We start with a finite dimensional unitary representation over $\mathbb C$ of some (compact) group $G$. In the matrix algebra, take a unitary matrix $U$ that is invariant under the conjugation action of $G$. Is it always possible to find a log of $U$ that is also invariant under the action of $G$? I know taking log of a unitary without regard to the action is easy since $U$ is diagonalizable. Thanks
How about infinite dimensions?