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104
Regularity properties in the classification program for separable amenable
 C ∗ algebras, Bull. Amer. Math. Soc
"... Abstract. We report on recent progress in the program to classify separable amenable C ∗algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Zstability, and on the importance of the Cuntz semigr ..."
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Cited by 31 (7 self)
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Abstract. We report on recent progress in the program to classify separable amenable C ∗algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Zstability, and on the importance of the Cuntz semigroup. We include a brief history of the program’s successes since 1989, a more detailed look at the Villadsentype algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup. 1.
On the classification problem for nuclear C ∗ algebras
"... Abstract. We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C ∗algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of ..."
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Cited by 19 (8 self)
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Abstract. We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C ∗algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C ∗algebras are farreaching: one has, among other things, that existing results on the classification of simple, unital AH algebras via the Elliott invariant of Ktheoretic data are the best possible, and that these cannot be improved by the addition of continuous homotopy invariant functors to the Elliott invariant. 1.
A homeomorphism invariant for substitution tiling spaces
, 2000
"... We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed dire ..."
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Cited by 16 (10 self)
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We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Čech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheellike tiling spaces. We also introduce a module structure on cohomology which is very convenient as well as of intuitive value.
Representations of distributive semilattices in ideal lattices of various algebraic structures, Algebra Universalis 45
"... Abstract. We study the relationships among existing results about representations of distributive semilattices by ideals in dimension groups, von Neumann regular rings, C*algebras, and complemented modular lattices. We prove additional representation results which exhibit further connections with t ..."
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Cited by 16 (13 self)
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Abstract. We study the relationships among existing results about representations of distributive semilattices by ideals in dimension groups, von Neumann regular rings, C*algebras, and complemented modular lattices. We prove additional representation results which exhibit further connections with the scattered literature on these different topics.
Strongly selfabsorbing C∗algebras
 PREPRINT, MATH. ARCHIVE MATH.OA/0502211
, 2005
"... Say that a separable, unital C ∗algebra D ≇ C is strongly selfabsorbing if there exists an isomorphism ϕ: D → D ⊗ D such that ϕ and idD ⊗1D are approximately unitarily equivalent ∗homomorphisms. We study this class of algebras, which includes the Cuntz algebras O2, O∞, the UHF algebras of infinite ..."
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Cited by 15 (7 self)
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Say that a separable, unital C ∗algebra D ≇ C is strongly selfabsorbing if there exists an isomorphism ϕ: D → D ⊗ D such that ϕ and idD ⊗1D are approximately unitarily equivalent ∗homomorphisms. We study this class of algebras, which includes the Cuntz algebras O2, O∞, the UHF algebras of infinite type, the Jiang–Su algebra Z and tensor products of O∞ with UHF algebras of infinite type. Given a strongly selfabsorbing C ∗algebra D we characterise when a separable C ∗algebra absorbs D tensorially (i.e., is Dstable), and prove closure properties for the class of separable Dstable C ∗algebras. Finally, we compute the possible Kgroups and prove a number of classification results which suggest that the examples listed above are the only strongly selfabsorbing C ∗algebras.
The Elliott conjecture for Villadsen algebras of the first type
, 2006
"... We study the class of simple C∗algebras introduced by Villadsen in his pioneering work on perforated ordered Ktheory. We establish six equivalent characterisations of the proper subclass which satisfies the strong form of Elliott’s classification conjecture: two C∗algebraic (Zstability and appro ..."
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Cited by 15 (3 self)
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We study the class of simple C∗algebras introduced by Villadsen in his pioneering work on perforated ordered Ktheory. We establish six equivalent characterisations of the proper subclass which satisfies the strong form of Elliott’s classification conjecture: two C∗algebraic (Zstability and approximate divisibility), one Ktheoretic (strict comparison of positive elements), and three topological (finite decomposition rank, slow dimension growth, and bounded dimension growth). The equivalence of Zstability and strict comparison constitutes a stably finite version of Kirchberg’s characterisation of purely infinite C∗algebras. The other equivalences confirm, for Villadsen’s algebras, heretofore conjectural relationships between various notions of good behaviour for nuclear C∗algebras.
The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(Z)
, 2006
"... Let F ⊆ SL2(Z) be a finite subgroup (necessarily isomorphic to one of Z2, Z3, Z4, or Z6), and let F act on the irrational rotational algebra Aθ via the restriction of the canonical action of SL2(Z). Then the crossed product Aθ ⋊α F and the fixed point algebra AF θ are AF algebras. The same is true ..."
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Cited by 14 (8 self)
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Let F ⊆ SL2(Z) be a finite subgroup (necessarily isomorphic to one of Z2, Z3, Z4, or Z6), and let F act on the irrational rotational algebra Aθ via the restriction of the canonical action of SL2(Z). Then the crossed product Aθ ⋊α F and the fixed point algebra AF θ are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z2 on any simple ddimensional noncommutative torus AΘ. Along the way, we prove a number of general results which should have useful applications in other situations.
Decidability of the isomorphism problem for stationary AFalgebras
, 1999
"... The notion of isomorphism of stable AFC ∗algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C∗isomorphism induces an equivalence relation on these matrices, call ..."
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Cited by 13 (3 self)
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The notion of isomorphism of stable AFC ∗algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C∗isomorphism induces an equivalence relation on these matrices, called C ∗equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e., there is an algorithm that can be used to check in a finite number of steps whether two given primitive nonsingular matrices are C∗equivalent or not.