Equivariant K-theory for proper actions of non-compact Lie groups

Generalizing a construction of Wolfgang L"uck and Bob Oliver, we define a good equivariant cohomology theory on the category of proper G-CW complexes when G is an arbitrary Lie group (possibly non-compact). This is done by constructing an appropriate classifying space that arises from a Gamma-G-space. It is proven that this theory effectively generalizes Segal’s equivariant K-theory when G is compact.