On the geometry of projective tensor products

Giladi, Ohad; Prochno, Joscha; Schütt, Carsten; Tomczak-Jaegermann, Nicole; Werner, Elisabeth

Abstract

© 2017 Elsevier Inc. In this work, we study the volume ratio of the projective tensor products ℓpn⊗πℓqn⊗πℓrnwith 1≤p≤q≤r≤∞. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kašin type whenever 1≤p≤q≤r≤2 or 1≤p≤2≤r≤∞ and q=2. Also, from the Bourgain–Milman bound on the volume ratio of Banach spaces in terms of their cotype 2 constant, we obtain information on the cotype of these 3-fold projective tensor products. Our results naturally generalize to k-fold products ℓp1n⊗π…⊗πℓpknwith k∈N and 1≤p1≤…≤pk≤∞.

Citation

Giladi, O., Prochno, J., Schütt, C., Tomczak-Jaegermann, N., & Werner, E. (2017). On the geometry of projective tensor products. Journal of functional analysis, 273(2), 471-495. https://doi.org/10.1016/j.jfa.2017.03.019

• Journal Article Type: Article
• Acceptance Date: Mar 31, 2017
• Online Publication Date: Apr 10, 2017
• Publication Date: Jul 15, 2017
• Deposit Date: Jun 29, 2018
• Publicly Available Date: Oct 27, 2022
• Journal: Journal of Functional Analysis
• Print ISSN: 0022-1236
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 273
• Issue: 2
• Pages: 471-495
• DOI: https://doi.org/10.1016/j.jfa.2017.03.019
• Keywords: Tensor product; Volume ratio; Cotype
• Public URL: https://hull-repository.worktribe.com/output/900231
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0022123617301489
• Additional Information: This article is maintained by: Elsevier; Article Title: On the geometry of projective tensor products; Journal Title: Journal of Functional Analysis; CrossRef DOI link to publisher maintained version: http://dx.doi.org/10.1016/j.jfa.2017.03.019; Content Type: article; Copyright: © 2017 Elsevier Inc. All rights reserved.
• Contract Date: Jun 29, 2018