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  1. ; (, Comptes Rendus. Mathématique)
    We initiate the study of the small scale geometry of operator spaces. The authors have previously shown that a map between operator spaces which is completely coarse (that is, the sequence of its amplifications is equi-coarse) must be ℝ -linear. We obtain a generalization of the aforementioned result to completely coarse maps defined on the unit ball of an operator space. By relaxing the condition to a small scale one, we prove that there are many non-linear examples of maps which are completely Lipschitz in small scale. We define a geometric parameter for homogeneous Hilbertian operator spaces which imposes restrictions on the existence of such maps. 
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  2. ; ; (, Mathematische Annalen)
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  3. ; (, Journal of Topology and Analysis)
    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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  4. ; (, Proceedings of the American Mathematical Society)
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  5. ; (, Linear Algebra and its Applications)
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