Group C*-algebras, metrics and an operator theoretic inequality

Details Group C*-algebras, metrics and an operator theoretic inequality Group C*-algebras, metrics and an operator theoretic inequality

2002-11-20

arxiv.org/abs/math/0211312v1

Mathematics

Operator Algebras

15 pages

Scientific paper

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.

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