  We identify and characterize unital completely positive (UCP) maps on finite
dimensional $C^*$-algebras for which the Choi-Effros product extended to the
space generated by peripheral eigenvectors matches with the original product.
We analyze a decomposition of general UCP maps in finite dimensions into
persistent and transient parts. It is shown that UCP maps on finite dimensional
$C^*$-algebras with spectrum contained in the unit circle are
$\ast$-automorphisms.
