**MSC:**- 46L89 Other ``noncommutative'' mathematics based on $C^*$-algebra theory, See also {58B30, 58G12}
- 19K56 Index theory, See also {58G12}
- 19D55 $K$-theory and homology; cyclic homology and cohomology, See also {18G60}

pairwise transverse Lagrangian submodules of $A^{2n}$ where $A$ is a $C^*$-algebra

and $A^{2n}$ is endowed with a nondegenerated skewhermitian form.

Our Maslov index is an element of the $K$-theory of $A$.

Furthermore we define a noncommutative eta-form associated to a pair

of Lagrangian submodules of $A^{2n}$. Then we prove a formula expressing

the Chern character of the Maslov index in terms of eta-forms in the de Rham

homology of certain dense subalgebras of $A$. The formula and its proof are modelled

on a result of Bunke and Koch who studied the family case: The Maslov index

is the index of a Dirac operator on a two dimensional manifold with boundary

and cylindric ends. The noncommutative index problem is solved by adapting

the superconnection formalism. For the construction and the study of the heat kernels

the theory of holomorphic semigroups is used.

This is part I, part II will appear in Mathematica Gottingensis 12