Decomposability of Bimodule Maps
Keyword(s):
Consider a unital $C^*$-algebra $A$, a von Neumann algebra $M$, a unital sub-$C^*$-algebra $C\subset A$ and a unital $*$-homomorphism $\pi\colon C\to M$. Let $u\colon A\to M$ be a decomposable map (i.e. a linear combination of completely positive maps) which is a $C$-bimodule map with respect to $\pi$. We show that $u$ is a linear combination of $C$-bimodule completely positive maps if and only if there exists a projection $e\in \pi(C)'$ such that $u$ is valued in $\mathit{e\mkern0.5muMe}$ and $e\pi({\cdot})e$ has a completely positive extension $A\to \mathit{e\mkern0.5muMe}$. We also show that this condition is always fulfilled when $C$ has the weak expectation property.
2013 ◽
Vol 16
(04)
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pp. 1350031
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2004 ◽
Vol 15
(03)
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pp. 289-312
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1978 ◽
Vol 21
(4)
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pp. 415-418
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1989 ◽
Vol 106
(2)
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pp. 313-323
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1992 ◽
Vol 03
(02)
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pp. 185-204
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1990 ◽
Vol 33
(4)
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pp. 434-441
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1997 ◽
Vol 55
(1)
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pp. 193-208
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2009 ◽
Vol 147
(2)
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pp. 323-337
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