Is this really the first time someone bothered to figure out the real K-theory of CP^n? I find it hard to believe, it looks like something that should have been done much earlier...

@DenisNardin certainly real projective spaces are ‘harder’ and done in Adams’s vector fields on spheres paper. It also follows from the case G=S^1 of the real version of the Atiyah-Segal completion theorem from 1969

@DylanWilson Right, this is just the AS completion theorem. From algebraic K-theory considerations I'd expect KO∧CP^n to be either KO⊕KU^{n/2} or KO⊕KU^{(n-1)/2}⊕Σ^{2n}KO (depending on the parity of n). Do you know if we can get it cheaply from AS?

@DenisNardin dunno, but maybe you could prove the result you're interested in by using induction and Wood's theorem? maybe it's not even hard to show directly that the K(1)-localization of CP^n splits as a bunch of cones on eta and maybe an extra sphere (since you can probably argue using just complex K-theory), and then the result would follow

er... actually maybe I'm skeptical about the K(1)-local statement, but you still ought to be able to use induction and identify the attaching map after smashing with KO as \eta or zero on the top bit, and zero everywhere else.

the achievement in that paper was the KO^*-algebra structure. for a less structured answer, he cites [3] - M. Fujii, K_0-groups of projective space, Osaka J. of Math. 4 (1967), 141–149, which is from the era you're thinking of