A probabilistic index for completely positive maps and an application

The probabilistic index of a completely positive map is defined in analogy with a formula of M. Pimsner and S. Popa for conditional expectations. As an application, we describe a new strategy for computing the Jones index of the range of certain endomorphisms. We use extended transition operators to associate to an endomorphism a unital completely positive map acting on a finite dimensional matrix algebra. Then the index to be computed equals the probabilistic index of this map. For a class of examples we get a complete classification.