  We give new characterizations to ensure that a free product of groups with
amalgamation has a simple reduced group C*-algebra, and provide a concrete
example of an amalgam with trivial kernel, such that its reduced group
C*-algebra has a unique tracial state, but is not simple. Moreover, we show
that there is a radical class of groups for which the reduced group C*-algebra
of any group is simple precisely when the group has a trivial radical
corresponding to this class.
