Assume that $(X, B, G)$ is a $G$- principal bundle where $G$ is a compact topological or Lie group. The normalized Haar measure of (each fiber of ) $X$ is denoted by $\mu$. The space of continuos complex valued function on $X$ is denoted by $C(X)$ which is equiped with the sup norm and the standard operation.
To this principal bundle, we associate a bounded linear operator $T: C(X) \to C(B)$ with $T(f)(b)=\int_{q^{-1}(b)} fd\mu$.
The above linear map $T$ has a right invers $S_q :C(B) \to C(X)$ with $S_q (g)= g\circ q$. Moreover $S_q$ is a linear isometry.
Question: Is it true that every isometric right inverse of $T$ is a multiplicative operator?If yes,are there some right inverse $S_p$ for some continuous $p:X \to B$ but $p$ is not homotopic to $q$?In particular what is the answer to the above question for the hopf fibration $S^3 \to S^2$?
