The Isometry Group of an RCD* Space is Lie

We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by R C D (au)-spaces and, under extra assumptions, by C D-spaces, C D (au) P-spaces. However, we show that the M C C P-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups.