Abstract
The nonlinear geometry of operator spaces has recently started to be investigated. Many notions of nonlinear embeddability have been introduced so far, but, as noticed before by other authors, it was not clear whether they could be considered “correct notions”. The main goal of these notes is to provide the missing evidence to support that almost complete coarse embeddability is “a correct notion”. This is done by proving results about the complete isomorphic theory of \(\ell _1\)-sums of certain operators spaces. Several results on the complete isomorphic theory of \(c_0\)-sums of operator spaces are also obtained.
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Notes
The definition of a coarse map between Banach spaces is precisely this one for \(n=1\).
We call a sequence \((g_n:X_n\rightarrow Y_n)_n\) of isomorphic embeddings equi-isomorphic if \(\sup _n\Vert g_n\Vert ,\sup _{n\in {\mathbb {N}}}\Vert g_n^{-1}\Vert <\infty \), where \(g^{-1}_n\) is defined only on the image of \(g_n\).
Recall, a family of maps \((f_n:X_n\rightarrow Y_n)\) between metric spaces \((X_n,d_n)\) and \((Y_n,\partial _n)\) are equi-coarse embeddings if for all \(r>0\) there is \(s>0\) such that (1) \(d_n(x,z)\le r\) implies \(\partial _n(f_n(x),f_n(z))\le s\) and (2) \(d_n(x,z)\ge s\) implies \(\partial _n(f_n(x),f_n(z))\ge r\).
Notice that \(B_{{\textrm{M}}_n(X)}\) is contained in \({\textrm{M}}_n(B_X)\), so \(f^n_n\restriction _{n\cdot B_{{\textrm{M}}_n(X)}}\) is well defined.
See Sect. 2 for the precise definition of those operator spaces.
Recall, an operator space X is Hilbertian if it is isomorphic (as a Banach space) to \(\ell _2\). Also, given \(\lambda \ge 1\), X is \(\lambda \)-homogeneous if \(\Vert u\Vert _{cb}\le \lambda \Vert u\Vert \) for all operators \(u:X\rightarrow X\). We then say X is homogeneous if it is \(\lambda \)-homogeneous for some \(\lambda \ge 1\).
Recall, an operator \(u:X\rightarrow Y\) between operator spaces is completely contractive if \(\Vert u\Vert _{cb} \le 1\).
We point out that authors interested in the isometric theory of operator spaces often use the term minimal to refer to what we are calling a 1-minimal operator space.
Recall, an operator \(u:X\rightarrow Y\) between Banach spaces is strictly singular if none of its restrictions to an infinite dimensional subspaces of X is an isomorphic embedding. For more on strictly singular operators, see [11, Sect. 2.c].
If E is an operator space, then \({\textrm{M}}_{1,k}(E)\) denotes the subspace of \({\textrm{M}}_k(E)\) consisting of all operators whose only nonzero rows are their first one. The space \({\textrm{M}}_{k,1}(E)\) is defined similarly, but with the word “columns” substituting “rows”.
I.e., \(x_k^*(x_k)=1\) and \(x^*_k(x_j)=0\) for all \(k\ne j\).
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The first and second named authors were partially supported by NSF DMS awards 2054860 and 1912897, respectively. B. M. Braga was also partially supported by FAPERJ (Proc. E-26/200.167/2023) and by CNPq (Proc. 303571/2022-5). The authors are grateful to the anonymous referee for numerous helpful suggestions.
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Braga, B.M., Oikhberg, T. Coarse geometry of operator spaces and complete isomorphic embeddings into \(\ell _1\) and \(c_0\)-sums of operator spaces. Math. Z. 304, 52 (2023). https://doi.org/10.1007/s00209-023-03314-6
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DOI: https://doi.org/10.1007/s00209-023-03314-6