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Noncommutative Maslov Index and Eta-Forms
Charlotte Wahl, Virginia Polytechnic Institute and State University, Blacksburg, VA
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Memoirs of the American Mathematical Society
2007; 118 pp; softcover
Volume: 189
ISBN-10: 0-8218-3997-7
ISBN-13: 978-0-8218-3997-3
List Price: US$63 Individual Members: US$37.80
Institutional Members: US\$50.40
Order Code: MEMO/189/887

The author defines and proves 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. The noncommutative Maslov index, defined for modules over a $$C^*$$-algebra $$\mathcal{A}$$, is an element in $$K_0(\mathcal{A})$$.

The generalized formula calculates its Chern character in the de Rham homology of certain dense subalgebras of $$\mathcal{A}$$. The proof is a noncommutative Atiyah-Patodi-Singer index theorem for a particular Dirac operator twisted by an $$\mathcal{A}$$-vector bundle. The author develops an analytic framework for this type of index problem.