Journal of Operator Theory

Volume 54, Issue 2, Fall 2005 pp. 339-361.

A probabilistic index for completely positive maps and an application

Authors: Rolf Gohm
Author institution: Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Berkshire, RG6 6AX, UK

Summary: 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.

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