I'm reading Atiyah and Segal's article "Equivariant K-theory and Completion" and need a little help understanding the notation they use. At various points in the paper they talk about objects of the form $K_G(X)$, $K^q_G(X)$ (where $q$ presumably signifies an integer), and $K^*_G(X)$.

This third notation confuses me: I've read papers that define it as the direct sum of the first two $K$-rings, but in the first paragraph they quote an isomorphism between $R(G)^{\wedge}$ (I'm assuming this signifies the completion of $R(G)$ at the augmentation ideal $I_G$) and a ring they call $K^*(B_G)$, where $B_G$ is a classifying space for the group $G$. As far as I can see, taking $X$ to be a point would then give $K^*(X)$ as the direct sum of two copies of this ring since the suspension of a point is an interval, and in particular is contractible.

I've also seen sources that define $K^*_G(X)$ simply as the system of rings $\{K^n_G(X): n\in\mathbb{Z}\}$, which in the complex case consists of only two rings. There is also a comment on page 7 following a lemma about $K_G(X)$ saying can be replaced by $K^*_G(X)$ by considering the product $X\times S^1$ instead of X.

Any help with this would be greatly appreciated.