Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^\*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I believe that it is true that $A=\oplus\_{\chi\in \hat{G}} A\_\chi$, with $A\_\chi$ being the subspace made of elements $x\in A$ such that $\alpha\_g(x)=\chi(g)x$. Moreover, I think that there exist a family of linear maps (indexed by the characters of the group) $E\_\chi: A\to A\_\chi$
$$
E\_\chi(x)=\int\_G \overline{\chi(g)}\alpha\_g(x) 
$$
Can someone tell me if it is true and give me reference to a book or paper? Thanks for the help.



Spectral decomposition of a C$^*$algebra with respect to an action of a compact abelian group
The facts that $A$ is a $C^\*$-algebra and that $G$ is abelian are irrelevant: such a decomposition holds more generally for any Banach space $A$ with a continuous linear action of a compact group. The reference I know for this is *Representations of a compact group on a Banach space* by Shiga, available 
[here](https://projecteuclid.org/euclid.jmsj/1261414131). There are probably earlier references for abelian groups, but I do not know them.


