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Full-Text Articles in Social and Behavioral Sciences
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.