Strong classification of purely infinite Cuntz-Krieger algebras

Abstract

Abstract. In 2006, Restorff completed the classification of all Cuntz-Krieger algebras with finitely many ideals (i.e., those that are purely infinite) up to stable isomorphism. He left open the questions concerning strong classification up to stable isomorphism and unital classification. In this paper, we address both questions. We show that any isomorphism between the reduced filtered K-theory of two Cuntz-Krieger algebras with finitely many ideals lifts to a * -isomorphism between the stabilized Cuntz-Krieger algebras. As a result, we also obtain strong unital classification.

The complete classification of unital graph $C^*$-algebras: Geometric and strong

“…An unexpected obstruction for obtaining strong classification was identified in [ARR14], as indeed the full filtered K-theory fails to have this property even for Cuntz-Krieger algebras. This focused the efforts on establishing strong classification for the standard reduced invariant, and in [CRR16], a complete description was given of the action on K-theory by the moves constituting Cuntz move equivalence. After obtaining such a description for Move (P) as well, we are able to conclude that our classification result is indeed strong: every given isomorphism at the level of the invariant is in fact induced by the moves in an appropriate sense.…”

Invariance of the Cuntz splice

“…In a similar way as in the above mentioned cases, this result is a key step in the recent development in the geometric classification of general Cuntz-Krieger algebras and of unital graph C * -algebras ( [ERRS16a,ERRS16b]) as well as in the question of strong classification of general Cuntz-Krieger algebras and of unital graph C * -algebras ( [CRR17,ERRS16b]). In fact, using the results as well as the proof methods of the present paper, these papers close the classification problem for all Cuntz-Krieger algebras and for all unital graph C * -algebras.…”

“…In this paper we show in general that Cuntz splicing a vertex that supports two distinct return paths yields stably isomorphic graph C * -algebras -only assuming that the graph is countable. The results, the proofs and methods of this paper are important for recent development in the geometric classification of general Cuntz-Krieger algebras and of unital graph C * -algebras ( [ERRS16a], [ERRS16b]) as well as for the question of strong classification of general Cuntz-Krieger algebras and of unital graph C * -algebras ( [CRR16], [ERRS16b]).…”

The nuclear dimension of $\mathcal O_\infty$-stable $C^*$-algebras

“…We will prove the second assertion, about θ with decomposition rank at most n. The nuclear dimension statement follows from a very similar small modification to the corresponding statement for algebras of finite nuclear dimension [59, Proposition 3.2]. 8 Fix a finite subset F ⊆ A of positive elements of norm 1 and 0 < ǫ < 1. Assume that θ has decomposition rank at most n. Using the Stinespring calculation of [33,Lemma 3.4], it suffices to find an n-decomposable approximation (F, ψ, η) of θ for F up to ǫ with η contractive such that…”

“…Tensorial absorbtion of O ∞ also has profound implications for non-simple C * -algebras, including Kirchberg and Rørdam's algebraic characterisation of O ∞ -stable nuclear C * -algebras ( [32]) and Kirchberg's classification of separable nuclear O ∞ -stable C * -algebras via ideal-related bivariant K-theory ( [25]) as outlined in [27]. This has played an important role in the classification of non-simple Cuntz-Krieger algebras and their automorphisms ( [8,14]) and other non-simple algebras with small ideal lattices ( [1]). So it is natural to seek to compute the nuclear dimension of O ∞ -stable C * -algebras.…”