Group algebras with a unique C∗-norm

AbstractCall a locally compact group G, C∗-unique, if L1(G) has exactly one (separating) C∗-norm. It is easy to see that a ∗-regular group G is C∗-unique and that a C∗-unique group is amenable. For connected groups G it is proved that G is C∗-unique, if the interior R(G)0 of a certain part R(G) of Prim(G), called the regular part of Prim(G), is dense in Prim(G), and that C∗-uniqueness of G implies the density of R(G) in Prim(G). From this it is derived that a connected group of type I is C∗-unique if and only if R(G)0 is dense in Prim(G). For exponential G, a quite explicit version of this result in terms of the Lie algebra of G is given. As an easy consequence, examples of amenable groups, which are not C∗-unique, and C∗-unique groups, which are not ∗-regular are obtained. Furthermore it is shown that a connected locally compact group G is amenable if and only if L1(G) has exactly one C∗-norm, which is invariant under the isometric ∗-automorphisms of L1(G)