Completely Bounded Norms of $k$-positive Maps

• URL: http://arxiv.org/abs/2401.12352v2
• Date: Tue, 7 May 2024 08:22:14 GMT
• Title: Completely Bounded Norms of $k$-positive Maps
• Authors: Guillaume Aubrun, Kenneth R. Davidson, Alexander Müller-Hermes, Vern I. Paulsen, Mizanur Rahaman,
• Abstract summary: Given an operator system $mathcalS$, we define the parameters $r_k(mathcalS)$ (resp. $d_k(mathcalS)$) We show that the sequence $(r_k( mathcalS))$ tends to $1$ if and only if $mathcalS$ is exact and that the sequence $(d_k(mathcalS))$ tends to $1$ if and only if $mathcalS$ has the lifting
• Score: 41.78224056793453
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Given an operator system $\mathcal{S}$, we define the parameters $r_k(\mathcal{S})$ (resp. $d_k(\mathcal{S})$) defined as the maximal value of the completely bounded norm of a unital $k$-positive map from an arbitrary operator system into $\mathcal{S}$ (resp. from $\mathcal{S}$ 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 $\mathcal{S}$ is a finite-dimensional operator system, adapting recent results of Passer and the 4th author, we show that the sequence $(r_k( \mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ is exact and that the sequence $(d_k(\mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ has the lifting property.

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