Abstract :
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 the reduced group C∗-algebra. Recent studies by M.A. Rieffel raise several questions with respect to such a metric on the state space. We propose a relaxation in the way a length function is used in the construction of a metric, and we then show that for groups of rapid decay there are many metrics related to a length function which have all the expected properties. At the end we show that this notion allows a non-commutative version of the Arzelà–Ascoli theorem.
