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24
Regularity properties in the classification program for separable amenable C∗algebras
 BULL. AMER. MATH. SOC
, 2008
"... 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 in ..."
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Cited by 40 (7 self)
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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.
About the QWEP conjecture
, 2003
"... This is a detailed survey on the QWEP conjecture and Connes’ embedding problem. Most of contents are taken from Kirchberg’s paper [Invent. Math. 112 (1993)]. ..."
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Cited by 24 (0 self)
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This is a detailed survey on the QWEP conjecture and Connes’ embedding problem. Most of contents are taken from Kirchberg’s paper [Invent. Math. 112 (1993)].
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.
Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property
, 2004
"... We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Let A be ..."
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Cited by 14 (7 self)
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We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Let A be a stably finite simple unital C*algebra, and let α be an automorphism of A which has the tracial Rokhlin property. Suppose A has real rank zero and stable rank one, and suppose that the order on projections over A is determined by traces. Then the crossed product algebra C ∗ (Z, A, α) also has these three properties. We also present examples of C*algebras A with automorphisms α which satisfy the above assumptions, but such that C ∗ (Z, A, α) does not have tracial rank zero.
Furstenberg transformations on irrational rotation algebras
, 2004
"... We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by ..."
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Cited by 12 (7 self)
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We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by traces (Blackadar’s Second Fundamental Comparability Question). We also prove that several classes of simple quotients of the C*algebras of discrete subgroups of five dimensional nilpotent Lie groups, considered by Milnes and Walters, are crossed products of simple C*algebras (C*algebras of minimal ordinary Furstenberg transformations) by automorphisms which have the tracial Rokhlin property. It follows that these algebras also have stable rank one, real rank zero, and order on projections determined by traces.
On the classification of simple Zstable C ∗ algebras with real rank zero and finite decomposition rank
 Department of Mathematics and Statistics, University of New
"... Abstract. We show that, if A is a separable simple unital C ∗algebra which absorbs the Jiang–Su algebraZ tensorially and which has real rank zero and finite decomposition rank, then A is tracially AF in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elli ..."
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Abstract. We show that, if A is a separable simple unital C ∗algebra which absorbs the Jiang–Su algebraZ tensorially and which has real rank zero and finite decomposition rank, then A is tracially AF in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of C ∗algebras as above which, additionally, satisfy the Universal Coefficients Theorem. In particular, such algebras are completely determined by their ordered Ktheory. They are approximately homogeneous of topological dimension less than or equal to 3, approximately subhomogeneous of topological dimension at most 2 and their decomposition rank also is no greater than 2. It is the aim of the Elliott classification program to find complete Ktheoretic invariants for separable simple nuclear C ∗algebras. For purely infinite C ∗algebras, this task was accomplished by Kirchberg and Phillips, and there are numerous classification results for special inductive limits of (sub)homogeneous C ∗algebras,
Approximate Homotopy of Homomorphisms from C(X) into a Simple C ∗algebra
, 2006
"... Abstract. Let X be a finite CW complex and let h1, h2: C(X) → A be two unital homomorphisms, where A is a unital C ∗algebra. We study the problem when h1 and h2 are approximately homotopy. We present a Ktheoretical necessary and sufficient condition for them to be approximately homotopy under the ..."
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Cited by 11 (8 self)
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Abstract. Let X be a finite CW complex and let h1, h2: C(X) → A be two unital homomorphisms, where A is a unital C ∗algebra. We study the problem when h1 and h2 are approximately homotopy. We present a Ktheoretical necessary and sufficient condition for them to be approximately homotopy under the assumption that A is a unital separable simple C ∗algebra of tracial rank zero, or A is a unital purely infinite simple C ∗algebra. When they are approximately homotopy, we also give a bound for the length of the homotopy. These results are also extended to the case that h1 and h2 are approximately multiplicative contractive completely positive linear maps. Suppose that h: C(X) → A is a monomorphism and u ∈ A is a unitary (with [u] = {0} in K1(A)). We prove that, for any ǫ> 0, and any compact subset F ⊂ C(X), there exists δ> 0 and a finite subset G ⊂ C(X) satisfying the following: if ‖[h(f), u] ‖ < δ and Bott(h, u) = {0}, then there exists a continuous rectifiable path {ut: t ∈ [0, 1]} such that u0 = u, u1 = 1A and ‖[h(g), ut] ‖ < ǫ for all g ∈ F and t ∈ [0,1]. (e 0.1)
Zstable ASH algebras
"... Abstract. The Jiang–Su algebra Z has come to prominence in the classification program for nuclear C ∗algebras of late, due primarily to the fact that Elliott’s classification conjecture predicts that all simple, separable, and nuclear C ∗algebras with unperforated Ktheory will absorb Z tensoriall ..."
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Abstract. The Jiang–Su algebra Z has come to prominence in the classification program for nuclear C ∗algebras of late, due primarily to the fact that Elliott’s classification conjecture predicts that all simple, separable, and nuclear C ∗algebras with unperforated Ktheory will absorb Z tensorially (i.e., will be Zstable). There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable andZstable C ∗algebras. We prove that virtually all classes of nuclear C ∗algebras for which the Elliott conjecture has been confirmed so far, consist of Zstable C ∗algebras. This result follows in large part from the following theorem, also proved herein: separable and approximately divisible C ∗algebras areZstable. The Jiang–Su algebra Z is a simple, separable, unital and nuclear C ∗algebra KKequivalent to C ([12]). Since its discovery in 1995 there has been a steady accumulation of evidence linking Z to Elliott’s program to classify separable, nuclear C ∗algebras via Ktheoretic invariants: in [12], Jiang and Su prove that simple,
Simple C ∗ algebras with locally finite decomposition rank, in preparation
"... Abstract. We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C ∗algebras. The concept is particularly relevant for Elliott’s program to classify nuclear C ∗algebras by Ktheory data. We study some of its properties and show that ..."
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Cited by 10 (7 self)
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Abstract. We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C ∗algebras. The concept is particularly relevant for Elliott’s program to classify nuclear C ∗algebras by Ktheory data. We study some of its properties and show that a simple unital C ∗algebra, which has locally finite decomposition rank, real rank zero and which absorbs the Jiang–Su algebra Z tensorially, has tracial rank zero in the sense of Lin. As a consequence, any such C ∗algebra, if it additionally satisfies the Universal Coefficients Theorem, is approximately homogeneous of topological dimension at most 3. Our result in particular confirms the Elliott conjecture for the class of simple unital Zstable ASH algebras with real rank zero. Moreover, it implies that simple unitalZstable AH algebras with real rank zero not only have slow dimension growth in the ASH sense, but even in the AH sense. This note is concerned with the stably finite real rank zero case of Elliott’s program