An action of a compact group on an AF algebra is called an
inductive limit action, if there is a sequence of finite-dimensional
subalgebras, each of which is invariant under the action, whose union is
dense. If the restrictions on the finite-dimensional subalgebras are inner
(induced by representations of the group), then such actions are
classified by equivariant K-theory, by the result of David Handelman and
Wulf Rossmann. I shall show that equivariant K-theory is not enough to
classify if the restrictions on finite dimensional subalgebras are not
inner, and give a complete classification of a more general class of
inductive limit actions by introducing a new invariant.
I will only assume some basic knowledge of representation theory for
finite groups and matrix algebras.