For a smooth complex algebraic X, there are purely algebraic models for computing the singular cohomology of X. I will discuss one such construction, called periodic cyclic homology, which extracts the cohomology of X from the category of vector bundles on X. The Hodge structure on the cohomology of a smooth projective variety arises very naturally from this perspective. I will describe how this story works when the complexification of a compact Lie group acts on X. The topological invariant which pops out is known as equivariant K-theory, and in some cases this invariant receives a Hodge structure as well.