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Let G be a connected semisimple real-rank one Lie group with finite center and let C*(G) denote the regular group C*-algebra of G. In this paper a complete description of the structure of C*(G) is obtained. 1. Introduction. Let G be a connected semisimple real-rank one Lie group with finite center and Lie algebra g. If Gc is the simply connected, complex analytic group corresponding to gc, we assume, in addition, that G is the real analytic subgroup of Gc corresponding to g. Let C*(G) denote the regular group C*-algebra of G, i.e., the completion of LX(G) with respect to the norm ll/llp = IIIp(/)III where p is the left regular representation of G and |||p(/)||| denotes the norm of p(f) as a left convolution operator on L2(G). The purpose of this paper is to give a complete description of the structure of C*(G) and thus give a partial answer (one for the above G) to a question raised in (6) as to an intrinsic characterization of C0(G). Throughout this paper H will denote a fixed separable infinite-dimensional Hilbert space and %(H) will denote the compact operators on H. We assume, in addition, that H has been identified with //©//. When T is a locally compact Hausdorff space, we denote by Cb(T, %(H)) the C*-algebra of all norm-continuous bounded functions tt-* x(t) of T into %(H) and by C°(T, %(H)) the C*-algebra of functions in C(T, %(H)) such that ||x(/)|| vanishes at infinity. The underlying hull-kernel topology on the spectrum of C*(G), Gr, plays a key role in describing the structure of C*(G) « C0(G). The main difficulty occurs when Gr is not Hausdorff. When Gr is Hausdorff e.g.,

The regular group 𝐶*-algebra for real-rank one groups