Completely coarse maps are \mathbb R-linear

2020/06/01 by Bruno M. Braga, Braga, Bruno M., Javier Alejandro Chávez‐Domínguez +1
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2006.00948

openalex publication_date 2020/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 OH, then X must be completely isomorphic to OH.

Related

• Towards a theory of coarse geometry of operator spaces
• Coarse geometry of operator spaces and complete isomorphic embeddings into ℓ₁ and c₀-sums of operator spaces
• On the small scale nonlinear theory of operator spaces
• On the equivalence between coarse and uniform embeddability of\n quasi-Banach spaces into a Hilbert space
• On the expansiveness of coarse maps between Banach spaces and geometry preservation