$$f$$ -algebras with a $$\sigma$$ -bounded approximate unit

$$f$$ -algebras with a $$\sigma$$ -bounded approximate unit Let $$X$$ be a lattice ordered algebra ( $$\ell$$ -algebra). A positive element $$x\in$$ $$X$$ is said to be totally bounded if $$x^{2}\le x$$ . The $$\ell$$ -algebra $$X$$ is said to have a $$\sigma$$ -bounded approximate unit if for each positive linear functional $$f$$ on $$X$$ the set $$\left\{ f(x)\text{: } x \text{ totally } \text{ bounded }\right\}$$ is bounded in $$\mathbb R$$ . In this paper we study the class of $$f$$ -algebras with a $$\sigma$$ -bounded approximate unit which contains the class of all unital $$f$$ -algebras. In particular It is shown that an $$f$$ -algebra $$X$$ has a $$\sigma$$ -bounded approximate unit if and only if the order bidual $$X^{\sim \sim }$$ is a unital $$f$$ -algebra. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

$$f$$ -algebras with a $$\sigma$$ -bounded approximate unit

, Volume 18 (1) – Apr 11, 2013
10 pages

/lp/springer_journal/f-algebras-with-a-sigma-bounded-approximate-unit-QckVxmoEe7
Publisher
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/s11117-013-0237-5
Publisher site
See Article on Publisher Site

Abstract

Let $$X$$ be a lattice ordered algebra ( $$\ell$$ -algebra). A positive element $$x\in$$ $$X$$ is said to be totally bounded if $$x^{2}\le x$$ . The $$\ell$$ -algebra $$X$$ is said to have a $$\sigma$$ -bounded approximate unit if for each positive linear functional $$f$$ on $$X$$ the set $$\left\{ f(x)\text{: } x \text{ totally } \text{ bounded }\right\}$$ is bounded in $$\mathbb R$$ . In this paper we study the class of $$f$$ -algebras with a $$\sigma$$ -bounded approximate unit which contains the class of all unital $$f$$ -algebras. In particular It is shown that an $$f$$ -algebra $$X$$ has a $$\sigma$$ -bounded approximate unit if and only if the order bidual $$X^{\sim \sim }$$ is a unital $$f$$ -algebra.

Journal

PositivitySpringer Journals

Published: Apr 11, 2013

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