Completely coarse maps are $\mathbb R$-linear

TL;DR

This paper proves that completely coarse maps between operator spaces are necessarily real-linear and introduces a weaker embeddability notion that still preserves key structural properties.

Contribution

It establishes that completely coarse maps are $ ext{R}$-linear and introduces a weaker embeddability concept with significant implications for operator space theory.

Findings

All completely coarse maps are $ ext{R}$-linear.

The introduced embeddability is weaker than complete $ ext{R}$-isomorphic embeddability.

If an infinite-dimensional operator space embeds into $ ext{OH}$ in this weaker sense, it is completely isomorphic to $ ext{OH}$.

Abstract

A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be $\mathbb R$-linear. On the opposite direction of this result, we introduce a notion of embeddability between operator spaces and show that this notion is strictly weaker than complete $\mathbb R$-isomorphic embeddability (in particular, weaker than complete $\mathbb C$-isomorphic embeddability). Although weaker, this notion is strong enough for some applications. For instance, we show that if an infinite dimensional operator space $X$ embeds in this weaker sense into Pisier's operator space $\mathrm{OH}$, then $X$ must be completely isomorphic to $\mathrm{OH}$.