I posted this question on math.se, but I think it may be more appropriate here, sorry if I am wrong about that.

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.

I posted this question on math.se here:https://math.stackexchange.com/q/2996787/482732, but I think it may be more appropriate here, sorry if I am wrong about that.

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.