| 1. |
G.A.Elliott, Y. Sato, Rationally AF algebras and KMS states of Z-absorbing C*-algebras, arXiv:2207.11653, 2022.09.  |
| 2. |
Yasuhiko Sato, Certain aperiodic automorphisms of unital simple projectionless C* -algebras, International Journal of Mathematics, 10.1142/S0129167X09005741, 20, 10, 1233-1261, 2009.10, 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.. |
| 3. |
Joan Bosa, Nathanial P. Brown, Yasuhiko Sato, Aaron Tikuisis, Stuart White, Wilhelm Winter, Covering dimension of C∗-algebras and 2-coloured classification, Memoirs of the American Mathematical Society, 10.10.1090/memo/1233, 257, 1233, 1-112, 2019.01, 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.. |
| 4. |
Yasuhiko Sato, Actions of amenable groups and crossed products of Z-absorbing C*-algebras, Advanced Studies in Pure mathematics, 80, 5, 189-210, 2019. |
| 5. |
Progress in the classification of C* algebras. |
| 6. |
Yasuhiko Sato, Stuart White, Wilhelm Winter, Nuclear dimension and Z -stability, Inventiones Mathematicae, 10.1007/s00222-015-0580-1, 202, 2, 893-921, 2015.11, 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.. |
| 7. |
Narutaka Ozawa, Mikael Rørdam, Yasuhiko Sato, Elementary amenable groups are quasidiagonal, Geometric and Functional Analysis, 10.1007/s00039-015-0315-x, 25, 1, 307-316, 2015.01, 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.. |
| 8. |
Hiroki Matui, Yasuhiko Sato, Ƶ-stability of crossed products by strongly outer actions II, American Journal of Mathematics, 10.1353/ajm.2014.0043, 136, 6, 1441-1496, 2014.12, 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.. |
| 9. |
Hiroki Matui, Yasuhiko Sato, Decomposition rank of UHF-absorbing c* -algebras, Duke Mathematical Journal, 10.1215/00127094-2826908, 163, 14, 2687-2708, 2014.01, 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.. |
| 10. |
Hiroki Matui, Yasuhiko Sato, Strict comparison and Z-absorption of nuclear C *-algebras, Acta Mathematica, 10.1007/s11511-012-0084-4, 209, 1, 179-196, 2012.10. |
| 11. |
Hiroki Matui, Yasuhiko Sato, Z-Stability of Crossed Products by Strongly Outer Actions, Communications in Mathematical Physics, 10.1007/s00220-011-1392-9, 314, 1, 193-228, 2012.07, 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.. |
| 12. |
Yasuhiko Sato, The Rohlin property for automorphisms of the Jiang-Su algebra, Journal of Functional Analysis, 10.1016/j.jfa.2010.04.006, 259, 2, 453-476, 2010.07, 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.. |
| 13. |
Yasuhiko Sato, A generalization of the Jiang-Su construction, arXiv:0903.5286, 2009.07. |
| 14. |
Yasuhiko Sato, Discrete amenable group actions on von Neumann algebras and invariant nuclear C*-subalgebras, arXiv:1104.4339, 2011.10. |
| 15. |
Yasuhiko Sato, Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary, arXiv:1209.3000, 2012.10. |
| 16. |
Yasuhiko Sato, 2-positive almost order zero maps and decomposition rank, Journal of Operator Theory, 2020.12, We consider 2-positive almost order zero (disjointness preserving) maps on C*-algebras. Generalizing the argument of M. Choi for multiplicative domains, we give an internal characterization of almost order zero for 2-positive maps. It is also shown that complete positivity can be reduced to 2-positivity in the definition of decomposition rank for unital separable C*-algebras.. |
