Two $E_\infty$ structures on infinite matrices

Question

Let $O$ be the infinite orthogonal group. By taking a colimit of the diagram of topological groups $O(1) \to O(2) \to O(3) \to \ldots$, we know $O$ has a continuous group operation given by matrix multiplication.

On the other hand, $O$ is an infinite loop space. The $E_\infty$ structure is often written as Whitney sum of matrices. Probably the cleanest model is to take the linear isometries operad as your model for the $E_\infty$ operad.

Here is my question: Is there a sense in which the two operations above are equivalent? In other words, is $O$ under Whitney sum equivalent as a group-like $E_\infty$ space to $O$ under matrix multiplication? (It is not even obvious that the latter is $E_\infty$ as opposed to $E_1$.)

I tried looking through May's works but could not decipher whether an answer is in his papers.

Answer 1

First note that if $f\colon\mathbb{R}^\infty\to\mathbb{R}^\infty$ is a linear isometric embedding and $u\in O(n)\leq O$ we can define $f_*(u)\in O$ by $f_*(u)(f(x))=f(u(x))$ when $x\in\mathbb{R}^n$ and $f_*(u)(y)=y$ when $y\in f(\mathbb{R}^n)^\perp$. This gives a map $f_*\colon O\to O$.

Let $\mathcal{L}$ be the linear isometries operad, so $\mathcal{L}(n)$ can be described as the space of $n$-tuples $(f_1,\dotsc,f_n)$, where each $f_i$ is a linear isometric embedding $\mathbb{R}^\infty\to\mathbb{R}^\infty$ such that $f_i(\mathbb{R}^\infty)$ is orthogonal to $f_j(\mathbb{R}^\infty)$ whenever $i\neq j$. Let $\mathcal{L}'$ be the similar operad defined without the orthogonality condition. Then both $\mathcal{L}(n)$ and $\mathcal{L}'(n)$ are contractible, and $\Sigma_n$ acts freely on $\mathcal{L}(n)$, but not on $\mathcal{L}'(n)$. There is a natural map $$ \alpha\colon \mathcal{L}'(n) \times O^n \to O $$ given by $$ \alpha(f_1,\dotsc,f_n,u_1,\dotsc,u_n)= (f_1)_*(u_1) \cdot \dotsb (f_n)_*(u_n). $$ Because multiplication in $O$ is not commutative, this does not factor through $\mathcal{L}'(n)\times_{\Sigma_n}O^n$. It gives an action of $\mathcal{L}'$ on $O$ in the sense of non-$\Sigma$-operads but not in the world of genuine operads. The operation $\alpha(\text{id},\text{id})\colon O^2\to O$ is just the multiplication map for the group structure of $O$.

If we restrict to the supoperad $\mathcal{L}<\mathcal{L}'$ we find that the maps $(f_i)_*(u_i)$ have orthogonal support in an evident sense, so they commute, so the map $\alpha\colon\mathcal{L}(n)\times O^n\to O$ factors through $\mathcal{L}(n)\times_{\Sigma_n}O^n$, so we have a genuine operad action. This is of course the standard operad action used to regard $O$ as an infinite loop space.