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Lipschitz geometry of operator spaces and Lipschitz-free operator spaces

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Abstract

We show that there is an operator space notion of Lipschitz embeddability between operator spaces which is strictly weaker than its linear counterpart but which is still strong enough to impose linear restrictions on operator space structures. This shows that there is a nontrivial theory of nonlinear geometry for operator spaces and it answers a question in Braga et al. (Proc Am Math Soc 149(3):1139–1149, 2021). For that, we introduce the operator space version of Lipschitz-free Banach spaces and prove several properties of it. In particular, we show that separable operator spaces satisfy a sort of isometric Lipschitz-lifting property in the sense of Godefroy and Kalton. Gateaux differentiability of Lipschitz maps in the operator space category is also studied.

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Notes

  1. Notice that, if \(f:X\rightarrow Y\) is a map between Banach spaces, the following implications hold: f is linear and bounded \(\Rightarrow \) f is Lipschitz \(\Rightarrow \) f is uniformly continuous \(\Rightarrow \) f is coarse.

  2. Throughout this introduction, all operator spaces are considered to be over the complex field. We point out however that all the main results of this paper remain valid for real operator spaces with unchanged proofs.

  3. Theorem 1.2 and Proposition 4.4 of [6] are actually stronger as they only demand the embeddings to be almost completely coarse embeddings [6, Definition 4.1].

  4. As seen in Sect. 2.1, an operator metric space X is defined as a subset of \({\mathcal {B}}(H)\) for some Hilbert space H.

  5. A map \(f:X\rightarrow Y\) between \({\mathbb {K}}\)-vectors spaces is called \({\mathbb {K}}\)-affine if \(g=f-f(0)\) is \({\mathbb {K}}\)-linear.

  6. See [8, Definition 6.17] for the definition of a Gaussian measure on a Banach space and the comments after [8, Proposition 6.20] for a proof of this statement.

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Authors and Affiliations

  1. IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil

    Bruno M. Braga

  2. Department of Mathematics, University of Oklahoma, Norman, OK, 73019-3103, USA

    Javier Alejandro Chávez-Domínguez

  3. Mathematics Department, Purdue University, 150 N. University Street, West Lafayette, IN, 47907-2067, USA

    Thomas Sinclair

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  1. Bruno M. Braga
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  2. Javier Alejandro Chávez-Domínguez
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  3. Thomas Sinclair
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Bruno M. Braga was partially supported by NSF grant DMS-2054860. Javier Alejandro Chávez-Domínguez was partially supported by NSF grant DMS-1900985. Thomas Sinclair was partially supported by NSF grants DMS-1600857 and DMS-2055155.

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Braga, B.M., Chávez-Domínguez, J.A. & Sinclair, T. Lipschitz geometry of operator spaces and Lipschitz-free operator spaces. Math. Ann. 388, 1053–1090 (2024). https://doi.org/10.1007/s00208-022-02518-1

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