Let $X$ be a connected, based CW complex. Then the James splitting of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra $$ \Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad \Sigma^{\infty} (S^0 \vee X \vee X^{[2]} \vee X^{[3]} \vee \cdots ) , $$ where $X^{[n]}$ denotes the $n$-fold smash product of $X$ with itself. Each side of this equivalence have the structure of $A_\infty$-ring spectra (the structure on the right side given by the tensor algebra over the sphere spectrum on the points of $X$).

Now, my understanding$^\dagger$ is that the Cartan formula for Hopf invariants implies that the above splitting is multiplicative up to homotopy.

Question: Can the above splitting be enriched to an equivalence of $A_\infty$-rings?

If so, can anyone provide me with a reference?

$^\dagger\tiny \text{From being once upon a time in Bill Richter's orbit.}$

Let $X$ be a connected, based CW complex. Then the James splitting of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra $$ \Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad \Sigma^{\infty} (S^0 \vee X \vee X^{[2]} \vee X^{[3]} \vee \cdots ) , $$ where $X^{[n]}$ denotes the $n$-fold smash product of $X$ with itself. Each side of this equivalence has the structure of $A_\infty$-ring spectrum (the structure on the right side is induced by concatenation $X^{[n]} \wedge X^{[m]} \cong X^{[m+n]}$, and the right side can be seen as the tensor algebra over the sphere spectrum on the points of $X$).

Now, my understanding$^\dagger$ is that the Cartan formula for Hopf invariants implies that the above splitting is multiplicative up to homotopy.

Question: Can the above splitting be enriched to an equivalence of $A_\infty$-rings?

If so, can anyone provide me with a reference?

$^\dagger\tiny \text{From being once upon a time in Bill Richter's orbit.}$

Let $X$ be a connected, based CW complex. Then the James splitting of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra $$ \Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad \Sigma^{\infty} (S^0 \vee X \vee X^{[2]} \vee X^{[3]} \vee \cdots ) , $$ where $X^{[n]}$ denotes the $n$-fold smash product of $X$ with itself. Each side of this equivalence have the structure of $A_\infty$-ring spectra (the structure on the right side given by the tensor algebra over the sphere spectrum on the points of $X$).

Now, my understanding$^\dagger$ is that the Cartan formula for Hopf invariants implies that the above splitting is multiplicative up to homotopy.

Question: Is the above splitting map of $A_\infty$-rings?

If so, can anyone provide me with a reference?

$^\dagger\tiny \text{From being once upon a time in Bill Richter's orbit.}$

Let $X$ be a connected, based CW complex. Then the James splitting of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra $$ \Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad \Sigma^{\infty} (S^0 \vee X \vee X^{[2]} \vee X^{[3]} \vee \cdots ) , $$ where $X^{[n]}$ denotes the $n$-fold smash product of $X$ with itself. Each side of this equivalence have the structure of $A_\infty$-ring spectra (the structure on the right side given by the tensor algebra over the sphere spectrum on the points of $X$).

Now, my understanding$^\dagger$ is that the Cartan formula for Hopf invariants implies that the above splitting is multiplicative up to homotopy.

Question: Can the above splitting be enriched to an equivalence of $A_\infty$-rings?

If so, can anyone provide me with a reference?

$^\dagger\tiny \text{From being once upon a time in Bill Richter's orbit.}$