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We prove that the class of reflexive asymptotic-$$c_{0}$$ Banach spaces is coarsely rigid, meaning that if a Banach space $$X$$ coarsely embeds into a reflexive asymptotic-$$c_{0}$$ space $$Y$$, then $$X$$ is also reflexive and asymptotic-$$c_{0}$$. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$$c_{0}$$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.
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Asymptotic dimension and coarse embeddings in the quantum setting
We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver [A von Neumann algebra approach to quantum metrics, Mem. Am. Math. Soc. 215(1010) (2012) 1–80]. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isoperimetric one from [K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf and F. Verstraete, The [Formula: see text]-divergence and mixing times of quantum Markov processes, J. Math. Phys. 51(12) (2010) 122201].
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- Award ID(s):
- 1900985
- PAR ID:
- 10636875
- Publisher / Repository:
- World Scientific
- Date Published:
- Journal Name:
- Journal of Topology and Analysis
- Volume:
- 15
- Issue:
- 02
- ISSN:
- 1793-5253
- Page Range / eLocation ID:
- 545 to 565
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S1793525321500382
