On Isometric Embedding $\ell_p^m\to S_\infty^n$ and Unique operator space structure

We study existence of linear isometric embedding of $\ell_p^m$ into $S_\infty,$ for $1\leq p< \infty$ and unique operator space structure on two dimensional Banach spaces. For $p\in(2,\infty)\cup\{1\},$ we show that indeed $\ell_p^2$ does not embed isometrically into $S_\infty$. This verifies a guess of Pisier and broadly generalizes the main result of \cite{GUR18}. We also show that $S_1^m$ does not embed isometrically into $S_p^n$ for all $1<p<\infty$ and $m\geq 2$. As a consequence, we establish noncommutative analogue of some of the results in \cite{LYS04}. We also show that $(\mathbb{C}^2,\|.\|_{B_{p,q}})$ does not embed isometrically into $S_\infty$ for $2<p,q<\infty.$ The main ingredients in our proofs are notions of Birkhoff-James orthogonality and norm parallelism for operators on Hilbert spaces. These enable us to deploy `infinite descent' type of arguments to obtain contradictions. Our approach is new even in the commutative case. We prove that $(\mathbb{C}^2,\|.\|_{B_{p,q}})$ does not have unique operator space structure whenever $(p,q)\in(1,\infty)\times[1,\infty)\cup[1,\infty)\times(1,\infty)$ by showing that they do not have Property P or two summing property. In view of \cite{MIPV19}, this produces genuinely new examples of two dimensional Banach spaces without unique operator space structure, providing a partial answer to a question of Paulsen. In this case, we derive our result by transferring the problem to real case and applying known results of \cite{ARFJS95}.

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