Let $U$ denote the limiting group of the chain $U(1) \to U(2) \to U(3) \to \cdots$

I wish to compute the group $K^{-1}\mathbb{C}/\mathbb{Z}(BU \times \mathbb{Z})$. For this, we have the long exact sequence

$\cdots \to K^{-1}(M) \to H^{odd}(M;\mathbb{C}) \to K^{-1}\mathbb{C}/\mathbb{Z}(M) \to K(M) \xrightarrow{ch \otimes \mathbb{C}} H^{even}(M;\mathbb{C}) \to \cdots$ .

It therefore helps to know the groups $K^{\pm 1}(BU \times \mathbb{Z})$ and $H^*(BU \times \mathbb{Z};\mathbb{C})$.

However I am unable to compute these groups. Could anyone please suggest some references where these groups may have been explicitly computed ? Or may be some hints as to how I may proceed ? Thanks so much !

Characteristic Classes. I should know something about $K^\ast(BU)$ off the top of my head, but I don't. But googling "K-theory of BU" leads to this wikipedia page, which calculates both your groups in section 3 and 4 respectively, referring to Adams'Stable Homotopy and Generalized Homologyfor the $K$-theory computation. I'm still confused by the notation $K^n\mathbb C / \mathbb Z$... $\endgroup$ – Tim Campion Oct 14 at 19:10