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

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 O ∞ -absorption theorem in the non-separable setting.2000 Mathematics Subject Classification. Primary 46L05, 46L10.

Non-amenable tight squeezes by Kirchberg algebras

“…Since A ⊗ O ∞ is simple, purely infinite, and α ⊗ β is outer (because of its amenability and the fact that F ∞ has no non-trivial amenable normal subgroup), it follows from Kishimoto's theorem [32] that C is purely infinite and simple (see e.g. Lemma 6.3 of [55] for details). Since A ⊗ O ∞ is nuclear, so is C by the amenability of α ⊗ β.…”

“…(However the resulting ambient algebras would be mysterious, cf. [55]). In particular, the reduced group C * -algebras of uncountable free groups admit a nuclear minimal ambient C * -algebra.…”

On pathological properties of fixed point algebras in Kirchberg algebras

“…By Kishimoto's theorem [18], one can conclude that C r,γ Γ, (C ⊗ Γ O ∞ ) r,θ Γ, and D are simple and purely infinite (see e.g. lemma 6.3 of [34] for details). By Cuntz's theorem [6], one can take a projection p in C r,γ Γ which represents a generator of K 0 (C r,γ Γ).…”