Towards a theory of coarse geometry of operator spaces

TL;DR

This paper introduces new notions of coarse embeddability between operator spaces, demonstrating they are weaker than isomorphic embeddability but still impose restrictions on the linear structures of the spaces.

Contribution

It defines two novel coarse embeddability concepts for operator spaces and explores their properties and implications, expanding the understanding of nonlinear embeddings in operator space theory.

Findings

New notions of coarse embeddability are strictly weaker than isomorphic embeddability.

Existence of such embeddings restricts the linear operator space structures.

Examples of nonlinear equivalences between operator spaces are provided.

Abstract

We introduce two notions of coarse embeddability between operator spaces: almost complete coarse embeddability of bounded subsets and spherically-complete coarse embeddability. We provide examples showing that these notions are strictly weaker than complete isomorphic embeddability - in fact, they do not even imply isomorphic embeddability. On the other hand, we show that, despite their nonlinearity, the existence of such embeddings provides restrictions on the linear operator space structures of the spaces. Examples of nonlinear equivalences between operator spaces are also provided.