Fixed points of normal completely positive maps on $\mathrm{B}(\mathcal{H})$

Abstract

Given a sequence of bounded operators $a_j$ on a Hilbert space $\mathcal{H}$ with ${\sum }_{j=1}^{\infty }{a}_j^{⁎}{a}_j=1={\sum }_{j=1}^{\infty }{a}_j{a}_j^{⁎}$, we study the map Ψ defined on $\mathrm{B}(\mathcal{H})$ by $\Psi (x)={\sum }_{j=1}^{\infty }{a}_j^{⁎}x{a}_j$ and its restriction Φ to the Hilbert–Schmidt class ${\mathrm{C}}^2(\mathcal{H})$. In the case when the sum ${\sum }_{j=1}^{\infty }{a}_j^{⁎}{a}_j$ is norm-convergent we show in particular that the operator $\Phi -1$ is not invertible if and only if the ${\mathrm{C}}^{⁎}$-algebra A generated by ${\{a_j\}}_{j=1}^{\infty }$ has an amenable trace. This is used to show that Ψ may have fixed points in $\mathrm{B}(\mathcal{H})$ 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 Ψ are in $A^{\prime }$ even if the operators $a_j$ are not positive.