# Commutativity of the Arens product in lattice ordered algebras

Commutativity of the Arens product in lattice ordered algebras Let $${\text{A}}$$ be an Abelian Archimedean lattice ordered algebra. The order bidual $${A''}$$ furnished with the Arens product is again a lattice ordered algebra. We show that the order continuous order bidual $$(A')'_n$$ is Abelian. This solves an open problem and improves a result of Scheffold, who proved it for the case of normed lattice ordered algebras. The proof is based on the ‘up-down-up’ approximation of positive elements in the order continuous order bidual $$(A')'_n$$ by elements in the canonical image $${\widehat A}$$ of $$A$$ in $$(A')'_n$$ Components of positive elements in $${\widehat A}$$ are characterized and the result is applied to the Arens product of $$f$$ -and almost $$f$$ -algebras. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Commutativity of the Arens product in lattice ordered algebras

Positivity, Volume 3 (4) – Oct 22, 2004
8 pages

/lp/springer_journal/commutativity-of-the-arens-product-in-lattice-ordered-algebras-DTVj0vckIE
Publisher
Springer Journals
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.
10.1023/A:1009880911903
Publisher site
See Article on Publisher Site

### Abstract

Let $${\text{A}}$$ be an Abelian Archimedean lattice ordered algebra. The order bidual $${A''}$$ furnished with the Arens product is again a lattice ordered algebra. We show that the order continuous order bidual $$(A')'_n$$ is Abelian. This solves an open problem and improves a result of Scheffold, who proved it for the case of normed lattice ordered algebras. The proof is based on the ‘up-down-up’ approximation of positive elements in the order continuous order bidual $$(A')'_n$$ by elements in the canonical image $${\widehat A}$$ of $$A$$ in $$(A')'_n$$ Components of positive elements in $${\widehat A}$$ are characterized and the result is applied to the Arens product of $$f$$ -and almost $$f$$ -algebras.

### Journal

PositivitySpringer Journals

Published: Oct 22, 2004

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