  In a unital C*-algebra with a faithful trace functional $\tau$, the set
$D_\tau(A)$ of positive $\rho\in A$ of trace \tau(\rho)=1 is an algebraic
analogue of the space of density matrices (the set of all positive matrices of
a fixed dimension of unit trace). Motivated by the literature concerning the
metric properties of the space of density matrices, the present paper studies
the density space $D_\tau(A)$ in terms of the Bures metric. Linear maps on A
that map $D_\tau(A)$ back into itself are positive and trace preserving, hence,
they may be viewed as an algebraic analogue of a quantum channel, which are
studied intensely in the literature on quantum computing and quantum
information theory.
  The main results in this paper are: (i) to establish that the Bures metric is
indeed a metric, (ii) to prove that channels induce nonexpansive maps of the
density space $D_\tau(A)$, (iii) to introduce and study channels on A that are
locally contractive maps (which we call Bures contractions) on the metric space
$D_\tau(A)$, and (iv) to analyse Bures contractions from the point of view of
the Frobenius theory of cone preserving linear maps.
  Although the focus is on unital C*-algebras, an important class of examples
is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra
results are established by first proving them for finite von Neumann algebras
and then proving them for C*-algebras by embedding a C*-algebra A into its
enveloping von Neumann algebra $A^{**}$.
