Multi-norms and Banach lattices

Abstract

In 2012, Dales and Polyakov introduced the concepts of multi-norms and dual multi-norms based on a Banach space. Particular examples are the lattice multi-norm p} ̈ }Lnq and the dual lattice multi-norm p} ̈ } n q based on a Banach lattice. Here we extend these notions to cover ‘p–multi-norms’ for 1 ď p ď 8, where 8–multi-norms and 1–multi-norms correspond to multinorms and dual multi-norms, respectively. We shall prove two representation theorems. First we modify a theorem of Pisier to show that an arbitrary multi-normed space can be represented as ppY , } ̈ }Lnq : n P Nq, where Y is a closed subspace of a Banach lattice; we then give a version for certain p–multi-norms. Second, we obtain a dual version of this result, showing that an arbitrary dual multi-normed space can be represented as pppX{Y q, } ̈ } n q : n P Nq, where Y is a closed subspace of a Banach lattice X; again we give a version for certain p–multi-norms. We shall discuss several examples of p–multi-norms, including the weak p–summing norm and its dual and the canonical lattice p–multi-norm based on a Banach lattice. We shall determine the Banach spaces E such that the p–sum power-norm based on E is a p–multi-norm. This relies on a famous theorem of Kwapień; we shall present a simplified proof of this result. We shall relate p–multi-normed spaces to certain tensor products. Our representation theorems depend on the notion of ‘strong’ p–multi-norms, and we shall define these and discuss when p–multi-norms and strong p–multi-norms pass to subspaces, quotients, and duals; we shall also consider whether these multi-norms are preserved when we interpolate between couples of p–multi-normed spaces. We shall discuss multi-bounded operators between p–multi-normed spaces, and identify the classes of these spaces in some cases, in particular for spaces of operators between Banach lattices taken with their canonical lattice p–multi-norms. Acknowledgements. The authors are grateful to the London Mathematical Society for the award of Scheme 2 grant 21202 that allowed Troitsky to come to Lancaster in May 2013; to the Fields Institute in Toronto, for invitations to Dales, Laustsen, and Troitsky to participate in the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras in March and April, 2014; to the Lorentz Center in Leiden for invitations to Dales, Laustsen, and Troitsky to participate in a meeting on Ordered Banach Algebras in July, 2014. Oikhberg acknowledges with thanks the support of the Simons Foundation Travel Grant 210060, and Troitsky acknowledges with thanks the support of an NSERC grant. 2000 Mathematics Subject Classification: Primary 46B42, 46B20; secondary 46B28, 46B70.