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30
Permutations of strongly selfabsorbing C∗algebras
 TRANS. AM. MATH. SOC
"... Let G be a finite group acting on {1,..., n}. For any C∗algebra A, this defines an action of α of G on A⊗n. We show that if A tensorially absorbs a UHF algebra of infinite type, the JiangSu algebra, or is approximately divisible, then A×α G has the corresponding property as well. ..."
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Cited by 65 (16 self)
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Let G be a finite group acting on {1,..., n}. For any C∗algebra A, this defines an action of α of G on A⊗n. We show that if A tensorially absorbs a UHF algebra of infinite type, the JiangSu algebra, or is approximately divisible, then A×α G has the corresponding property as well.
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.
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
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
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.
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.
Cancellation for inclusions of C*algebras of finite depth
 Indiana Univ. Math. J
"... Abstract. Let 1 ∈ A ⊂ B be a pair of C*algebras with common unit. We prove that if E: B → A is a conditional expectation with indexfinite type and a quasibasis of n elements, then the topological stable rank satisfies tsr(B) ≤ tsr(A) + n − 1. As an application we show that if an inclusion 1 ∈ A ..."
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Cited by 4 (2 self)
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Abstract. Let 1 ∈ A ⊂ B be a pair of C*algebras with common unit. We prove that if E: B → A is a conditional expectation with indexfinite type and a quasibasis of n elements, then the topological stable rank satisfies tsr(B) ≤ tsr(A) + n − 1. As an application we show that if an inclusion 1 ∈ A ⊂ B of unital C*algebras has indexfinite type and finite depth, and A is a simple unital C*algebra with tsr(A) = 1 and Property (SP), then B has cancellation. In particular, if α an action of a finite group G on A, then the crossed product A ⋊α G has cancellation. For outer actions of Z, we obtain cancellation for A ⋊α Z under the additional condition that α ∗ = id on K0(A). Examples are given. 1.
Permanence properties for crossed products and fixed point algebras of finite groups
 Trans. Amer. Math. Soc
"... Abstract. Let α: G → Aut(A) be an action of a finite group G on a C*algebra A. We present some conditions under which properties of A pass to the crossed product C∗(G,A, α) or the fixed point algebra Aα. We mostly consider the ideal property, the projection property, topological dimension zero, and ..."
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Cited by 4 (1 self)
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Abstract. Let α: G → Aut(A) be an action of a finite group G on a C*algebra A. We present some conditions under which properties of A pass to the crossed product C∗(G,A, α) or the fixed point algebra Aα. We mostly consider the ideal property, the projection property, topological dimension zero, and pure infiniteness. In many of our results, additional conditions are necessary on the group, the algebra, or the action. Sometimes the action must be strongly pointwise outer, and in a few results it must have the Rokhlin property. When G is finite abelian, we prove that crossed products and fixed point algebras by G preserve topological dimension zero with no condition on the action. We give an example to show that the ideal property and the projection property do not pass to fixed point algebras (even when the group is Z2). The construction also gives an example of a C*algebra B which does not have the ideal property but such that M2(B) does have the ideal property; in fact, M2(B) has the projection property.