Buskes, Gerard; Page, Robert A positive note on a counterexample by Arens. (English) Zbl 1080.47003 Quaest. Math. 28, No. 1, 117-121 (2005). Summary: In his seminal paper [Proc. Am. Math. Soc. 2, 839–848 (1952; Zbl 0044.32601)], R. Arens gave an example of a positive bilinear map \(P:\ell^1 \times\ell^1\to \mathbb R\) that is not regular. In this note, we show that for Banach lattices \(E\) such a bilinear non-regular map \(E\times E\to \mathbb R\) exists if and only if \(\ell^1\) does not embed into \(E\). Cited in 2 Documents MSC: 47A07 Forms (bilinear, sesquilinear, multilinear) 47B60 Linear operators on ordered spaces 46B42 Banach lattices 46J99 Commutative Banach algebras and commutative topological algebras 47B65 Positive linear operators and order-bounded operators Keywords:Arens triadjoint; bilinear map of order bounded variation; Banach lattices Citations:Zbl 0044.32601 PDFBibTeX XMLCite \textit{G. Buskes} and \textit{R. Page}, Quaest. Math. 28, No. 1, 117--121 (2005; Zbl 1080.47003) Full Text: DOI