  We examine a special case of an approximation of the joint spectral radius
given by Blondel and Nesterov, which we call the outer spectral radius. The
outer spectral radius is given by the square root of the ordinary spectral
radius of the $n^2$ by $n^2$ matrix $\sum{\overline{X_i}}\otimes{X_i}.$ We give
an analogue of the spectral radius formula for the outer spectral radius which
can be used to quickly obtain the error bounds in methods based on the work of
Blondel and Nesterov. The outer spectral radius is used to analyze the iterates
of a completely postive map, including the special case of quantum channels.
The average of the iterates of a completely positive map approach to a
completely positive map where the Kraus operators span an ideal in the algebra
generated by the Kraus operators of the original completely positive map. We
also give an elementary treatment of Popescu's theorems on similarity to row
contractions in the matrix case, describe connections to the Parrilo-Jadbabaie
relaxation, and give a detailed analysis of the maximal spectrum of a
completely positive map.
