Let $X, Y$ be infinite loop spaces: $X = QA$ and $Y = QB$, where $A,B$ are connected topological spaces, and $Q$ stands for $\Omega^\infty S^\infty.$ Let $f:X \to Y$ be a continuous map such that $\Omega f: \Omega X \to \Omega Y$ has a left inverse, i.e. there is a map $g: \Omega Y \to \Omega X$ such that $g \circ \Omega f =$ identity. (*)
Is it true that $f$ itself also has a left inverse?
The "proof" would be: Apply to the equality (*) the functor $B$, associating to an $H$-space $Z$ its classifying space $BZ.$ Seemingly we get what we wanted: The map $Bg$ would be the left inverse of $B\Omega f$, which is equal to $f$.(Is it?)
The problem with this argument is that if $g$ is not an $H$-space homomorphism, then the map $Bg$ makes no sense.
Is it true that any map $g:\Omega QA \to \Omega QB$ is homotopic to an $H$-homomorphism? (Then the above proof seems to work.)