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Fixed points of normal completely positive maps on B$$(\mathcal H)$$. (English) Zbl 1266.47059
Author’s abstract: Given a sequence of bounded operators $$a_{j}$$ on a Hilbert space $$\mathcal{H}$$ with $$\sum_{j=1}^{\infty }a_{j}^{\ast }a_{j}=1=\sum_{j=1}^{\infty }a_{j}a_{j}^{\ast }$$, we study the map $$\Psi$$ defined on $$B\left( \mathcal{H}\right)$$ by $$\Psi \left( x\right) =\sum_{j=1}^{\infty }a_{j}^{\ast }xa_{j}$$ and its restriction $$\Phi$$ to the Hilbert-Schmidt class $$C^{2}\left( \mathcal{H}\right)$$. In the case when the sum $$\sum_{j=1}^{\infty }a_{j}^{\ast }a_{j}$$ is norm-convergent, we show, in particular, that the operator $$\Phi -1$$ is not invertible if and only if the $$C^{\ast }$$-algebra $$A$$ generated by $$\left\{ a_{j}\right\} _{j=1}^{\infty }$$ has an amenable trace. This is used to show that $$\Psi$$ may have fixed points in $$B\left( \mathcal{H}\right)$$ which are not in the commutant $$A^{\prime }$$ of $$A$$ even in the case when the weak$$^*$$ closure of $$A$$ is injective. However, if $$A$$ is abelian, then all fixed points of $$\Psi$$ are in $$A^{\prime }$$ even if the operators $$a_{j}$$ are not positive.

##### MSC:
 47B48 Linear operators on Banach algebras 47L20 Operator ideals
##### Keywords:
quantum operation; fixed point; amenable trace; $$C^*$$-algebra
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