Rigid sides of approximately finite dimensional simple operator algebras in non-separable category

TL;DR

This paper constructs new examples of prime, tensorially prime, and AFD operator algebras in the non-separable setting, answering longstanding open questions and revealing limitations of existing absorption theorems.

Contribution

It introduces novel methods to build prime and tensorially prime operator algebras in non-separable categories, extending the scope of operator algebra theory.

Findings

Construction of amenable prime group factors with no infinite dimensional regular abelian subalgebras.

First examples of prime AFD factors and tensorially prime simple AF-algebras in ZFC.

Demonstration of the failure of Kirchberg's $ ext{O}_ ext{infty}$-absorption in non-separable cases.

Abstract

Applying Popa's orthogonality method to a new class of groups, we construct amenable group factors which are prime and have no infinite dimensional regular abelian *-subalgebras. By adjusting Farah--Katsura's solution of Dixmier's problem to the von Neumann algebra setting, we obtain the first examples of prime AFD factors and tensorially prime simple AF-algebras. Our results are proved in 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.