Noncommutative Maslov Index and Eta-Forms (Memoirs of the American Mathematical Society)

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.