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Non Commutative Geometry of GLp-Bundles

  • Akira Asada
Part of the Mathematics and Its Applications book series (MAIA, volume 350)

Abstract

Let H be a separable Hilbert space with a polarization ε, GL(H) the group of bounded linear operation on H. The group GL p is defined to be {T ∈ GL(H) | [ε, T]I p , the p-th Schatten ideal}. The group of GL p consisted by unitary operators is denoted by U p . If X is a compact spin manifold and G a linear Lie group, then Map(X,G) is a subgroup of GL p , p > dim .X/2. Map(X,G) is the structure group of the tangent bundle of Map(X,M), if the structure group of the tangent bundle of M is G, so the study of GL p -bundles has meanings. Theory of string classes ([3],[3]’) shows usual connection is insufficient to the study of GL p bundles. A supplementary tool is the non-commutative connection which is an adaptation of Connes’ non-commutative geometry to GLp-bundles. By using non-commutative connections, we have the following results: (i) Vanishing Theorem. If a U p -bundle has an Hermitian non-commutative connection invertible curvature, then it is trivial, (ii) Non-commutative Poincare Lemma. A necessary and sufficient condition to the local existence of non-commutative connection whose curvature becomes given operator valued function, is given. By this condition, we get a complete obstruction class in some twisted cohomology to the triviality of a GL p -bundle. (iii) Reduction Theorem. A U p -bundle is equivalent to a U 1-bundle. Reduction Theorem may follow from the topological argument. But our proof shows this reduction closely related to Kato-Rellich Theorem in perturbation theory ([11],[11′],[17]).

Keywords

String Class Reduction Theorem Maximal Abelian Subgroup Vanishing Theorem Finite Rank Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Akira Asada
    • 1
  1. 1.Department of MathematicsFaculty of Science Sinsyu UniversityMatumoto 390Japan

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