In a unital C*-algebra with a faithful trace functional τ, the set D_τ(A) of positive ρ∈A of trace τ(ρ)=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_τ(A) in terms of the Bures metric. Linear maps on A that map D_τ(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_τ(A); (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space D_τ(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^**.

The first author was supported in part by an NSERC Discovery Grant. The second author was supported part by a University of Regina Graduate Research Fellowship