Banach Journal of Mathematical Analysis
- Banach J. Math. Anal.
- Volume 11, Number 1 (2017), 108-129.
Duality for ideals of Lipschitz maps
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 -summing maps, maps admitting Lipschitz factorization through subsets of -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.
Banach J. Math. Anal., Volume 11, Number 1 (2017), 108-129.
Received: 5 November 2015
Accepted: 25 February 2016
First available in Project Euclid: 10 November 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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