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I would like to see that the map of spectra $\Sigma^\infty \mathbb{C}P^\infty_+ \to KU$ is actually a map of ring spectra, where $KU$ denotes the complex K-theory spectrum and the map is given as the adjoint of the inclusion $\mathbb{C}P^\infty \to BU \times \{1\}$. Preferably in the framework of Adams stable category theory (as described in Adams' book "Stable homotopy and generalised homology" or Switzer's "Algebraic topology - homology and homotopy").

To this end: is there a reference showing that the adjunction of $\Sigma^\infty$ and $\Omega^\infty$ is monoidal in this setting? Or, alternatively, is there a reference showing the equivalence of Adams category to one of the more modern definitions (e.g. using symmetric spectra) and one showing the monoidalness of the adjunction in the latter setting? I found the paper "Model categories of diagram spectra" by Mandell, May, Schwede, and Shipley (which seems to be the usual reference for the equivalence of the different definitions of the stable homotopy category), but they do not seem to show anything about Adams' stable category.

Robin Stoll
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It is true that in Boardman's category of spectra that there is a natural map $$\Sigma^\infty(X \wedge Y) \to \Sigma^\infty X \wedge \Sigma^\infty Y$$ which additionally is an equivalence, i.e., $\Sigma^\infty$ is strong monoidal. It follows, e.g., from corollary 13.39 in Switzer. Moreover, I believe that one can deduce formally from this that $\Omega^\infty$ is at least lax monoidal. However, I don't think $\Omega^\infty$ can be strong monoidal, unless you break the condition that $\mathbb{S}$ is the unit for the smash product or the condition $\Omega^\infty \Sigma^\infty X \simeq QX$ from Lewis' theorem. My instinct is that if it were strong monoidal, then we would have $$QS^0 \simeq \Omega^\infty\mathbb{S} \simeq \Omega^\infty(\mathbb{S} \wedge \mathbb{S}) \simeq \Omega^\infty \mathbb{S} \wedge \Omega^\infty \mathbb{S} \simeq QS^0 \wedge QS^0.$$

As a bonus, let's consider the situation with the $S$-modules from EKMM. In this case, the suspension spectrum functor is again strong monoidal. However, we have $\Omega^\infty \Sigma^\infty X \simeq X$. So at least when restricted to suspension spectra $\Omega^\infty$ is also strong monoidal. I don't know about other spectra in general.

As a further bonus, here is a cute way to show that the map $\Sigma^\infty_+ \mathbb{C}P^\infty \to KU$ is a ring map. By Snaith's theorem, we can identity $KU \simeq \Sigma^\infty_+ \mathbb{C}P^\infty [\beta^{-1}]$, where $\beta$ is the Bott element. So the map $\Sigma^\infty_+ \mathbb{C}P^\infty \to KU$ is just a localization map, which is a ring map (e.g., by EKMM proposition V.2.3).

JHF
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