Let $A$ be a C*-algebra and let $\alpha$ be an action of the circle group $S_1$ on $A$ (Gauge action). We define the following map: $$E:A\rightarrow A;\quad E(a):=\int\alpha_t(a)\textrm{d} t.$$ My question is why $E$ is a conditional expectation into the fixed point algebra for the gauge action?

For example if we take the Cuntz algebra $O_n$ and the gauge action as $$\alpha_t(S_j)=e^{it}S_j\quad \forall j=1\dots n,$$ where $ S_j $ is a generating set of isometries for $O_n$, and extending the action to the whole of $O_n$.

What is the value of the integration after restricting to the *-algebra generated algebraically by the isometries?