Throughout this review \(G\) will always be a locally compact, Hausdorff, separable group. Let \(L^2(G)\) be the space of square integrable functions on \(G\) with respect to a left Haar measure on \(G\), and let \(L\) be a length function on \(G\). We shall say that \(L\) is locally bounded if for any compact set \(U\) in \(G\), \(\sup \{ L(u) : u \in U \} < \infty\). For \(k \geq 0\), define \[ H_L^k (G) = \{ f \in L^2(G) : \int_G (1 + L(x))^{2k} | f(x)| ^2 dx < \infty\}, \] and define \(H_L^{\infty} (G) = \bigcap_{k \geq 0} H_L^k (G)\). The space \(H_L^{\infty} (G)\) is called the space of rapidly decaying functions. For the rest of this review assume all length functions on \(G\) are locally bounded. The group \(G\) is said to have property \(RD\) if and only if the space of rapidly decaying functions is contained in the reduced group \(C^{\ast}\)-algebra of \(G\). In this paper the authors characterize the connected Lie groups that have property \(RD\). Before we state the main result of this paper we need one more definition, which we shall now give. A Lie algebra is of type \(R\) if all the weights of the adjoint representation are purely imaginary. The main result of this paper is the following theorem:
Let \(G\) be a connected Lie group with Lie algebra \(\mathfrak g\) and universal cover \(\widetilde{G}\). The following are equivalent:
(1) \(G\) has property \(RD\);
(2) \(\mathfrak g = \mathfrak s \times \mathfrak q\), where \(\mathfrak s\) is semisimple and \(\mathfrak q\) is an algebra of type \(R\);
(3) \(\widetilde{G} = \widetilde{S} \times \widetilde{Q}\), where the connected Lie groups \(\widetilde{S}\) and \(\widetilde{Q}\) are, respectively, semisimple and of polynomial volume growth.
The authors also give a necessary and sufficient condition for property \(RD\) on unimodular groups of the form \(G = PK\), where \(K\) is a compact subgroup and \(P\) is a closed and amenable subgroup. This condition involves the growth of the elementary spherical function \(\phi_0\). It is also shown that property \(RD\) is stable under central extensions having polynomially distorted center.