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.



Image of J splitting
my friend, I have an email! But I can offer the history.
First, although the $E\_{\infty}$ book was published in 1977, it is a 
shotgun marriage of a bunch of earlier preprints that were rejected 
for publication. Despite the ongoing development of infinite loop
space theory at the time, most algebraic topologists didn't care. You are referring to Corollary 4.2 (not 4.6) in Chapter VIII, which is joint
with Tornehave. As I wrote on p. 203, specifically referring to the
relevant Section 4, ``These results were first proven, quite differently,
by the second author'', that is, by Tornehave. He was already notorious 
for not publishing anything, and that is borne out by Math Reviews. 
I had a preprint from him, via his adviser (and my student) Ib Madsen.
I wrote Tornehave to ask permission to publish jointly and then reworked 
the material. Remark 4.6 (that is the 4.6 you probably had in mind) describes
his original proof. 


Detail: the splitting is of the identity component $SF$ of the $p$-local
sphere infinite loop space $QS^0$.


Incidentally, you refer to Mark's work at the prime $2$, and it is 
worth emphasizing that although $SF$ splits as a space at $p=2$, it does not 
split as an infinite loop space or even a $1$-fold loop space. There is a still open conjecture by Haynes
Miller and Stewart Priddy about what does happen; see ``On $G$ and the stable 
Adams conjecture". Geometric applications of homotopy theory (Proc. Conf., Evanston, 
Ill., 1977), II, pp. 331–348, Springer Lecture Notes in Math. Vol 658, 1978. 


