Let me start with the following


**Illustration:** Let $G$ be a compact group, and let $\pi:G\to H$ be its (surjective) continuous homomorphism onto a (compact) group $H$. So we can think that $H$ is the quotient group of $G$ modulo the kernel $K=\operatorname{Ker}\pi$ of the mapping $\pi$
$$
H=G/K
$$
(and $\pi$ is just the quotient mapping).


The map $\pi$ generates a (linear and continuous) mapping of the spaces of continuous functions:
$$
P:C(H)\to C(G).
$$
This operator is a coretraction in the category of Banach spaces (and, what is the same in this situation, in the category of locally convex spaces), since it has a left inverse operator,
$$
S:C(G)\to C(H),\qquad S\circ P=\operatorname{id}\_{C(H)}
$$
defined by the formula
$$
S(f)(t)=\int\_K f(t\cdot s)\ \mu\_K(d s),\qquad f\in C(G)
$$
where $\mu\_K$ is the normalized Haar measure on $K$. (This mapping turns each function $f\in C(G)$ into a function $S(f)\in C(G)$, invariant with respect to the shifts by the elements of $K$, and this means that $S(f)$ can be considered as a function on $H=G/K$).


Now the


**Question:** Let $X$ be a (Hausdorff) compact space, and let $\pi:X\to Y$ be a (surjective) continuous mapping onto a (compact) *metrizable* space $Y$. So we can think that $Y$ is the quotient space of $X$ modulo the equivalence relation
$$
x\sim x' \quad\Leftrightarrow \quad \pi(x)=\pi(x')
$$
(and $\pi$ is just the quotient mapping).


The map $\pi$ generates a (linear and continuous) mapping of the spaces of continuous functions:
$$
P:C(Y)\to C(X).
$$



> 
> Is this operator a coretraction in the category of Banach spaces (or, what is the same here, in the category of locally convex spaces)?
> 
> 
> 


In other words, does there exist an operator
$$
S:C(X)\to C(Y),
$$
such that
$$
S\circ P=\operatorname{id}\_{C(Y)}
$$
?


In my considerations the space $Y$ is metrizable, but I don't know, perhaps this condition is extra. Similarly, I don't know, perhaps the space $X$ need not to be compact, but just locally compact, or belong to some wider class.



Is there an operation in topology analogous to the operation of averaging over a compact subgroup in harmonic analysis?
As stated (i.e., without assuming metrizability of $X$), the answer is negative: Let $Y=\alpha\mathbb N$ be the Alexandrov (one point) compactification, $X=\beta\mathbb N$ the Stone-Cech compactification and $\pi:X \to Y$ the unique continuous extension to $\beta\mathbb N$ of the inclusion $\mathbb N\hookrightarrow\alpha\mathbb N$. $C(Y)$ is the space $c$ of convergent sequences whereas $C(X)=\ell^\infty$ is the space of bounded sequences and $P$ is the embedding $c\to\ell^\infty$. Phillips's theorem ($c\_0$ is not complemented in $\ell^\infty$) implies that $P$ does not have a continuous linear left inverse.




---


If $X$ is metrizable, such structural arguments are not very promising: On the one hand, Sobczyk's theorem ($c\_0$ is complemented in every separable superspace) covers the case where $C(Y)$ is isomorphic to $c\_0$ and on the other hand, Miljutin's theorem says that $C(X)$ is isomorphic to $C([0,1])$ for every uncountable compact metric space $X$.


