  We present some results on equivariant KK-theory in the context of tensor
triangular geometry. More specifically, for G a finite group, we show that the
spectrum of the tensor triangulated subcategory of KK^G generated by the tensor
unit contains as a retract a canonical copy of the Zariski spectrum of the
complex character ring of G. For G trivial, this inclusion is a homeomorphism.
  We also prove a general criterion for a support theory on a compactly
generated tensor triangulated category to provide the universal support datum,
in the sense of Paul Balmer, on its subcategory of compact objects.
