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I should mention that there is a real geometric relationship between the C*-algebra picture and Donovan-Karoubi one given by going between C*-algebras with spectrum X and some kind of Hilbert bundle over X. I don't understand this story well, but it's not just some coincidental isomorphism.
@QiaochuYuan yes. That's all true. Maycock's classifies Morita classes of complex C*-algebras and Moutuou's does the same for real C*-algebras. However, both also look a lot like the Donovan-Karoubi Brauer groups except that Maycock's is missing the $H^0(X;\mathbb{Z}/2)$ that it should have.
I should maybe say that from talking to Tyler Lawson, Charles Rezk and Kiran Luecke, I have learned that one can prove the non-triviality of the first two $k$-invariants of $\mathrm{pic}(KO)$ via the $J$-homomorphism. I don't think this extends to the $KU$ case.
When I started thinking about this (see e.g. arxiv.org/abs/1708.03042), my advisor made some vague comments about these X(n) somehow being akin to cyclotomic extensions of ℚ. I don't really understand this, but that memory, combined with your question, caused me to wonder if there's anything interesting to say about some kind of "derived ramification groups" of these intermediate Galois extensions 𝕊→X(n)→MU as they relate to the chromatic filtration.
