Completely positive interpolations of compact, trace-class and Schatten-p class operators

Extending Li and Poon's results on interpolation problems for matrices, we give characterizations of the existence of a completely positive linear map ΦcpΦcp between compact (or Schatten-p class) operators sending a particular operator A to another B. It is shown that such a map exists if a multiple of the numerical range of A contains the numerical range of B. Given two commutative families of compact (or Schatten-p   class) operators {Aα}{Aα} and {Bα}{Bα}, we provide sufficient and necessary conditions to ensure that we can choose a completely positive interpolation ΦcpΦcp to preserve trace and/or approximate units such that Φcp(Aα)=BαΦcp(Aα)=Bα for all α.