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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

$ K\sp n$-positive maps in $ C\sp \ast$-algebras


Author: Takashi Itoh
Journal: Proc. Amer. Math. Soc. 101 (1987), 76-80
MSC: Primary 46L05
MathSciNet review: 897073
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Abstract: Let $ {K^n}$ be the set of $ n$-positive maps of $ B(H)$ to $ B(H)$. A $ {K^n}$-positive map of a $ {C^ * }$-algebra $ A$ to $ B(H)$ is a positive linear map $ \phi $ such that $ \sum {{\text{Tr(}}\phi {\text{(}}} {a_i})b_i^t) \geq 0$ for any $ \sum {{a_i} \otimes {b_i} \in \{ x \in A{ \otimes _\gamma }T(H)\vert{K^n}} { \... ... }\alpha ,({\text{id}} \otimes \alpha {\text{)(}}x{\text{)}} \geq {\text{0}}\} $. It is shown that the following three statements are equivalent. (1) Every $ {K^n}$-positive map of $ A$ to $ B(H)$ is $ {K^{n + 1}}$-positive. (2) Every $ {K^n}$-positive map of $ A$ to $ B(H)$ is completely positive. (3) $ A$ is an $ n$-subhomogeneous $ {C^ * }$-algebra.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0897073-2
Keywords: $ n$-positive map, $ {K^n}$-positive map, completely positive map, $ n$-subhomogeneous $ {C^ * }$-algebra
Article copyright: © Copyright 1987 American Mathematical Society