TITLE: Infinite loop space structure of $BU^+$
QUESTION [10 upvotes]: It is well-known that $BU^+$ is homotopy equivalent to an infinite loop space where $U$ is the limit of the unitary groups $U(n)$ for $n \rightarrow \infty$ and $+$ denotes Quillen's Plus construction. 
On the other hand there is a tool to detect infinite loop spaces by the action from $E\Sigma_p$ ($\Sigma_p$ is the symmetric group).
My question is: how does the map $E\Sigma_p \times_{\Sigma_p} BU^p \rightarrow BU$ looks like.
Is it possible to write it down.

REPLY [10 votes]: There are a number of equivalent ways of seeing $BU\times Z$ or $BU$ as an infinite loop space.
The description that Neil gives is the action of the linear isometries operad $\mathcal{L}$ (complex version) on $BU$.  A large number of related spaces have such a structure, as explained in the first 
section of the first chapter of "$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra", SLN 577
(1977).  Morever, the Bott maps are maps of $\mathcal{L}$-spaces, which ties in Bott's original proof that $BU$ is an infinite loop space.  The fact that $\coprod_n U(n)$ is a permutative category gives 
a quite different operad action on $\coprod_n BU(n)$, whose associated infinite loop space is 
$BU\times \mathbf{Z}$.  It is not obvious a priori how to compare the infinite loop structures on
these two models.  The question is resolved in this and related examples (e.g. $BTop$) in my paper
"The spectra associated to $\mathcal{I}$-monoids".  Math. Proc. Camb. Phil. Soc. 84(1978), 313--322.