In Waldhausen's paper Algebraic K theory of Spaces(the long one) he proves the following:
$$A(X)\simeq \mathbb{Z}\times B\widehat{Gl}(\Omega^{\infty}\Sigma^\infty |G|)$$
Where $|G|$ is a loop group of $X$. My problem is that to apply $B$ we need $\widehat{Gl}(\Omega^{\infty}\Sigma^\infty |G|)$ to be $A^\infty$, but since $\Omega^{\infty}\Sigma^\infty |G|$ is only a ring up to homotopy this isn't obvious to me. Waldhausen just brushes past this point, so I was hoping to get a reference to somewhere that shows this in detail. Thanks.
Edit: I posted this question here:https://mathoverflow.net/q/315239/131196, since I think it might be more appropriate on that site, but just in case I will leave this up as well.
