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Full-Text Articles in Science and Technology Studies
The Local Index Formula In Semifinite Von Neumann Algebras Ii: The Even Case, Alan Carey, John Phillips, Adam Rennie, F Sukochev
The Local Index Formula In Semifinite Von Neumann Algebras Ii: The Even Case, Alan Carey, John Phillips, Adam Rennie, F Sukochev
Associate Professor Adam Rennie
We generalise the 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. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.
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 …
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.
Families Of Type Iii Kms States On A Class Of C-Algebras Containing On And Qn, A L. Carey, J Phillips, I F. Putnam, A Rennie
Families Of Type Iii Kms States On A Class Of C-Algebras Containing On And Qn, A L. Carey, J Phillips, I F. Putnam, A Rennie
Associate Professor Adam Rennie
We construct a family of purely infinite C¤-algebras, Q¸ for ¸ 2 (0, 1) that are classified by their K-groups. There is an action of the circle T with a unique KMS state à on each Q¸. For ¸ = 1/n, Q1/n »= On, with its usual T action and KMS state. For ¸ = p/q, rational in lowest terms, Q¸ »= On (n = q − p + 1) with UHF fixed point algebra of type (pq)1. For any n > 1, Q¸ »= On for infinitely many ¸ with distinct KMS states and UHF fixed-point algebras. For any ¸ …