The Isolated Points of Ĝ ρ and the w * ‐Strongly Exposed Points of P ρ ( G ) 0

Let G be a separable locally compact group and let Ĝ be its dual space with Fell's topology. It is well known that the set P ( G ) of continuous positive‐definite functions on G can be identified with the set of positive linear functionals on the group C * ‐algebra C * ( G ). We show that if π is discrete in Ĝ, then there exists a nonzero positive‐definite function ϕ π associated with π such that ϕ π is a w * ‐strongly exposed point of P ( G ) 0 , where P ( G ) 0 ={ f ∈ P ( G ): f ( e )⩽ 1. Conversely, if some nonzero positive‐definite function ϕ π associated with π is a w * ‐strongly exposed point of P ( G ) 0 , then π is isolated in Ĝ. Consequently, G is compact if and only if, for every π∈Ĝ, there exists a nonzero positive‐definite function associated with π that is a w * ‐strongly exposed point of P ( G ) 0 . If, in addition, G is unimodular and π∈Ĝ ρ , then π is isolated in Ĝ ρ if and only if some nonzero positive‐definite function associated with π is a w * ‐strongly exposed point of P ρ ( G ) 0 , where ρ is the left regular representation of G and Ĝ ρ is the reduced dual space of G . We prove that if B ρ ( G ) has the Radon–Nikodym property, then the set of isolated points of Ĝ ρ (so square‐integrable if G is unimodular) is dense in Ĝ ρ . It is also proved that if G is a separable SIN‐group, then G is amenable if and only if there exists a closed point in Ĝ ρ . In particular, for a countable discrete non‐amenable group G (for example the free group F 2 on two generators), there is no closed point in its reduced dual space Ĝ ρ .