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31
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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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.
The topological Ktheory of certain crystallographic groups
 JOURNAL OF NONCOMMUTATIVE GEOMETRY
, 2010
"... Let Γ be a semidirect product of the form Z n ⋊ρ Z/p where p is primeand the Zpaction ρ on Z n is free away from the origin. We will compute the topological Ktheory of the real and complex group C ∗algebra of Γ and show that Γ satisfies the unstable GromovLawsonRosenberg Conjecture. On the wa ..."
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Cited by 6 (4 self)
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Let Γ be a semidirect product of the form Z n ⋊ρ Z/p where p is primeand the Zpaction ρ on Z n is free away from the origin. We will compute the topological Ktheory of the real and complex group C ∗algebra of Γ and show that Γ satisfies the unstable GromovLawsonRosenberg Conjecture. On the way we will analyze the (co)homology and the topological Ktheory of the classifying spaces BΓ and BΓ. The latter is the quotient of the induced Z/paction on the torus T n.
Deformation of operator algebras by Borel cocycles
 J. Funct. Anal
"... Abstract. Assume that we are given a coaction δ of a locally compact group G on a C∗algebra A and a Tvalued Borel 2cocycle ω on G. Motivated by the approach of Kasprzak to Rieffel’s deformation we define a deformation Aω of A. Among other properties of Aω we show that Aω ⊗K(L2(G)) is canonically ..."
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Abstract. Assume that we are given a coaction δ of a locally compact group G on a C∗algebra A and a Tvalued Borel 2cocycle ω on G. Motivated by the approach of Kasprzak to Rieffel’s deformation we define a deformation Aω of A. Among other properties of Aω we show that Aω ⊗K(L2(G)) is canonically isomorphic to A oδ G ̂ oδ̂,ω G. This, together with a slight extension of a result of Echterhoff et al., implies that for groups satisfying the BaumConnes conjecture the Ktheory of Aω remains invariant under homotopies of ω.
FIBRATIONS WITH NONCOMMUTATIVE FIBERS
, 810
"... Abstract. We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C ∗algebra bundles. We then derive an analogue of the LeraySerre spectral sequence to compute the Ktheory of the fibration in terms of the cohomology of the base and the Ktheor ..."
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Abstract. We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C ∗algebra bundles. We then derive an analogue of the LeraySerre spectral sequence to compute the Ktheory of the fibration in terms of the cohomology of the base and the Ktheory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature. In recent years the study of the topological properties of C*algebra bundles plays a more and more prominent rôle in the field of Operator algebras. The main reason for this is twofold: on one side there are many important examples of C*algebras which do come with a canonical bundle structure. On the other
K and Ltheory of group rings
, 2010
"... This article will explore the K and Ltheory of group rings and their applications to algebra, geometry and topology. The FarrellJones Conjecture characterizes K and Ltheory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current statu ..."
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This article will explore the K and Ltheory of group rings and their applications to algebra, geometry and topology. The FarrellJones Conjecture characterizes K and Ltheory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its consequences are surveyed.
TOPOLOGICAL KTHEORY OF THE GROUP C∗ALGEBRA OF A SEMIDIRECT PRODUCT Z n ⋊Z/m FOR A FREE CONJUGATION ACTION
, 2011
"... We compute the topological Ktheory of the group C ∗algebra C ∗ r (Γ) for a group extension 1 → Zn → Γ → Z/m → 1 provided that the conjugation action of Z/m on Z n is free outside the origin. ..."
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We compute the topological Ktheory of the group C ∗algebra C ∗ r (Γ) for a group extension 1 → Zn → Γ → Z/m → 1 provided that the conjugation action of Z/m on Z n is free outside the origin.
Ktheory of noncommutative Bieberbach manifolds.
, 2014
"... We computeKtheory of noncommutative Bieberbach manifolds, which quotients of a threedimensional noncommutative torus by a free action of a cyclic group ZN, N = 2, 3, 4, 6. 1 ..."
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We computeKtheory of noncommutative Bieberbach manifolds, which quotients of a threedimensional noncommutative torus by a free action of a cyclic group ZN, N = 2, 3, 4, 6. 1
Research Statement
"... My research is in an area of functional analysis known as C ∗algebras. A C ∗algebra is an algebra A over C having an involution ∗: A → A and a norm satisfying the following properties. We write a ∗ for the image of a under ∗. The involution must be conjugate linear and satisfy (ab) ∗ = b ∗ a ∗ a ..."
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My research is in an area of functional analysis known as C ∗algebras. A C ∗algebra is an algebra A over C having an involution ∗: A → A and a norm satisfying the following properties. We write a ∗ for the image of a under ∗. The involution must be conjugate linear and satisfy (ab) ∗ = b ∗ a ∗ and (a ∗ ) ∗ = a for all a and b in A. The norm must satisfy ‖a ∗ a ‖ = ‖a ‖ 2 and the algebra must be complete in this norm. One example is the n by n matrices over C, Mn(C) with the involution given by the conjugate transpose. For a commutative example consider the continuous functions on any compact Hausdorff space X. The operations are pointwise and for the involution we use pointwise complex conjugation. In particular, for my research I have focused on the properties of crossed product C ∗algebras. Let A be a C ∗algebra and let α: G → Aut(A) be an action of a finite group G on A. Then, as a set, the crossed product C ∗ (G, A, α) is the group ring A[G]. However, the multiplication and involution are skewed by the action α of G on A. If G is not finite but is discrete, we must complete A[G] in a suitable norm. This construction has provided new examples of C ∗algebras, and new ways of looking at old and naturally occurring C ∗algebras. For example, the irrational rotation algebras Aθ were originally given by generators and relations but can also described as crossed products