Exotic group C * -algebras of simple Lie groups with real rank one

Abstract

Exotic group C * -algebras are C * -algebras that lie between the universal and the reduced group C * -algebra of a locally compact group. We consider simple Lie groups G with real rank one and investigate their exotic group C * -algebras C L p+ * (G), which are defined through L p -integrability properties of matrix coefficients of unitary representations. First, we show that the subset of equivalence classes of irreducible unitary L p+ -representations forms a closed ideal of the unitary dual of these groups. This result holds more generally for groups with the Kunze–Stein property. Second, for every classical simple Lie group G with real rank one and every 2≤q<p≤∞, we determine whether the canonical quotient map C L p+ * (G)↠C L q+ * (G) has non-trivial kernel. Our results generalize, with different methods, recent results of Samei and Wiersma on exotic group C * -algebras of SO 0 (n,1) and SU(n,1). In particular, our approach also works for groups with property (T).