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12
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.
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.
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.
Finite group actions on certain stably projectionless C∗algebras with the Rokhlin property
, 1308
"... ar ..."
Actions of Finite Groups on Substitution Tilings and Their Associated C*algebras
"... The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*algebras. Finite symmetry groups of the ..."
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The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*algebras respectively. Of particular interest are the crossed product C*algebras; we derive important structure results about them and compute their Ktheory. ii Acknowledgements First and foremost, I would like to thank my wife and the love of my life Anna. This would not have been possible without her love, support, encouragement and friendship. I am also eternally grateful to my wonderful supervisor Thierry Giordano for being generous with excellent advice (both mathematical and otherwise) and financial support throughout my PhD studies. I would also like to thank my family: my mother Susan, my brother Bob, my grandparents Noeline (Nana) and Peter, and my stepfather TJ. In particular I must thank my Nana, who taught me at a young age the importance of continual learning, inquisitiveness, curiosity, and compassion. Mathematically, I would like to thank Ian Putnam, Michael Whittaker, Daniel Gonçalves, David Handelman, and Siegfried Echterhoff for many extremely helpful conversations about this thesis. I would also like to thank NSERC for financial support for the first half of my PhD studies. iii Dedication In memory of my father Ray, who was taken from us before I could finish this degree. We love you, dad. iv
CODES AS FRACTALS AND NONCOMMUTATIVE SPACES
, 1107
"... Abstract. We consider the CSS algorithm relating selforthogonal classical linear codes to qary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational ..."
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Abstract. We consider the CSS algorithm relating selforthogonal classical linear codes to qary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational noncommutative tori and finite abelian group actions on Cuntz algebras and fractals associated to the classical codes. 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