Free group $C^{\ast }$-algebras associated with ${\ell }_p$

Abstract

For every $p\ge 2$, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to the positive linear functionals on the free group $C^{\ast }$-algebra associated with the ideal ${\ell }_p$. This is a generalization of Haagerup's characterization for the case of the reduced free group $C^{\ast }$-algebra. As a consequence, the associated $C^{\ast }$-algebras are mutually non-isomorphic, and they have a unique tracial state.

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• Exotic group C*-algebras