Recently, [Ching and Salvatore](https://arxiv.org/abs/2002.03878 "Koszul duality for topological E_n-operads") have proven that the $E\_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E\_n$ operad, I realized there is an obvious first question: does the spectrum $\Sigma^\infty\_+ O(n)^\vee$ have an $A\_\infty$-structure? Perhaps worth noting, since $\Sigma^\infty\_+ O(n)$ is self dual, it suffices to show $\Sigma^{-n(n-1)/2}\_+ O(n)$ has an $A\_\infty$ structure.


Edit: I realized this question is very basic; the Spanier-Whitehead dual of any space is canonically $E\_\infty$ via the diagonal.



Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\infty$-ring spectrum?
The question as stated probably requires clarification. If


$X$ is a space, then the S-dual $D\_+(X)$ (i.e., functions from $X\_+$ to the sphere) is always an $E\_\infty$-ring spectrum. In particular, it will also be an $A\_\infty$-ring.


Perhaps what is being asked is whether $D\_+(G)$ is an **$A\_\infty$-coalgebra** when $G$ is a topological group. The answer is yes, because $G$ is an $A\_\infty$-space.


