I am trying to understand a proof in [this paper](https://arxiv.org/abs/1512.01669) (specifically theorem 5.4). In it, a fact is used that every element of the $W^\*$-algebra $A$ is a linear combination of exponential unitaries.


I've tried to reason this fact out, but I don't follow where this fact comes from or why it necessarily holds true. I do know that any element can be written as a sum of four unitaries (as explained [here](https://math.stackexchange.com/questions/2256498/prove-that-every-element-a-of-a-c-algebra-a-is-a-finite-linear-combination)), but I can't seem to find any information about being a sum of specifically exponential unitaries, which is important to the logic of the proof.


Why is this true, and is this known specifically about $W^\*$-algebras or is it a more general fact? Any information on this would be appreciated.



Every element of a $W^*$-algebra is a linear combination of exponential unitaries?
It is true in any unital $C^\ast$-algebra. Any element $x$ can be written as a linear combination of self-adjoints, by $x=\tfrac{1}{2}(x+x^\ast)+\tfrac{1}{2i}i(x−x^\ast)$. Moreover, if $y$ is a self-adjoint element with $\| y\| \leq 1$, and $\arccos\colon [−1,1]\to [0,π]$, then $u=\exp(i \arccos(y))$ is an exponential unitary such that $y=\tfrac{1}{2}(u+u^\ast)$. By combining these two results one gets the desired fact.


