  We define the profinite completion of a C*-algebra, which is a
pro-C*-algebra, as well as the pro-C*-algebra of a profinite group. We show
that the continuous representations of the pro-C*-algebra of a profinite group
correspond to the unitary representations of the group which factor through a
finite group. We define natural homomorphisms from the C*-algebra of a locally
compact group and its profinite completion to the pro-C*-algebra of the
profinite completion of the group. We give some conditions for injectivity or
surjectivity of these homomorphisms, but an important question remains open.
