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Articles 1 - 23 of 23
Full-Text Articles in Science and Technology Studies
Fractal Spectral Triples On Kellendonk's C*-Algebra Of A Substitution Tiling, Michael Mampusti, Michael F. Whittaker
Fractal Spectral Triples On Kellendonk's C*-Algebra Of A Substitution Tiling, Michael Mampusti, Michael F. Whittaker
Faculty of Engineering and Information Sciences - Papers: Part A
.We introduce a new class of noncommutative spectral triples on Kellendonk's C*-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perro-Frobenius eigenvalue. We show that each spectral triple is θ-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it …
Nonunital Spectral Triples And Metric Completeness In Unbounded Kk-Theory, Bram Mesland, Adam C. Rennie
Nonunital Spectral Triples And Metric Completeness In Unbounded Kk-Theory, Bram Mesland, Adam C. Rennie
Faculty of Engineering and Information Sciences - Papers: Part A
We consider the general properties of bounded approximate units in non-self-adjoint operator algebras. Such algebras arise naturally from the differential structure of spectral triples and unbounded Kasparov modules. Our results allow us to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, self-adjointness and regularity of induced operators on tensor product C⁎-modules and the lifting of Kasparov products to the unbounded category. In particular, we prove novel existence results for quasicentral approximate units in non-self-adjoint operator algebras, allowing us to strengthen Kasparov's technical theorem and extend it to this realm. Finally, we show …
Summability For Nonunital Spectral Triples, Adam C. Rennie
Summability For Nonunital Spectral Triples, Adam C. Rennie
Associate Professor Adam Rennie
This paper examines the issue of summability for spectral triples for the class of nonunital algebras. For the case of (p, -) summability, we prove that the Dixmier trace can be used to define a (semifinite) trace on the algebra of the spectral triple. We show this trace is well-behaved, and provide a criteria for measurability of an operator in terms of zeta functions. We also show that all our hypotheses are satisfied by spectral triples arising from eodesically complete Riemannian manifolds. In addition, we indicate how the Local Index Theorem of Connes-Moscovici extends to our nonunital setting.
Smoothness And Locality For Nonunital Spectral Triples, Adam C. Rennie
Smoothness And Locality For Nonunital Spectral Triples, Adam C. Rennie
Associate Professor Adam Rennie
To deal with technical issues in noncommuntative geometry for nonunital algebras, we introduce a useful class of algebras and their modules. Thes algebras and modules allo us to extend all of the smoothness results for spectral triples to the nonunital case. In addition, we show that smooth spectral tiples are closed under the C- functional calculus of self-adjoint elements. In the final section we show that our algebras allow the formulation of Poincare Duality and that the algebras of smooth spectral triples are H-unital.
Dense Domains, Symmetric Operators And Spectral Triples, Iain Forsyth, B Mesland, Adam Rennie
Dense Domains, Symmetric Operators And Spectral Triples, Iain Forsyth, B Mesland, Adam Rennie
Associate Professor Adam Rennie
This article is about erroneous attempts to weaken the standard definition of unbounded Kasparov module (or spectral triple). This issue has been addressed previously, but here we present concrete counterexamples to claims in the literature that Fredholm modules can be obtained from these weaker variations of spectral triple. Our counterexamples are constructed using self-adjoint extensions of symmetric operators.
Dense Domains, Symmetric Operators And Spectral Triples, Iain G. Forsyth, B Mesland, Adam C. Rennie
Dense Domains, Symmetric Operators And Spectral Triples, Iain G. Forsyth, B Mesland, Adam C. Rennie
Faculty of Engineering and Information Sciences - Papers: Part A
This article is about erroneous attempts to weaken the standard definition of unbounded Kasparov module (or spectral triple). This issue has been addressed previously, but here we present concrete counterexamples to claims in the literature that Fredholm modules can be obtained from these weaker variations of spectral triple. Our counterexamples are constructed using self-adjoint extensions of symmetric operators.
The Chern Character Of Semifinite Spectral Triples, Alan L. Carey, John Phillips, Adam C. Rennie, Fyodor A. Sukochev
The Chern Character Of Semifinite Spectral Triples, Alan L. Carey, John Phillips, Adam C. Rennie, Fyodor A. Sukochev
Associate Professor Adam Rennie
In previous work we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a ∗-subalgebra A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and rely on the introduction of a function valued cocycle (called the resolvent cocycle) which is 'almost' a (b,B)-cocycle in the cyclic cohomology of A. In this paper we show that this resolvent cocycle 'almost' represents the Chern character, and assuming analytic continuation properties for zeta functions, we show that the associated residue cocycle, …
Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie
Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie
Associate Professor Adam Rennie
We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable, we construct a class [D] ? KK1(A, K(N)). For a unitary u ? A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.
The Local Index Formula In Semifinite Von Neumann Algebras I: Spectral Flow, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev
The Local Index Formula In Semifinite Von Neumann Algebras I: Spectral Flow, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev
Associate Professor Adam Rennie
We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a ∗-subalgebra A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is 'almost' a (b,B)-cocycle in the cyclic cohomology of A.
An Analytic Approach To Spectral Flow In Von Neumann Algebras, M-T Benameur, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev, K Wojciechowski
An Analytic Approach To Spectral Flow In Von Neumann Algebras, M-T Benameur, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev, K Wojciechowski
Associate Professor Adam Rennie
The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by Breuer-Fredholm operators in a semifinite von Neumann algebra. The latter have continuous spectrum so that the notion of spectral flow turns out to be rather more difficult to deal with. However quite remarkably there is a uniform approach in which the proofs do not depend on discreteness of the spectrum of the operators in question. The first part of this paper …
Spectral Flow Invariants And Twisted Cyclic Theory For The Haar State On Suq(2), A L. Carey, A Rennie, K Tong
Spectral Flow Invariants And Twisted Cyclic Theory For The Haar State On Suq(2), A L. Carey, A Rennie, K Tong
Associate Professor Adam Rennie
In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SUq.2/ and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of …
Spectral Triples: Examples And Index Theory, Alan L. Carey, John Phillips, Adam C. Rennie
Spectral Triples: Examples And Index Theory, Alan L. Carey, John Phillips, Adam C. Rennie
Associate Professor Adam Rennie
The main objective of these notes is to give some intuition about spectral triples and the role they play in index theory. The notes are basically a road map, with much detail omitted. To give a complete account of all the topics covered would require at least a book, so we have opted for a sketch.
Semifinite Spectral Triples Associated With Graph C*-Algebras, Alan L. Carey, John Phillips, Adam Rennie
Semifinite Spectral Triples Associated With Graph C*-Algebras, Alan L. Carey, John Phillips, Adam Rennie
Associate Professor Adam Rennie
We review the recent construction of semifinite spectral triples for graph C^*-algebras. These examples have inspired many other developments and we review some of these such as the relation between the semifinite index and the Kasparov product, examples of noncommutative manifolds, and an index theorem in twisted cyclic theory using a KMS state.
Spectral Triples For Hyperbolic Dynamical Systems, Michael F. Whittaker
Spectral Triples For Hyperbolic Dynamical Systems, Michael F. Whittaker
Faculty of Engineering and Information Sciences - Papers: Part A
Spectral triples are defined for C-algebras associated with hyperbolic dynam- ical systems known as Smale spaces. The spectral dimension of one of these spectral triples is shown to recover the topological entropy of the Smale space.
Spectral Triples: Examples And Index Theory, Alan L. Carey, John Phillips, Adam C. Rennie
Spectral Triples: Examples And Index Theory, Alan L. Carey, John Phillips, Adam C. Rennie
Faculty of Engineering and Information Sciences - Papers: Part A
The main objective of these notes is to give some intuition about spectral triples and the role they play in index theory. The notes are basically a road map, with much detail omitted. To give a complete account of all the topics covered would require at least a book, so we have opted for a sketch.
Efficient Spectral Feature Selection With Minimum Redundancy, Zheng Zhao, Lei Wang, Huan Liu
Efficient Spectral Feature Selection With Minimum Redundancy, Zheng Zhao, Lei Wang, Huan Liu
Faculty of Engineering and Information Sciences - Papers: Part A
Spectral feature selection identifies relevant features by measuring their capability of preserving sample similarity. It provides a powerful framework for both supervised and unsupervised feature selection, and has been proven to be effective in many real-world applications. One common drawback associated with most existing spectral feature selection algorithms is that they evaluate features individually and cannot identify redundant features. Since redundant features can have significant adverse effect on learning performance, it is necessary to address this limitation for spectral feature selection. To this end, we propose a novel spectral feature selection algorithm to handle feature redundancy, adopting an embedded model. …
Spectral Flow Invariants And Twisted Cyclic Theory For The Haar State On Suq(2), A L. Carey, A Rennie, K Tong
Spectral Flow Invariants And Twisted Cyclic Theory For The Haar State On Suq(2), A L. Carey, A Rennie, K Tong
Faculty of Engineering and Information Sciences - Papers: Part A
In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SUq.2/ and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of …
The Chern Character Of Semifinite Spectral Triples, Alan L. Carey, John Phillips, Adam C. Rennie, Fyodor A. Sukochev
The Chern Character Of Semifinite Spectral Triples, Alan L. Carey, John Phillips, Adam C. Rennie, Fyodor A. Sukochev
Faculty of Engineering and Information Sciences - Papers: Part A
In previous work we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a ∗-subalgebra A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and rely on the introduction of a function valued cocycle (called the resolvent cocycle) which is 'almost' a (b,B)-cocycle in the cyclic cohomology of A. In this paper we show that this resolvent cocycle 'almost' represents the Chern character, and assuming analytic continuation properties for zeta functions, we show that the associated residue cocycle, …
Semifinite Spectral Triples Associated With Graph C*-Algebras, Alan L. Carey, John Phillips, Adam Rennie
Semifinite Spectral Triples Associated With Graph C*-Algebras, Alan L. Carey, John Phillips, Adam Rennie
Faculty of Engineering and Information Sciences - Papers: Part A
We review the recent construction of semifinite spectral triples for graph C^*-algebras. These examples have inspired many other developments and we review some of these such as the relation between the semifinite index and the Kasparov product, examples of noncommutative manifolds, and an index theorem in twisted cyclic theory using a KMS state.
Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie
Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie
Faculty of Engineering and Information Sciences - Papers: Part A
We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable, we construct a class [D] ? KK1(A, K(N)). For a unitary u ? A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.
An Analytic Approach To Spectral Flow In Von Neumann Algebras, M-T Benameur, Alan L. Carey, John Phillips, Adam C. Rennie, Fyodor A. Sukochev, K P. Wojciechowski
An Analytic Approach To Spectral Flow In Von Neumann Algebras, M-T Benameur, Alan L. Carey, John Phillips, Adam C. Rennie, Fyodor A. Sukochev, K P. Wojciechowski
Faculty of Engineering and Information Sciences - Papers: Part A
The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by Breuer-Fredholm operators in a semifinite von Neumann algebra. The latter have continuous spectrum so that the notion of spectral flow turns out to be rather more difficult to deal with. However quite remarkably there is a uniform approach in which the proofs do not depend on discreteness of the spectrum of the operators in question. The first part of this paper …
Summability For Nonunital Spectral Triples, Adam C. Rennie
Summability For Nonunital Spectral Triples, Adam C. Rennie
Faculty of Engineering and Information Sciences - Papers: Part A
This paper examines the issue of summability for spectral triples for the class of nonunital algebras. For the case of (p, -) summability, we prove that the Dixmier trace can be used to define a (semifinite) trace on the algebra of the spectral triple. We show this trace is well-behaved, and provide a criteria for measurability of an operator in terms of zeta functions. We also show that all our hypotheses are satisfied by spectral triples arising from eodesically complete Riemannian manifolds. In addition, we indicate how the Local Index Theorem of Connes-Moscovici extends to our nonunital setting.
Smoothness And Locality For Nonunital Spectral Triples, Adam C. Rennie
Smoothness And Locality For Nonunital Spectral Triples, Adam C. Rennie
Faculty of Engineering and Information Sciences - Papers: Part A
To deal with technical issues in noncommuntative geometry for nonunital algebras, we introduce a useful class of algebras and their modules. Thes algebras and modules allo us to extend all of the smoothness results for spectral triples to the nonunital case. In addition, we show that smooth spectral tiples are closed under the C- functional calculus of self-adjoint elements. In the final section we show that our algebras allow the formulation of Poincare Duality and that the algebras of smooth spectral triples are H-unital.