Non Commutative Geometry of GLp-Bundles

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


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]).


String Class Reduction Theorem Maximal Abelian Subgroup Vanishing Theorem Finite Rank Operator 
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© Kluwer Academic Publishers 1996

Authors and Affiliations

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

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