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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.
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.
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, …
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.