Noncommutative Maslov Index and Eta Forms

Details Noncommutative Maslov Index and Eta Forms Noncommutative Maslov Index and Eta Forms

2003-09-19

arxiv.org/abs/math/0309323v2

Mathematics

K-Theory and Homology

122 pages, 1 figure; based on the author's PhD-thesis; Changes in Introduction, \S 1.3, \S 4.5 added, new notion of trace clas

Scientific paper

We define and prove a noncommutative generalization of a formula relating the Maslov index of a triple of Lagrangian subspaces of a symplectic vector space to eta-invariants associated to a pair of Lagrangian subspaces generalizing a result of Bunke and Koch in the family case. The noncommutative Maslov index, defined for modules over a $C^*$-algebra $\A$, is an element in $K_0(\A)$. The generalized formula calculates its Chern character in the de Rham homology of certain dense subalgebras of $\A$. The proof, modelled on the proof by Bunke and Koch, is a noncommutative Atiyah-Patodi-Singer index theorem for a particular Dirac operator twisted by an $\A$-vector bundle. We develop an analytic framework for this type of index problem.

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