© 2002 by London Mathematical Society
© The London Mathematical Society
The Isolated Points of 
and the w*-Strongly Exposed Points of P(G)0
Department of Mathematical Sciences, Lakehead University Thunder Bay, Ontario, Canada P7E 5E1, tmiao{at}thunder.lakeheadu.ca
Received 20 December 2000.
Let G be a separable locally compact group and let 
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 
is discrete in , then there exists a nonzero positive-definite function 
associated with such that 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 associated with is a w*-strongly exposed point of P(G)0, then is isolated in . Consequently, G is compact if and only if, for every , there exists a nonzero positive-definite function associated with that is a w*-strongly exposed point of P(G)0. If, in addition, G is unimodular and 
, then is isolated in if and only if some nonzero positive-definite function associated with is a w*-strongly exposed point of P(G)0, where is the left regular representation of G and is the reduced dual space of G. We prove that if B(G) has the Radon–Nikodym property, then the set of isolated points of (so square-integrable if G is unimodular) is dense in . 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 . In particular, for a countable discrete non-amenable group G (for example the free group F2 on two generators), there is no closed point in its reduced dual space .