Open Access
January 2017 Duality for ideals of Lipschitz maps
M. G. Cabrera-Padilla, J. A. Chávez-Domínguez, A. Jiménez-Vargas, Moisés Villegas-Vallecillos
Banach J. Math. Anal. 11(1): 108-129 (January 2017). DOI: 10.1215/17358787-3764290


We develop a systematic approach to the study of ideals of Lipschitz maps from a metric space to a Banach space, inspired by classical theory on using Lipschitz tensor products to relate ideals of operator/tensor norms for Banach spaces. We study spaces of Lipschitz maps from a metric space to a dual Banach space that can be represented canonically as the dual of a Lipschitz tensor product endowed with a Lipschitz cross-norm, and we show that several known examples of ideals of Lipschitz maps (Lipschitz maps, Lipschitz p-summing maps, maps admitting Lipschitz factorization through subsets of Lp-space) admit such a representation. Generally, we characterize when the space of a Lipschitz map from a metric space to a dual Banach space is in canonical duality with a Lipschitz cross-norm. Finally, we introduce a concept of operators which are approximable with respect to one of these ideals of Lipschitz maps, and we identify them in terms of tensor-product notions.


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M. G. Cabrera-Padilla. J. A. Chávez-Domínguez. A. Jiménez-Vargas. Moisés Villegas-Vallecillos. "Duality for ideals of Lipschitz maps." Banach J. Math. Anal. 11 (1) 108 - 129, January 2017.


Received: 5 November 2015; Accepted: 25 February 2016; Published: January 2017
First available in Project Euclid: 10 November 2016

zbMATH: 1368.46022
MathSciNet: MR3571147
Digital Object Identifier: 10.1215/17358787-3764290

Primary: 46B28
Secondary: 26A16 , ‎46E15 , 47L20

Keywords: $p$-summing operator , Duality , ideal , Lipschitz map , tensor product

Rights: Copyright © 2017 Tusi Mathematical Research Group

Vol.11 • No. 1 • January 2017
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