AN ANALYTIC APPROACH TO SPECTRAL FLOW IN VON NEUMANN ALGEBRAS

Moulay-Tahar Benameur, Alan L. Carey, John Phillips, Adam Rennie, Fedor Sukochev, Krzysztof P. Wojciechowski

Abstract

Analysis, Geometry and Topology of Elliptic Operators, pp. 297-352 (2006) No AccessAN ANALYTIC APPROACH TO SPECTRAL FLOW IN VON NEUMANN ALGEBRASMOULAY-TAHAR BENAMEUR, ALAN L. CAREY, JOHN PHILLIPS, ADAM RENNIE, FYODOR A. SUKOCHEV, and KRZYSZTOF P. WOJCIECHOWSKIMOULAY-TAHAR BENAMEURUMR 7122 du CNRS, Université de Metz, Ile du Saulcy, Metz, France, ALAN L. CAREYMathematical Sciences Institute, Australian National University, Canberra ACT, 0200, Australia, JOHN PHILLIPSDepartment of Mathematics and Statistics, University of Victoria, Victoria, B.C. V8W 3P4, Canada, ADAM RENNIEInstitute for Mathematical Sciences, Department of Mathematics, Universitetsparken 5, DK-2100 Copenhagen, Denmark, FYODOR A. SUKOCHEVSchool of Informatics and Engineering, Flinders University, Bedford Park S.A 5042, Australia, and KRZYSZTOF P. WOJCIECHOWSKIDepartment of Mathematics, IUPUI (Indiana/Purdue), Indianapolis, IN, 46202-3216, USAhttps://doi.org/10.1142/9789812773609_0012Cited by:16 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: 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 gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow first analysed in the classical setting by Getzler and extended to Breuer-Fredholm operators by some of the current authors. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (non-compact) covering spaces of a compact spin manifold using the adiabatic method. This answers a question of Mathai connecting Atiyah's L2-index theorem to our analytic spectral flow. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on RN using some ideas of Shubin. This provides us with an opportunity to describe the deepest part of the theory namely the semifinite local index theorem in noncommutative geometry. This theorem, which gives a formula for spectral flow was recently proved by some of the present authors. It provides a far-reaching generalisation of the original 1995 result of Connes and Moscovici. This research is supported by the ARC (Australia), NSERC (Canada) and an early career grant from the University of Newcastle. Dedication: Dedicated to Krzysztof P. Wojciechowski on his 50th birthday from his co–authorsKeywords: Not availableAMSC: Primary 19K56, Secondary 58J20, Secondary 46L80, Secondary 58J30 FiguresReferencesRelatedDetailsCited By 16The Witten index and the spectral shift functionAlan Carey, Galina Levitina, Denis Potapov, and Fedor Sukochev17 February 2022 | Reviews in Mathematical Physics, Vol. 34, No. 05Spectral Flow of Monopole Insertion in Topological InsulatorsAlan L. Carey and Hermann Schulz-Baldes31 January 2019 | Communications in Mathematical Physics, Vol. 370, No. 3Oscillation Theory for the Density of States of High Dimensional Random OperatorsJulian Groß mann, Hermann Schulz-Baldes and Carlos Villegas-Blas12 October 2017 | International Mathematics Research Notices, Vol. 2019, No. 15Index Theory and Topological Phases of Aperiodic LatticesC. Bourne and B. Mesland6 February 2019 | Annales Henri Poincaré, Vol. 20, No. 6Spectral Flow Argument Localizing an Odd Index PairingTerry A. 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