The theorem then states that the projection map induces an isomorphism of prorings Here, the induced map has as domain the completion of the G-equivariant K-theory of X with respect to I, where I denotes the augmentation ideal of the representation ring of G. In the special case of X being a point, the theorem specializes to give an isomorphism
between the K-theory of the classifying space of G and the completion of the representation ring.
The theorem can be interpreted as giving a comparison between the geometrical process of taking the homotopy quotient of a G-space, by making the action free before passing to the quotient, and the algebraic process of completing with respect to an ideal.
[1] The theorem was first proved for finite groups by Michael Atiyah in 1961,[2] and a proof of the general case was published by Atiyah together with Graeme Segal in 1969.
[3] Different proofs have since appeared generalizing the theorem to completion with respect to families of subgroups.