Let A be C*-algebra that is acted upon by a compact abelian group. We show that if the fixed-point algebra of the action contains a Cartan subalgebra D satisfying an appropriate regularity condition, then A is the reduced C*-algebra of a groupoid twist. We further show that the embedding D ↪ A is uniquely determined by the twist. These results generalize Renault's results on Cartan subalgebras of C*-algebras.

Whilst conducting this research, JHB and AHF made use of meeting space made available by the Columbus Metropolitan Library. We would like to thank CML for their important work in the community. AHF would like to thank Christopher Schafhauser for patiently answering his questions on [4].

This work was partially supported by grants from the Simons Foundation (#316952 to David Pitts and #360563 to Sarah Reznikoff) and by the American Institute of Mathematics SQuaREs Program.