Let $KU$ be the spectrum representing complex $K$-theory and $KO$ be the spectrum representing real $K$-theory. Complex-conjugation of complex vector bundles lifts to a $C_2$-action on $KU$. It's a well-known fact that $$KU^{hC_2} \simeq KO.$$
I would like to have a reasonably self-contained proof of this fact that does not require any knowledge of $\pi_* KO$. (If I knew $\pi_* KO$, I could run the homotopy fixed point spectral sequence for $KU^{hC_2}$ and show that they have the same homotopy groups, etc.)
The proof ought to need some geometric input since I've defined $KU$ and $KO$ geometrically, and I think Atiyah's $KR$-theory is the appropriate tool to use. Recall, $KR$-theory interpolates between $KO$ and $KU$:
- Over spaces $X$ with trivial involution, $KR(X) \cong KO(X)$.
- Over spaces of the form $X \sqcup X$ with the swap involution, $KR(X \sqcup X) \cong KU(X)$.
How do I pass from these facts about the cohomology theory $KR(-)$ to facts about spectra $KU, KO$ and their homotopy fixed points?