Recently, Ching and Salvatore 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.
Best Answer
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.