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

TL;DR

This paper investigates fixed points of certain completely positive maps on bounded operators, linking their existence to properties of the generated C*-algebra, and distinguishes cases based on algebraic commutativity.

Contribution

It characterizes fixed points of these maps in relation to the amenability and commutativity of the associated C*-algebra.

Findings

Fixed points can exist outside the commutant when the algebra is non-abelian.

If the algebra is abelian, all fixed points lie in the commutant.

The invertibility of the operator \\Phi - 1\\) relates to the existence of an amenable trace.

Abstract

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 $\Psi$ defined on $B(H)$ by $\Psi(x)=\sum a_j^*xa_j$ and its restriction $\Phi$ 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 $\Phi-1$ 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 $\Psi$ 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 $\Psi$ are in $A'$ even if the operators $a_j$ are not positive.