈ A . In this paper we make a detailed study of the orbite space $$\Phi \left[ x\right] , x\in X$$ Φ x , x ∈ X . It turn out that they are vector lattices with a weak order unit. Moreover, It is proved that for any representation $$\Phi :A\longrightarrow \mathcal {L}_{b}(X)$$ Φ : A ⟶ L b ( X ) can be extended to a representation of the order bidual $$A^{\sim }{}^{\sim }$$ A ∼ ∼ of $$A$$ A .

Journal

PositivitySpringer Journals

Published: Jan 29, 2015

References

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