Direct proof that $U$ is an $E_\infty$-space

Question

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.

Answer 1

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 $U( \infty)\times\dots\times U(\infty)\to U(\infty)$.