Eigenvalues of Completely Nuclear Maps and Completely Bounded Projection Constants

We investigate the distribution of eigenvalues of completely nuclear maps on an operator space. We prove that eigenvalues of completely nuclear maps are square-summable in general and summable if the underlying operator space is Hilbertian and homogeneous. Conversely, if eigenvalues are summable for all completely nuclear maps, then every finite dimensional subspace of the underlying operator space is uniformly completely complemented, and consequently Hilbertian and homogeneous if the underlying operator space is infinite dimensional by the result of G. Pisier and T. Oikhberg. As an application we consider an estimate of completely bounded projection constants of n-dimensional operator spaces.