The image of J space is defined for $p$ odd as the homotopy fibre $J_{(p)}$ of the self map $$\psi^k - 1: BU_{(p)} \to BU_{(p)}.$$ Here, $\psi^k$ is an Adams operation, and $k \in \mathbb{N}$ descends to a topological generator of the $p$-adic units. For $p=2$, $J_{(2)}$ is defined as the homotopy fibre of a lift of an orthogonal form of this map, $\psi^3-1: BO_{(2)} \to BSpin_{(2)}$.
It is known that $J_{(p)}$ splits off of the $p$-local sphere, $Q S^0_{(p)}$. A map $e: Q S^0_{(p)} \to J_{(p)}$ is given by the unit of the ring spectrum whose zeroth space is $J_{(p)}$. A one-sided inverse to $e$ can be constructed from the solution of the Adams conjecture. My question is: who is responsible for the proof of this result?
Mahowald's "The order of the image of the J-homomorphism" constructs the map $e$ and proves the resulting surjection $\pi_* S^0_{(p)} \to \pi_* J_{(p)}$, but does not phrase the result in terms of a splitting of the infinite loop spaces. Also, he works only at $p=2$. The result is stated as Corollary 4.6 in May-Quinn-Ray-Tornehave's "$E_\infty$ ring spaces and $E_\infty$ ring spectra," but since that book appeared 7 years after Quillen's proof of the Adams conjecture and Mahowald's paper, I wonder if the proof appeared somewhere else in between these publications.
(Not so) secretly, I'm hoping that Peter will jump in and fill us in on the history of this question.
