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A positive note on a counterexample by Arens. (English) Zbl 1080.47003

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 \(E\).

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

Citations:

Zbl 0044.32601
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