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Pré-Publication, Document De Travail Année : 2024

Completely Bounded Norms of $k$-positive Maps

Résumé

Given an operator system $\cl S$, we define the parameters $r_k(\cl S)$ (resp.\ $d_k(\cl S)$) defined as the maximal value of the completely bounded norm of a unital $k$-positive map from an arbitrary operator system into $\cl S$ (resp.\ from $\cl S$ into an arbitrary operator system). In the case of the matrix algebras $M_n$, for $1 \leq k \leq n$, we compute the exact value $r_k(M_n) = \frac{2n-k}{k}$ and show upper and lower bounds on the parameters $d_k(\M_n)$. Moreover, when $\cl S$ is a finite-dimensional operator system, adapting results of Passer and the 4th author \cite{PaPa}, we show that the sequence $(r_k(\cl S))$ tends to $1$ if and only if $\cl S$ is exact and that the sequence $(d_k(\cl S))$ tends to $1$ if and only if $\cl S$ has the lifting property.

Dates et versions

hal-04454872 , version 1 (13-02-2024)

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Guillaume Aubrun, Kenneth Davidson, Alexander Müller-Hermes, Vern Paulsen, Mizanur Rahaman. Completely Bounded Norms of $k$-positive Maps. 2024. ⟨hal-04454872⟩
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