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We give necessary and sufficient conditions that show that both the group ofisometries and the group of measure-preserving isometries are Lie groups for alarge class of metric measure spaces. In addition we study, among otherexamples, whether spaces having a generalized lower Ricci curvature boundfulfill these requirements. The conditions are satisfied by RCD*-spaces and,under extra assumptions, by CD-spaces, CD*-spaces, and MCP-spaces. However, weshow that the MCP-condition by itself is not enough to guarantee a smoothbehavior of these automorphism groups. More generally we show that spaces withgood optimal transport properties meet as well the hypotheses.

The isometry group of an $\mathsf{RCD}^*$-space is Lie