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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?

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  • $\begingroup$ Is not in your example $Y=(0,1]$? Because $f(-x)=-f(x)$? $\endgroup$ Oct 15, 2014 at 6:01
  • $\begingroup$ @მამუკაჯიბლაძე No the algebra $A$ is the non unital $C^{*}$ algebra generated by odd function. So it is equal to the algebra of all continuous functions $f$ with f(0)=0. $\endgroup$ Oct 15, 2014 at 6:06
  • $\begingroup$ @მამუკაჯიბლაძე So it is the ideal of continuoes function with compact supoert in $[-1,\;1]-\{0\}$. Am I correct? $\endgroup$ Oct 15, 2014 at 6:31
  • $\begingroup$ Yes sorry, you are right. $\endgroup$ Oct 15, 2014 at 10:59
  • $\begingroup$ @მამუკაჯიბლაძე thank you very much for your comment on consideration of action of G on itself. As you said, for G finite the space $Y=X-\emptyset=X$. Now I try to concentrate on infinite group. What do you think about $G=S^{1}$? $\endgroup$ Oct 15, 2014 at 11:31

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