Language：English   Publishing type：Research paper (scientific journal)  

We introduce the concept of finitely coloured equivalence for unital ∗-homomorphisms between C∗-algebras, for which unitary equivalence is the 1- coloured case. We use this notion to classify ∗-homomorphisms from separable, unital, nuclear C∗-algebras into ultrapowers of simple, unital, nuclear, Z-stable C∗- algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C∗-algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a "homotopy equivalence implies isomorphism" result for large classes of C∗-algebras with finite nuclear dimension.

DOI： 10.10.1090/memo/1233

Language：English   Publishing type：Research paper (scientific journal)  

Language：Japanese   Publishing type：Research paper (scientific journal)  

Progress in the classification of C^* algebras

DOI： 10.11429/sugaku.0701044

Language：English   Publishing type：Research paper (scientific journal)  

Simple, separable, unital, monotracial and nuclear $$\mathrm {C}^{*}$$C∗-algebras are shown to have finite nuclear dimension whenever they absorb the Jiang–Su algebra $$\mathcal {Z}$$Z tensorially. This completes the proof of the Toms–Winter conjecture in the unique trace case.

DOI： 10.1007/s00222-015-0580-1

Language：English   Publishing type：Research paper (scientific journal)  

We show that the group C*-algebra of any elementary amenable group is quasidiagonal. This is an offspring of recent progress in the classification theory of nuclear C*-algebras.

DOI： 10.1007/s00039-015-0315-x

Language：English   Publishing type：Research paper (scientific journal)  

We consider a crossed product of a unital simple separable nuclear stably finite Ƶ-stable C∗-algebra A by a strongly outer cocycle action of a discrete countable amenable group Γ. Under the assumption that A has finitely many extremal tracial states and Γ is elementary amenable, we show that the twisted crossed product C∗-algebra is Ƶ-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Ƶ are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite C∗-algebras.

DOI： 10.1353/ajm.2014.0043

Language：English   Publishing type：Research paper (scientific journal)  

Let Abe a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal uniformly hyperfinite (UHF) algebra has decomposition rank at most one. We then prove that A is nuclear, quasidiagonal, and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF algebra has tracial rank zero. Using this result, we obtain a counterexample to the Powers-Sakai conjecture.

DOI： 10.1215/00127094-2826908

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s11511-012-0084-4

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

We consider a certain class of unital simple stably finite C*-algebras which absorb the Jiang-Su algebra Z tensorially. Under a mild assumption, we show that the crossed product of a C*-algebra in this class by a strongly outer action of ℤ ^N or a finite group is Z -stable. As an application, we also prove that all strongly outer actions of ℤ ² on Z are mutually cocycle conjugate.

DOI： 10.1007/s00220-011-1392-9

Language：English   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

For projectionless C*-algebras absorbing the Jiang-Su algebra tensorially, we study a kind of the Rohlin property for automorphisms. We show that the crossed products obtained by automorphisms with this Rohlin property also absorb the Jiang-Su algebra tensorially under a mild technical condition on the C*-algebras. In particular, for the Jiang-Su algebra we show the uniqueness up to outer conjugacy of the automorphism with this Rohlin property.

DOI： 10.1016/j.jfa.2010.04.006

Language：English   Publishing type：Research paper (scientific journal)  

Let G be an inductive limit of finite cyclic groups, and A be a unital simple projectionless C*-algebra with K₁(A) ≅ G and a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in Aut(A)/WInn(A) are conjugate, where WInn(A) means the subgroup of Aut(A) consisting of automorphisms which are inner in the tracial representation. In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple A-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and crossed products by actions of with the Rohlin property. Let A be a TAF-algebra in this class. We show that for any automorphism α of A there exists an automorphism of A with the Rohlin property such that ∼ α and α are asymptotically unitarily equivalent. For the proof we use an aperiodic automorphism of the Jiang-Su algebra.

DOI： 10.1142/S0129167X09005741

Language：English   Publishing type：Research paper (scientific journal)