A New Topology on the Space of Unbounded Selfadjoint Operators, K-theory and Spectral Flow

Wahl, Charlotte

doi:10.1007/978-3-7643-8604-7_16

Cited by 12 publications

(23 citation statements)

References 7 publications

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“…This topology is usually called the Riesz topology and it is different from the graph norm topology used in [12]. A more detailed discussion of topologies on the set of unbounded self-adjoint Fredholm operators and the relevance of these for spectral flow may be found in [36].…”

Section: Spectral Flowmentioning

confidence: 99%

The Spectral Shift Function and Spectral Flow

Azamov

Carey

Sukochev

2007

Commun. Math. Phys.

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At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer's proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general semifinite von Neumann algebra N and the perturbation V ∈ N . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein's spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein's theory to the setting of semifinite spectral triples where D has compact resolvent belonging to N and V is any bounded selfadjoint operator in N . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer's idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8,9,10]. One surprising conclusion that follows from our results is that the Krein spectral shift function 1 2 is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].

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Section: Spectral Flowmentioning

confidence: 99%

The Spectral Shift Function and Spectral Flow

Azamov

Carey

Sukochev

2007

Commun. Math. Phys.

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“…In parallel, new definitions of the (ordinary) spectral flow for paths of unbounded selfadjoint Fredholm operators have been given [5,26]. The straightforward way is to define the spectral flow for unbounded operators as the spectral flow of the bounded transform.…”

Section: Introductionmentioning

confidence: 99%

Spectral Flow and Winding Number in Von Neumann Algebras

Wahl¹

2008

JMJ

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Abstract. We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips' analytical definition and derive integral formulas for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey-Phillips. Furthermore we prove the homotopy invariance of the real-valued index. As an example we consider invariant symmetric elliptic differential operators on Galois coverings.

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“…There are several different but equivalent definitions of the spectral flow with various degrees of generality that have appeared in the literature during the last decades. Here we just want to mention [BW85], [Fl88], [RS95], [FPR99], [BLP05], [Wah08], which is probably far away from being exhaustive. In what follows, we use the definition of [BLP05] which applies to any gap-continuous path of (generally) unbounded selfadjoint Fredholm operators on a separable Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

Waterstraat

2015

Proc. London Math. Soc.

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We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one-parameter families of nonautonomous Hamiltonian vector fields.

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A New Topology on the Space of Unbounded Selfadjoint Operators, K-theory and Spectral Flow

Cited by 12 publications

References 7 publications

The Spectral Shift Function and Spectral Flow

The Spectral Shift Function and Spectral Flow

Spectral Flow and Winding Number in Von Neumann Algebras

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

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