On pathological properties of fixed point algebras in Kirchberg algebras

Abstract

We investigate how the fixed point algebra of a C * -dynamical system can differ from the underlying C * -algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra O 2 whose fixed point algebra is almost equal to the reduced group C * -algebra C * r (Γ). Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.2000 Mathematics Subject Classification. Primary 22D25, 46L55, Secondary 46L05.

Equivariant -absorption theorem for exact groups

“…Recently, in [Suz19a], we discovered actions of non-amenable groups on simple -algebras with properties which should be regarded as amenability of -dynamical systems. A few applications of such actions are already found in -algebra theory; see [Suz20a, Suz20b, Suz19b]. We believe that this novel phenomenon provides a new rich field in -algebra theory.…”

Non-amenable tight squeezes by Kirchberg algebras

“…Let β : F ∞ ↷ O ∞ be an amenable action obtained in Corollary 4.5. As shown in the proofs of the Proposition in [56] and Theorem 5.1 in [54] (by using [30], [41]), the crossed product…”

“…We remark that Inn(A) in Lemma 2.3 is not replaceable by its pointwise norm closure (the group of approximately inner automorphisms). In fact, when the acting group is a non-commutative free group, by [33], any C * -dynamical system on a simple separable C * -algebra admits a non-amenable approximately inner perturbation (see the Proposition in [56] for details and an application).…”

“…Let A be a Kirchberg algebra. We have constructed, in the proof of the Proposition in [56] (see also the proof of Theorem 5.1 in [54]), an action α : F ∞ ↷ C on a unital Kirchberg algebra C in the bootstrap class whose reduced crossed product C ⋊ r,α F ∞ is non-nuclear, purely infinite simple, and KK-equivalent to O ∞ . Let β : F ∞ ↷ O ∞ be an amenable action obtained in Corollary 4.5.…”

“…□ Remark 5.2. By a similar method to the Proposition in [56] (by using [39]), one can arrange the smallest algebra in Theorem B not having the completely bounded approximation property (see Section 12.3 of [9] for the definition). is simple.…”