Completely coarse maps are ℝ-linear

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 R {\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 R {\mathbb {R}} -isomorphic embeddability (in particular, weaker than complete C {\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 X embeds in this weaker sense into Pisier’s operator space O H \mathrm {OH} , then X X must be completely isomorphic to O H \mathrm {OH} .

Towards a theory of coarse geometry of operator spaces

“…This paper continues the investigation of the nonlinear theory of operator spaces which was initiated in [BCD21] and continued in [BCDS21]. Operator spaces are Banach subspaces of the space of bounded operators on some (complex) Hilbert space H, denoted by B(H); so operator space theory is often regarded as noncommutative Banach space theory.…”

“…Mn(Y ) ≤ s for all n ∈ N and all [x ij ] n i,j=1 , [z ij ] n i,j=1 ∈ M n (X) -for n = 1, this is precisely the definition of a coarse map between Banach spaces. As shown in [BCD21], the study of completely coarse maps between operator spaces does not lead to an interesting theory. Precisely: Theorem 1.1.…”

“…One way to overcome this issue is, instead of considering a single function with equi-coarse amplifications, to consider a sequence (f n : X → Y ) n whose amplifications (f n n : M n (X) → M n (Y )) n are equi-coarse -i.e., each f n does not need to be completely coarse, but only to "behave well" until its n-th amplification. Consider the following linear notion (see [BCD21,Definition 4.1]): X almost completely isomorphically embeds into Y if there is a sequence (f n : X → Y ) n of linear maps whose amplifications (f n n : M n (X) → M n (Y )) n are equi-isomorphic embeddings, i.e., there is D ≥ 1 such that…”

“…for all n ∈ N and all [x ij ] n i,j=1 ∈ M n (X). This notion is clearly weaker than complete isomorphic embeddability and, as shown in [BCD21,Theorem 4.2], it is actually strictly weaker. Moreover, this has a natural nonlinearization: X almost completely coarsely embeds into Y if there is a sequence (f n : X → Y ) n whose amplifications (f n n : M n (X) → M n (Y )) n are equi-coarse embeddings, i.e., (f n n ) n is both equi-coarse and equi-expanding: for each s > 0 there is r > 0 such that…”

“…(I) the nonlinear embedding must still be strong enough to recover some linear aspects of the operator space structures of the spaces, and (II) the nonlinear embedding must be strictly weaker than (almost) complete coarse embeddability. The results of [BCD21,BCDS21] show that almost complete coarse embeddability satisfies both (I) and (II). 1 As a sample example for (I): it was shown in [BCD21, Theorem 1.2] that if an infinite dimensional operator space X almost completely coarsely embeds into G. Pisier's operator Hilbert space OH, then X must be completely isomorphic to OH.…”