There is an action of $\Gamma$ on a compact Hausdorff space $X$. The question is to find an iff condition for the existence of a $\Gamma$ invariant sub-algebra of $C(X)$. I started out with the following:
Suppose that $\Gamma$ acts on a compact Hausdroff space $X$. There exists a $\Gamma$-invariant subalgebra of $C(X)$ iff there exists an injective $\Gamma$-equivariant map from a $\Gamma$-space $A$ to $C(X)$.
Suppose that $B$ is an invariant subalgebra of $C(X)$. Let $j: B \to C(X)$ be the inclusion map. Then $j$ is injective. Moreover $j(s.b)=s.b=s.j(b)$. So $j$ is $\Gamma$-equivariant.
On the other hand suppose that there is an injective $\Gamma$-equivariant map $\phi$ from a $\Gamma$-space $A$ to $C(X)$. Then $\phi(A)$ is a subset of $C(X)$. Moreover $s.\phi(a)=\phi(s.a) \in \phi(A) \subset C(X)$. Thus $\phi(A)$ is $\Gamma$-invariant.
But then I realized that $\phi(A)$ is not necessarily a sub algebra of $C(X)$. To make it a subalgebra I have to impose a condition on $\phi$. So I thought of making $\phi$ a linear homomorphism. Then this establishes the claim.
I want to know if this is optimal in the sense that if there can be any other condition weaker than what I have got here.
Thanks for the help!!