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Journal of Operator Theory

Volume 73, Issue 2, Spring 2015  pp. 443-463.

Transition probabilities of positive functionals on $*$-algebras

Authors:  Konrad Schmuedgen
Author institution:Mathematisches Institut, Universitaet Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany

Summary: The transition probability $P_A(f,g)$ of positive linear functionals $f$ and $g$ on a unital $*$-algebra $A$ was defined by A. Uhlmann, \textit{Rep. Math. Phys.} {\bf 9}(1976), 273--279. In this paper we study this notion in the context of {\it unbounded} Hilbert space representations of the $*$-algebra $A$ and derive a number of basic results. The main technical assumption is the essential self-adjointness of the GNS representations $\pi_f$ and $\pi_g$. Applications to functionals given by density matrices or by integrals and to vector functionals on the Weyl algebra are given.

Keywords: transition probability, non-commutative probability, unbounded representations

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