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Full-Text Articles in Science and Technology Studies
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.
Index Theory For Locally Compact Noncommutative Geometries, Alan Carey, V Gayral, Adam Rennie, F Sukochev
Index Theory For Locally Compact Noncommutative Geometries, Alan Carey, V Gayral, Adam Rennie, F Sukochev
Associate Professor Adam Rennie
Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra.
A Residue Formula For The Fundamental Hochschild 3-Cocycle For Suq(2), Ulrich Krahmer, Adam Rennie, Roger Senior
A Residue Formula For The Fundamental Hochschild 3-Cocycle For Suq(2), Ulrich Krahmer, Adam Rennie, Roger Senior
Associate Professor Adam Rennie
An analogue of a spectral triple over SUq(2) is constructed for which the usual assumption of bounded commutators with the Dirac operator fails. An analytic expression analogous to that for the Hochschild class of the Chern character for spectral triples yields a non-trivial twisted Hochschild 3-cocycle. The problems arising from the unbounded commutators are overcome by defining a residue functional using projections to cut down the Hilbert space.
The Resolvent Cocycle In Twisted Cyclic Cohomology And A Local Index Formula For The Podle's Sphere, Adam Rennie, Roger Senior
The Resolvent Cocycle In Twisted Cyclic Cohomology And A Local Index Formula For The Podle's Sphere, Adam Rennie, Roger Senior
Associate Professor Adam Rennie
We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of 'modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class which is in dimension 2.
The K-Theory Of Heegaard Quantum Lens Spaces, Piotr M. Hajac, Adam Rennie, Bartosz Zielinski
The K-Theory Of Heegaard Quantum Lens Spaces, Piotr M. Hajac, Adam Rennie, Bartosz Zielinski
Associate Professor Adam Rennie
Representing Z/NZ as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/NZ, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/NZ to construct an associated complex loine bundle. This paper proves the stable non-triviality of these line bundles over any of the quantum lens spaces we consider. We use the pullback structure of the C*-algebra of the lens space to compute its K-theory via the Mayer-Vietoris sequence, and an explicit form of the Bass connecting homomorphism to prove the stable non-triviality of the bundles. On the algebraic …
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.