Positivity 4: 227–231, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. Almost f -algebras: Commutativity and the Cauchy-Schwarz Inequality 1 2 G. BUSKES and A. VAN ROOIJ Department of Mathematics, University of Mississippi, Mississippi 38677, USA Mathematical Institute, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands In 1981 Scheffold in  proved that every normed almost f -algebra is commut- ative. Basly and Triki in  were able to do away with the norm condition. Both the proof of Scheffold and the proof of Basly and Triki make use of the Axiom of Choice by using certain nonconstructive representation theorems. Bernau and Huijsmans in  gave a constructive proof. It is long and quite involved, how- ever. In this paper we present a short and constructive proof. Interestingly, it deals with bilinear maps rather than algebra multiplication and does not make use of associativity. Theorem 1, that is basic for our results, is an extension of Satz 1 in . All Riesz spaces occurring in this paper are Archimedean. 1 is the constant function with value 1. If E is a Riesz space, a bilinear map A of E E into a vector space F
Positivity – Springer Journals
Published: Oct 16, 2004
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