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36
CANCELLATION AND STABLE RANK FOR DIRECT LIMITS OF RECURSIVE SUBHOMOGENEOUS ALGEBRAS
, 2001
"... Abstract. We prove the following results for a unital simple direct limit A of recursive subhomogeneous algebras with no dimension growth: (1) tsr(A) = 1. (2) The projections in M∞(A) satisfy cancellation: if e ⊕ q ∼ f ⊕ q, then e ∼ f. (3) A satisfies Blackadar’s Second Fundamental Comparability Qu ..."
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Cited by 35 (10 self)
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Abstract. We prove the following results for a unital simple direct limit A of recursive subhomogeneous algebras with no dimension growth: (1) tsr(A) = 1. (2) The projections in M∞(A) satisfy cancellation: if e ⊕ q ∼ f ⊕ q, then e ∼ f. (3) A satisfies Blackadar’s Second Fundamental Comparability Question: if p, q ∈ M∞(A) are projections such that τ(p) < τ(q) for all normalized traces τ on A, then p � q. (4) K0(A) is unperforated for the strict order: if η ∈ K0(A) and there is n> 0 such that nη> 0, then η> 0. The last three of these results hold under certain weaker dimension growth conditions and without assuming simplicity. We use these results to obtain previously unknown information on the ordered K-theory of the crossed product C ∗ (Z, X, h) obtained from a minimal homeomorphism of an infinite finite dimensional compact metric space X. Specifically, K0(C ∗ (Z, X, h)) is unperforated
RECURSIVE SUBHOMOGENEOUS ALGEBRAS
, 2001
"... Abstract. We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which al ..."
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Cited by 29 (1 self)
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Abstract. We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form C(X, Mn) many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies in particular that if A is a separable C*-algebra whose irreducible representations all have dimension at most N <∞, and if for each n the space of n-dimensional irreducible representations has finite covering dimension, then A is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it. Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in a separate paper.
THE K-THEORY OF HEEGAARD-TYPE QUANTUM 3-SPHERES Dedicated to the memory of Olaf Richter.
, 2004
"... We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C ∗-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C ∗ ..."
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Cited by 20 (6 self)
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We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C ∗-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C ∗-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts. 1
Crossed products by finite group actions with the Rokhlin property
, 2009
"... We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: • AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. • Sim ..."
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Cited by 19 (7 self)
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We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: • AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. • Simple unital AH algebras with slow dimension growth and real rank zero. • C*-algebras with real rank zero or stable rank one. • Simple C*-algebras for which the order on projections is determined by traces. • C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. • C*-algebras with a unique tracial state. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.
On embeddings of full amalgamated free product of C∗-algebras
- Proc. Amer. Math. Soc
"... Abstract. We examine the question of when the ∗–homomorphism λ: A∗DB → A ̃ ∗ eD B ̃ of full amalgamated free product C∗–algebras, arising from compatible inclusions of C∗–algebras A ⊆ Ã, B ⊆ B ̃ and D ⊆ D̃, is an embedding. Results giving sufficient conditions for λ to be injective, as well of cla ..."
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Cited by 10 (0 self)
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Abstract. We examine the question of when the ∗–homomorphism λ: A∗DB → A ̃ ∗ eD B ̃ of full amalgamated free product C∗–algebras, arising from compatible inclusions of C∗–algebras A ⊆ Ã, B ⊆ B ̃ and D ⊆ D̃, is an embedding. Results giving sufficient conditions for λ to be injective, as well of classes of examples where λ fails to be injective, are obtained. As an application, we give necessary and sufficient condition for the full amalgamated free product of finite dimensional C∗–algebras to be residually finite dimensional. 1.
On Rørdam’s classification of certain C∗-algebras with one non-trivial ideal, II
, 2007
"... In this paper we extend the classification results obtained by Rørdam in the paper [16]. We prove a strong classification theorem for the unital essential extensions of Kirchberg algebras, a classification theorem for the non-stable, non-unital essential extensions of Kirchberg algebras, and we char ..."
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Cited by 10 (8 self)
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In this paper we extend the classification results obtained by Rørdam in the paper [16]. We prove a strong classification theorem for the unital essential extensions of Kirchberg algebras, a classification theorem for the non-stable, non-unital essential extensions of Kirchberg algebras, and we characterize the range in both cases. The invariants are cyclic six term exact sequences together with the class of some unit.
Amalgamated free products of C∗-bundles
, 2007
"... Given two unital continuous C∗-bundles A and B over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X). ..."
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Cited by 7 (5 self)
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Given two unital continuous C∗-bundles A and B over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X).
Decomposition rank of subhomogeneous C∗-algebras
- Proc. London Math. Soc
"... In [4], E. Kirchberg and the author introduced the decomposition rank; this is a non-commutative generalization of topological covering dimension. If A is a nuclear C-algebra, the decomposition rank of A, drA, is de0ned by imposing a certain condition on systems of completely positive (c.p.) approxi ..."
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Cited by 6 (3 self)
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In [4], E. Kirchberg and the author introduced the decomposition rank; this is a non-commutative generalization of topological covering dimension. If A is a nuclear C-algebra, the decomposition rank of A, drA, is de0ned by imposing a certain condition on systems of completely positive (c.p.) approximations of A; see
