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18
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 66 (8 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.
The tracial Rokhlin property for actions of finite groups on C*algebras
, 2008
"... We define “tracial” analogs of the Rokhlin property for actions of finite groups, approximate representability of actions of finite abelian groups, and of approximate innerness. We prove the following four analogs of related “nontracial” results. • The crossed product of an infinite dimensional sim ..."
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Cited by 33 (10 self)
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We define “tracial” analogs of the Rokhlin property for actions of finite groups, approximate representability of actions of finite abelian groups, and of approximate innerness. We prove the following four analogs of related “nontracial” results. • The crossed product of an infinite dimensional simple separable unital C*algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero. • An outer action of a finite abelian group on an infinite dimensional simple separable unital C*algebra has the tracial Rokhlin property if and only if its dual is tracially approximately representable, and is tracially approximately representable if and only if its dual has the tracial Rokhlin property. • If a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital C*algebra has the tracial Rokhlin property, then it is tracially approximately representable. • An automorphism of an infinite dimensional simple separable unital C*algebra A with tracial rank zero is tracially approximately inner if and only if it is the identity on K0(A) mod infinitesimals. 0. Introduction. Tracially AF C*algebras, now known as C*algebras with tracial rank zero, were introduced in [14]. Roughly speaking, a C*algebra
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 33 (12 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.
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.
Finite cyclic group actions with the tracial Rokhlin property
, 2006
"... We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the foll ..."
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Cited by 13 (5 self)
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We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of Z/2Z on a UHF algebra has the tracial Rokhlin property; in particular, unlike for the strict Rokhlin property, every UHF algebra admits such an action. We prove that Blackadar’s action of Z/2Z on the 2 ∞ UHF algebra, whose crossed product is not AF because it has nontrivial K1group, has the tracial Rokhlin property, and we give an example of an action of Z/2Z on a simple unital AF algebra which has the tracial Rokhlin property and such that the K0group of the crossed product has torsion. In particular, the crossed product of a simple unital AF algebra by an action of Z/2Z with the tracial Rokhlin property need not be AF. We give examples of a tracially approximately representable action of Z/2Z on a simple unital AF algebra which is nontrivial on K0, and
FREENESS OF ACTIONS OF FINITE GROUPS ON C*ALGEBRAS
, 2009
"... We describe some of the forms of freeness of group actions on noncommutative C*algebras that have been used, with emphasis on actions of finite groups. We give some indications of their strengths, weaknesses, applications, and relationships to each other. The properties discussed include the Rokh ..."
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Cited by 12 (3 self)
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We describe some of the forms of freeness of group actions on noncommutative C*algebras that have been used, with emphasis on actions of finite groups. We give some indications of their strengths, weaknesses, applications, and relationships to each other. The properties discussed include the Rokhlin property, Ktheoretic freeness, the tracial Rokhlin property, pointwise outerness, saturation, hereditary saturation, and the requirement that the strong Connes spectrum be the entire dual.
The Cuntz semigroup of continuous functions into certain simple C∗algebras
 INTERNAT. J. MATH
"... This paper contains computations of the Cuntz semigroup of separable C∗algebras of the form C0(X, A), where A is a unital, simple, Zstable ASH algebra. The computations describe the Cuntz semigroup in terms of Murrayvon Neumann semigroups of C(K, A) for compact subsets K of X. In particular, th ..."
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Cited by 11 (1 self)
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This paper contains computations of the Cuntz semigroup of separable C∗algebras of the form C0(X, A), where A is a unital, simple, Zstable ASH algebra. The computations describe the Cuntz semigroup in terms of Murrayvon Neumann semigroups of C(K, A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(T, A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of wellbehaved nonsimple C∗algebras.
Crossed product C*algebras by finite group actions with the tracial rokhlin property
"... Abstract. In this paper we introduce an analog of the tracial Rokhlin property, called the projection free tracial Rokhlin property, for C ∗algebras which may not have any nontrivial projections. Using this we show that if A is an infinite dimensional stably finite simple unital C ∗algebra with st ..."
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Cited by 8 (0 self)
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Abstract. In this paper we introduce an analog of the tracial Rokhlin property, called the projection free tracial Rokhlin property, for C ∗algebras which may not have any nontrivial projections. Using this we show that if A is an infinite dimensional stably finite simple unital C ∗algebra with stable rank one, with strict comparison of positive elements, with only finitely many extreme tracial states, and with the property that every 2quasitrace is a trace, and if α is an action of a finite group G with the projection free tracial Rokhlin property, then the crossed product C ∗ (G, A, α) also has stable rank one. 1.
Strong Morita equivalence of higherdimensional noncommutative tori
, 2005
"... Abstract. We show that two C ∗algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗algebras of arb ..."
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Cited by 7 (2 self)
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Abstract. We show that two C ∗algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗algebras of arbitrary finitely generated abelian groups. 1.