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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
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