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31
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
Zstability of crossed products by strongly outer actions
"... We consider a crossed product of a unital simple separable nuclear stably finite Zstable Calgebra A by a strongly outer cocycle action of a discrete countable amenable group Γ. Under the assumption that A has finitely many extremal tracial states and Γ is elementary amenable, we show that the twis ..."
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Cited by 24 (7 self)
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We consider a crossed product of a unital simple separable nuclear stably finite Zstable Calgebra A by a strongly outer cocycle action of a discrete countable amenable group Γ. Under the assumption that A has finitely many extremal tracial states and Γ is elementary amenable, we show that the twisted crossed product Calgebra is Zstable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Z are cocycle conjugate to each other. This is the first classification result for actions of nonabelian infinite groups on stably finite Calgebras. 1
C*ALGEBRAS ASSOCIATED WITH INTEGRAL DOMAINS AND CROSSED PRODUCTS BY ACTIONS ON ADELE SPACES
, 906
"... Abstract. We compute the Ktheory for C*algebras naturally associated with rings of integers in number fields. The main ingredient is a duality theorem for arbitrary global fields. It allows us to identify the crossed product arising from affine transformations on the finite adeles with the analogo ..."
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Cited by 21 (13 self)
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Abstract. We compute the Ktheory for C*algebras naturally associated with rings of integers in number fields. The main ingredient is a duality theorem for arbitrary global fields. It allows us to identify the crossed product arising from affine transformations on the finite adeles with the analogous crossed product algebra over the infinite adele space. 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.
Every simple higher dimensional noncommutative torus is an AT algebra
, 2006
"... We prove that every simple higher dimensional noncommutative torus is an AT algebra. ..."
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Cited by 18 (7 self)
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We prove that every simple higher dimensional noncommutative torus is an AT algebra.
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
Structure and Ktheory of crossed products by proper actions, Expo
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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.
On the Ktheory of crossed products by automorphic semigroup actions, in preparation
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