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42
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 65 (9 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.
Amenability for dual Banach algebras
 Run 2] [Sel] [Spr] [Woo 1] [Woo 2] V. Runde, Lectures on Amenability. Lecture Notes in Mathematics 1774
, 2002
"... We define a Banach algebra A to be dual if A = (A∗) ∗ for a closed submodule A ∗ of A ∗. The class of dual Banach algebras includes all W ∗algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Ban ..."
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Cited by 28 (7 self)
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We define a Banach algebra A to be dual if A = (A∗) ∗ for a closed submodule A ∗ of A ∗. The class of dual Banach algebras includes all W ∗algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already superamenable and thus finitedimensional. We then develop two notions of amenability — Connesamenability and strong Connesamenability — which take the w ∗topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra A to the (strong) Connesamenability of A ∗ ∗ ; as an application, we show that there are reflexive Banach spaces with the approximation property such that L(E) is not Connesamenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudomeasures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones which avoids the equivalence of amenability and nuclearity for C ∗algebras.
The Completely Bounded Approximation Property For Discrete Crossed Products
 Indiana Univ. Math. J
, 1997
"... We consider the relationship between the Haagerup constants for a C algebra and its crossed product by an amenable group. We prove equality when the group is discrete, and deduce equality when the group is compact and abelian. () Partially supported by a NATO collaborative research grant. (y) ..."
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Cited by 19 (3 self)
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We consider the relationship between the Haagerup constants for a C algebra and its crossed product by an amenable group. We prove equality when the group is discrete, and deduce equality when the group is compact and abelian. () Partially supported by a NATO collaborative research grant. (y) Partially supported by an NSF research grant. x1. Introduction In [10] Haagerup introduced an important isomorphism invariant (A) for a C  algebra A, and a corresponding one w (M) for a von Neumann algebra M. For C  algebras, (A) is the infimum of numbers ? 0 for which there exists a net fOE fi : A ! A; kOE fi k cb g fi2B of finite rank completely bounded maps converging to the identity in the point norm topology ((A) = 1 if no such net exists). When (A) is finite, A is said to have the completely bounded approximation property (CBAP ). For von Neumann algebras w (M) is defined similarly, but the maps are normal and convergence is in the point w topology. These two consta...
Dual Banach algebras: Connesamenability, normal, virtual diagonals, and injectivity of the predual bimodule
, 2003
"... ..."
Amenable and weakly amenable Banach algebras with compact multiplication
 J. Funct. Anal
"... compact multiplication ..."
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A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC TO A C ∗ALGEBRA
"... Abstract. It has been a longstanding problem whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C∗algebra. In this note, we give a nonseparable counterexample. The existence of a separable counterexample remains an open problem. We also initiate a general study of unit ..."
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Cited by 8 (5 self)
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Abstract. It has been a longstanding problem whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C∗algebra. In this note, we give a nonseparable counterexample. The existence of a separable counterexample remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in C∗algebras and show that our method cannot produce a separable counterexample. 1.
Perturbations of nuclear C∗algebras
, 2009
"... Kadison and Kastler introduced a natural metric on the collection of all C∗subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also co ..."
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Cited by 8 (3 self)
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Kadison and Kastler introduced a natural metric on the collection of all C∗subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider onesided versions of these notions, obtaining embeddings from certain near inclusions involving separable nuclear C∗algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.
A Survey of Hochschild Cohomology for von Neumann Algebras
"... Abstract. In this paper we survey the field of Hochschild cohomology for von Neumann algebras. We describe the basic definitions and results without assuming background knowledge beyond some familiarity with von Neumann algebra theory. We offer no formal proofs of results, although we give detailed ..."
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Cited by 7 (1 self)
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Abstract. In this paper we survey the field of Hochschild cohomology for von Neumann algebras. We describe the basic definitions and results without assuming background knowledge beyond some familiarity with von Neumann algebra theory. We offer no formal proofs of results, although we give detailed references to the literature. Instead, we concentrate on explaining the main techniques and how they are applied to obtain the significant theorems of the subject. 1.