Banach Journal of Mathematical Analysis

Duality for ideals of Lipschitz maps

M. G. Cabrera-Padilla, J. A. Chávez-Domínguez, A. Jiménez-Vargas, and Moisés Villegas-Vallecillos

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Abstract

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.

Article information

Source
Banach J. Math. Anal., Volume 11, Number 1 (2017), 108-129.

Dates
Received: 5 November 2015
Accepted: 25 February 2016
First available in Project Euclid: 10 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1478746989

Digital Object Identifier
doi:10.1215/17358787-3764290

Mathematical Reviews number (MathSciNet)
MR3571147

Zentralblatt MATH identifier
1368.46022

Subjects
Primary: 46B28: Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]
Secondary: 26A16: Lipschitz (Hölder) classes 46E15: Banach spaces of continuous, differentiable or analytic functions 47L20: Operator ideals [See also 47B10]

Keywords
Lipschitz map tensor product $p$-summing operator duality ideal

Citation

Cabrera-Padilla, M. G.; Chávez-Domínguez, J. A.; Jiménez-Vargas, A.; Villegas-Vallecillos, Moisés. Duality for ideals of Lipschitz maps. Banach J. Math. Anal. 11 (2017), no. 1, 108--129. doi:10.1215/17358787-3764290. https://projecteuclid.org/euclid.bjma/1478746989


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References

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