Suppose I take $R=\mathbb{P}(\Sigma^\infty\mathbb{CP}^\infty)$ the free commutative ring spectrum on complex projective space, and I invert the image of the Bott class $S^2\to \Sigma^\infty\mathbb{CP}^\infty\to R$. What do I get?

@CharlesRezk I wish I knew. I think this thing has a canonical map to $MUP$ (coming from the inclusions $BU(1)^n \to BU(n)$) which is a rational isomorphism. Could it be an isomorphism?

@SaalHardali If I understand correctly, the homology of this thing is the tensor product of the homologies of P(S^2)[beta^{-1}] and P(S^{2n}) for n > 1. This seems much too big to be the homology of MUP.

There is a splitting of loop spaces BUxA=QCP^infinity, where A is some loop space with torsion homotopy groups. When you invert the Bott element in Sigma^infinity_+ BU you will get MUP, so if you invert the Bott element in Sigma^infinity_+QCP^infinity you get the MUP homology of this A space. The space A is a bit of a messy object so I don't know why you'd want to do this

To make the splitting BU x A = QCP^infinity you have to make an interesting loop map QCP^infinity--->BU. Segal has a great paper that Hopkins always advertises that makes this map via the Brauer induction theorem. Probably the best modern way to see this map exists is via the Weiss calculus of functors tower for the functor V--->BU(V)

One can try to slowly deform QCP^infinity to BU by killing more and more of this A space. This interacts well with the chromatic filtration and is studied by Hopkins--Lawson

also the subject of Arone+Lesh's BU Whitehead conjecture