Paper detail

Fixed points of normal completely positive maps on B(H)

Given a sequence of bounded operators $a_j$ on a Hilbert space $H$ with $\sum a_j^*a_j=1=\sum a_ja_j^*$, we study the map $Ψ$ defined on $B(H)$ by $Ψ(x)=\sum a_j^*xa_j$ and its restriction $Φ$ to the Hilbert-Schmidt class $C^2(H)$. In the case when the sum $\sum a_j^*a_j$ is norm-convergent we show in particular that the operator $Φ-1$ is not invertible if and only if the C$^*$-algebra $A$ generated by $(a_j)$ has an amenable trace. This is used to show that $Ψ$ may have fixed points in $B(H)$ which are not in the commutant $A'$ 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 $A'$ even if the operators $a_j$ are not positive.