The stable uniqueness theorem for equivariant Kasparov theory

Abstract

This paper examines and strengthens the Cuntz-Thomsen picture of equivariant Kasparov theory for arbitrary second-countable locally compact groups, in which elements are given by certain pairs of cocycle representations between C * -dynamical systems. The main result is a stable uniqueness theorem that generalizes a fundamental characterization of ordinary KK-theory by Lin and Dadarlat-Eilers. Along the way, we prove an equivariant Cuntz-Thomsen picture analog of the fact that the equivalence relation of homotopy agrees with the (a priori stronger) equivalence relation of stable operator homotopy. The results proved in this paper will be employed as the technical centerpiece in forthcoming work of the authors to classify certain amenable group actions on Kirchberg algebras by equivariant Kasparov theory. Contents 12 3. Absorption of cocycle representations 19 4. Criteria for asymptotic unitary equivalence 32 5. The dynamical stable uniqueness theorem 38 References 40 2020 Mathematics Subject Classification. 46L55, 19K35.1 This is a tiny oversimplification. It captures how we reprove the original theorem as a special case of ours, but the former is actually a bit weaker. This boils down to the difference between proper and strong asymptotic unitary equivalence in Definition 4.1.

The dynamical Kirchberg–Phillips theorem

“…As is easily observed (see [38,Remark 1.4]), an action β: G↷B is strongly stable if and only if there is a sequence of isometries r n ∈M(B) β such that…”

“…An important consequence of this reduction principle is that it allows us to obtain explicit descriptions of examples of absorbing cocycle representations, which are highly relevant in light of the stable uniqueness theorem for equivariant KK-theory. We note here that in various places in both the second and third sections, many of the most non-trivial technical ingredients are imported as consequences of observations made in the context of studying general KK G -groups in our earlier article [38].…”

“…Although something similar had been demonstrated in Dadarlat-Eilers' original work, the recent work of the first author [37] includes a new proof of the original Kirchberg-Phillips theorem by exploiting the stable uniqueness theorem to its fullest. Motivated by the importance of such methods in the Elliott program, the authors developed in [38] a dynamical generalization of the stable uniqueness theorem in the context of equivariant KK-theory for arbitrary locally compact groups.…”

“…In a nutshell, an action β: G↷B can be seen to be isometrically shift-absorbing if B is locally approximated by L 2 (G, B) as an equivariant B -bimodule (Proposition 3.8), whereas β is amenable precisely when L 2 (G, B) admits a suitable net of approximate fixed points. This grants us access to the kind of averaging arguments that one usually only has with some kind of Rokhlin property, and becomes the key ingredient that allows us to apply our stable uniqueness theorem [38] in a fruitful way. All of this culminates in suitable existence and uniqueness theorems for the group actions under consideration, which we shall state here in an oversimplified form for the sake of readability.…”

Every countable group admits amenable actions on stably finite simple C*-algebras

“…For useful characterizations and basic properties of amenability of C * -dynamical systems, we refer the reader to [2], [6], [17]. A recent highlight around this subject is the successful classification results [22], [7], [8] of certain amenable actions on purely infinite simple C * -algebras.…”