  In this paper we study a natural decomposition of $G$-equivariant $K$-theory
of a proper $G$-space, when $G$ is a Lie group with a compact normal subgroup
$A$ acting trivially. Our decomposition could be understood as a generalization
of the theory known as Mackey machine under suitable hypotheses, since it
decomposes $G$-equivariant K-theory in terms of twisted equivariant K-theory
groups respect to some subgroups of $G/A$. Similar decompositions were known
for the case of a compact Lie group acting on a space, but our main result
applies to discrete, linear and almost connected groups. We also apply this
decomposition to study equivariant $K$-theory of spaces with only one isotropy
type. We provide a rich class of examples in order to expose the strength and
generality of our results. We also study the decomposition for equivariant
connective $K$-homology for actions of compact Lie groups using a suitable
configuration space model, based on previous papers published by the third
author.
