Completely Bounded Norms of $k$-positive Maps
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914838725263360 |
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| author | Aubrun, Guillaume Davidson, Kenneth R. Müller-Hermes, Alexander Paulsen, Vern I. Rahaman, Mizanur |
| author_facet | Aubrun, Guillaume Davidson, Kenneth R. Müller-Hermes, Alexander Paulsen, Vern I. Rahaman, Mizanur |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Completely Bounded Norms of $k$-positive Maps Aubrun, Guillaume Davidson, Kenneth R. Müller-Hermes, Alexander Paulsen, Vern I. Rahaman, Mizanur Operator Algebras Functional Analysis Probability Quantum Physics 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. |
| title | Completely Bounded Norms of $k$-positive Maps |
| topic | Operator Algebras Functional Analysis Probability Quantum Physics |
| url | https://arxiv.org/abs/2401.12352 |