An immediate consequence of Bott periodicity is that the infinite unitary space is an infinite loop space and so an $E\_\infty$-space. I wonder if there is a direct proof (not using $U = \Omega^2 U$) giving an explicit construction of an $E\_\infty$-operad actiong on $U$? I'm also interested in the case of the infinite orthogonal group. 



Direct proof that $U$ is an $E_\infty$-space
The linear isometries operad is the $E\_\infty$ operad that most naturally acts on $O(\infty):=colim\_n O(n)$.


The $n$-ary operations in that operad is the space of linear isometric maps (not necessarily surjective) from $\underbrace{\mathbb R^\infty\oplus\ldots \oplus\mathbb R^\infty}\_{n}$ to $\mathbb R^\infty$.


To get an action of that same operad on the unitary group $U(\infty)$, you need to first complexify a map $\mathbb R^\infty\oplus\ldots \oplus\mathbb R^\infty 
\to \mathbb R^\infty$ to $\mathbb C^\infty\oplus\ldots \oplus\mathbb C^\infty 
\to \mathbb C^\infty$ in order to construct the induced map $U(
\infty)\times\dots\times U(\infty)\to U(\infty)$.


