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?



Why is this a conditional expectation into the fixed point algebra?
For your first question, you can check that $E(a)$ is invariant with respect to the action, by using that the Haar measure is left-invariant. The other properties are also straightforward to check.


With respect to your second question, note that $\alpha\_t(S\_j^\*) = e^{-it} S\_j^\*$. Consider an element $S\_{i\_1} \cdots S\_{i\_k} S\_{j\_1}^\* \cdots S\_{j\_l}$ of the Cuntz algebra. Using the relation above you can see that this is invariant with respect to the gauge action if and only if $k=l$. These operators generate the fixed point algebra. I don't know if it's easy to see, for example, what the image of $S\_i$ is under this map.


