23 Apr 2003

Per Jakobsen "Conditional expectation for operator valued probability measures".

I will start with a short review of classical conditional probability. By generalizing the classical case we will be lead to define the category of spectral spaces. States will be seen to have a natural interpretation as arrows in this category. Using this notion of state we will in a very natural way be lead to a generalized probability theory where the probability measure is operator valued and random variables are Hilbert space valued maps. Such measures (but not random variables) have been investigated intensively in connection with phase-space quantization. In these investigations the group invariance is used to construct interesting measures (generalized systems of imprimitivity). Generalization of the Laplace principle to the case of operator valued probability measures will be formulated in terms of  group invariance and a nontrivial (nonconstant) solution will be calculated explicitly for the case of a 2-dimensional Hilbert space. Finally the notion of conditional expectation will be generalized to the operator valued context and will be seen to involve operator valued half densities in a very natural way. As a final topic I will discuss possible generalizations of the conditional product construction from the classical case.