# Isometries between Hilbertian homogeneous finite dimensional operator spaces

We know that if $i:R_n\rightarrow C_n$ is an isometry then for any $n$-dimensional operator space E, there is a factorization $i=uv$ with $v:R_n\rightarrow E$, $u:E\rightarrow C_n$ such that $\|u\|_{cb}\|v\|_{cb}=n^{1/2}$. The question is, if E is a homogeneous Hilbertian operator space can we find $u$ and $v$ such that they are isometries?

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Yes. See the proof of Proposition 10.1 in Pisier's book. The idea is simple: if you start from any factorization $uv$, by the polar decomposition you can assume that both maps are diagonal in an orthonormal basis. Averaging these diagonal maps over all permutations of the basis does not increase the cb norm, and yields maps that are multiple of the identity.