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Maximal Banach ideals of Lipschitz maps. (English) Zbl 1365.46016

Summary: There are known results showing a canonical association between Lipschitz cross-norms (norms on the Lipschitz tensor product of a metric space and a Banach space) and ideals of Lipschitz maps from a metric space to a dual Banach space. We extend this association, relating Lipschitz cross-norms to ideals of Lipschitz maps taking values in general Banach spaces. To do that, we prove a Lipschitz version of the representation theorem for maximal operator ideals. As a consequence, we obtain linear characterizations of some ideals of (nonlinear) Lipschitz maps between metric spaces.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
26A16 Lipschitz (Hölder) classes
46E15 Banach spaces of continuous, differentiable or analytic functions
47L20 Operator ideals