Asymptotic dimension and coarse embeddings in the quantum setting

Abstract

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].