Rigid Sides of Approximately Finite-Dimensional Simple Operator Algebras in Non-Separable Category

Abstract

Abstract Applying Popa’s orthogonality method to a new class of groups, we construct amenable group factors that are prime and have no infinite-dimensional regular abelian $\ast$-subalgebras. By adjusting Farah–Katsura’s solution of Dixmier’s problem to the von Neumann algebra setting, we obtain the 1st examples of prime approximately finite-dimensional factors and tensorially prime simple AF-algebras. Our results are proved in Zermelo--Fraenkel set theory with the axiom of choice (ZFC), thus in particular answering questions asked by Farah–Hathaway–Katsura–Tikuisis. We also directly determine central sequences of certain crossed products. This concludes the failure of the Kirchberg $\mathcal{O}_{\infty }$-absorption theorem in the non-separable setting.