  On a discrete group G a length function may implement a spectral triple on
the reduced group C*-algebra. Following A. Connes, the Dirac operator of the
triple then can induce a metric on the state space of reduced group C*-algebra.
Recent studies by M. Rieffel raise several questions with respect to such a
metric on the state space. Here it is proven that for a free non Abelian group,
the metric on the state space is bounded. Further we propose a relaxation in
the way a length function is used in the construction of a metric, and we show
that for groups of rapid decay there are many metrics related to a length
function which all have all the expected properties. The boundedness result for
free groups is based on an estimate of the completely bounded norm of a certain
Schur multiplier and on some techniques concerning free groups due to U.
Haagerup. At the end we have included a noncommutative version of the
Arzela-Ascoli Theorem.
