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| On Representations Associated with Completely $n$-Positive Linear Maps on Pro-$C*$-Algebras |
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Citation： |
Maria JOI\c{T}A.On Representations Associated with Completely $n$-Positive Linear Maps on Pro-$C*$-Algebras[J].Chinese Annals of Mathematics B,2008,29(1):55~64 |
| Page view： 1095
Net amount： 987 |
Authors： |
Maria JOI\c{T}A; |
Foundation： |
Project supported by the grant CNCSIS (Romanian National Council for Research in High Education)-code
A 1065/2006. |
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| Abstract： |
It is shown that an $n\times n$ matrix of continuous linear maps from a pro-$
C*$-algebra $A$ to $L(H)$, which verifies the condition of complete
positivity, is of the form $ [ V*Tij\Phi (\, \cdot\, ) V
] i,j=1n$, where $\Phi $ is a representation of $A$ on a Hilbert space $K$, $V$ is a bounded linear operator from $H$ to $K$, and $[Tij]i,j=1n$ is a positive element in the $C*$-algebra of all $n\times n$ matrices over the commutant of $\Phi ( A ) $ in $ L(K)$. This generalizes a result of C. Y. Suen in Proc. Amer. Math. Soc., 112(3), 1991, 709--712. Also, a covariant version of this construction is given. |
Keywords： |
Pro-$C*$-Algebra, Completely n-positive linear maps, Covariant completely n-positive linear maps |
Classification： |
46L05, 46L40 |
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