Embeddings of finite-dimensional operator spaces into the second dual
Volume 181 / 2007
Studia Mathematica 181 (2007), 181-198
MSC: 46L07, 47L25.
DOI: 10.4064/sm181-2-5
Abstract
We show that, if a a finite-dimensional operator space $E$ is such that $X$ contains $E$ $C$-completely isomorphically whenever $X^{**}$ contains $E$ completely isometrically, then $E$ is $2^{15} C^{11}$-completely isomorphic to $\mathbf{R}_m \oplus \mathbf{C}_n$ for some $n, m \in \mathbb N \cup \{0\}$. The converse is also true: if $X^{**}$ contains $\mathbf{R}_m \oplus \mathbf{C}_n$ $\lambda$-completely isomorphically, then $X$ contains $\mathbf{R}_m \oplus \mathbf{C}_n$ $(2\lambda+\varepsilon)$-completely isomorphically for any $\varepsilon > 0$.
