$$f$$ -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

Positivity , Volume 18 (1) – Apr 11, 2013

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Springer Basel
Copyright © 2013 by Springer Basel
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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