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Bounded module maps and pure completely positive maps. (English) Zbl 0791.46032
From the author’s introduction: Let $A$ and $B$ be two $C\sp*$-algebras, $\varphi$ a completely positive map from $B$ to $B$. The map $\varphi$ gives a Hilbert $B$-module $H\sb \varphi$ and a *-homomorphism $\pi\sb \varphi$ from $A$ into the $C\sp*$-algebra of all bounded $B$-module maps with adjoints on $H\sb \varphi$. In the case that $B=\bbfC$, it is well known that $\varphi$ is pure if and only if $\pi\sb \varphi$ is irreducible. It is natural to ask whether it is also true for general $C\sp*$-algebras $B$. In this note we give a negative answer to the problem in general. We also show that for many $C\sp*$-algebras $B$, the maps $\varphi$ are never pure and $\pi\sb \varphi$ are never irreducible. However, for some $C\sp*$-algebras $B$, the purity of $\varphi$ does imply the irreducibility of $\pi\sb \varphi$ and for some $C\sp*$-algebras $B$, the irreducibility of $\pi\sb \varphi$ implies $\varphi$ is pure.

46L05General theory of $C^*$-algebras
46H25Normed modules and Banach modules, topological modules