  Let T be the circle and A be a T-C*-algebra. Then the T-equivariant K-theory
of A is a module over the representation ring of the circle. The latter is a
Laurent polynomial ring. Using the support of the module as an invariant, and
techniques of Atiyah, Bott and Segal, we deduce that there are examples of
T-C*-algebras A not KK^T-equivalent to any commutative T-C*-algebra. This is in
contrast to the non-equivariant situation, in which any C*-algebra in the
boostrap category is KK-equivalent to a commutative one. Our examples arise
from dynamics, and include Cuntz-Krieger algebras with their usual circle
actions. Using similar techniques, we also prove an equivariant version of the
Lefschetz fixed-point formula. This is a special case of a result with Ralf
Meyer that applies to general compact connected groups. The Lefschetz theorem
equates the module trace of the module map of the T-equivariant K-theory of a
smooth compact manifold induced by an equivariant self-correspondence of the
manifold, with an appropriate Kasparov product; the Kasparov product is the
T-equivariant index of the Dirac operator on a suitable `coincidence manifold'
of the correspondence. Finally, we prove several results related to
localization and the Kunneth and universal coefficient theorems, and give an
essentially complete description of the T-equivariant K-theory of compact
spaces, by combining localization techniques of Atiyah and Segal and results of
Paul Baum and Alain Connes for equivariant K-theory of finite group actions.
