The unique representation of a selfadjoint bounded linear functional Academic Article

It is well known that every selfadjoint bounded linear functional on a C {C^ * } -algebra has a unique minimal decomposition [6, Theorem 3.2.5]. In this paper we prove that under some conditions a selfadjoint completely bounded linear map with a unique minimal decomposition is equivalent to the map with a unique commutant representation (up to unitary equivalence). Using the results, we generalize the Gelfand-Naimark-Segal construction.