  Josza's definition of fidelity for a pair of (mixed) quantum states is
studied in the context of two types of operator algebras. The first setting is
mainly algebraic in that it involves unital C$^*$-algebras $A$ that possess a
faithful trace functional $\tau$. In this context, the role of quantum states
(that is, density operators) in the classical quantum-mechanical framework is
assumed by positive elements $\rho\in A$ for which $\tau(\rho)=1$. The second
of our two settings is more operator theoretic: by fixing a faithful normal
semifinite trace $\tau$ on a semifinite von Neumann algebra $M$, we define and
consider the fidelity of pairs of positive operators in $M$ of unit trace. The
main results of this paper address monotonicity and preservation of fidelity
under the action of certain trace-preserving positive linear maps of $A$ or of
the predual $M_*$. Our results also yield a new proof of a theorem of Moln\'ar
on the structure of quantum channels on the trace-class operators that preserve
fidelity.
