I am looking for a decent account of the homotopy ring spectrum structure on $KU$ with $KU_0 = BU \times \mathbb{Z}$ that would be self-contained for a reader with good point-set topology background, but not involving $E_\infty$ or model category theory.

What I find in the literature is all sketchy, but maybe I am looking in the wrong places.

First, a discussion of the H-space structure on $BU\times \mathbb{Z}$ in the first place I find on p. 205 (213 of 251) in *A Concise Course in Algebraic Topology*. But for the crucial step it there only says:

we merely affirm that, by fairly elaborate arguments, one can pass to colimits to obtain a product

Is there a reference that would spell this out?

Next, for the proof of the homotopy ring spectrum structure on $KU$, the idea is indicated on the first page of

James McLure, *$H_\infty$-ring spectra via space-level homotopy theory* (pdf)

Is there a place where this would be spelled out in some detail?

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