Let $G$ be a compact abelian topological group with invariant measure $\mu$ which acts on a compact Hausdorff space $X$. A $G$-odd function is a continuous function $f:X\to \mathbb{C}$ such that $\int_{G} f(gx)d\mu =0\;\;\forall x\in X$. Let $A$ be the non unital $C^{*}$ subalgebra of $C(X)$ generated by odd functions.
$A$ is a (non unital) $C^{*}$ algebra, so it is isomorphic to $C_{0}(Y)$, for some locally compact hausdorff space $Y$.
What can be said about $A$ and $Y$.
Is $A$ an Ideal in $C(X)$? Is $Y$ embeddable in $X$?Is $Y$ homeomorphic to $(X-\text{fixed points of $G$})$?
Note that for the obvious action of $Z_{2}$ on the interval $[-1,\;1]$ the answers to above questions are affirmative. In fact $A=\{f\in C[-1,\;1]\mid f(0)=0\}$. So $Y=[-1,\;1]-\{0\}$.
Now assume that we have a non commutative dynamical system(G acts on a noncommutative $C^{*}$ algebra). What is an appropriate NC space $Y$, as a non commutativization of the above process?