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The eta invariant and families of pseudodifferential operators

**Richard B. Melrose**

### Abstract:

For a compact manifold without boundary a suspended algebra of
pseudodifferential operators is considered; it is an algebra of
pseudodifferential operators on, and translation-invariant in, an
additional real variable. It is shown that the eta invariant, as defined by
Atiyah, Patodi and Singer for admissible Dirac operators, extends to a
homomorphism from the ring of invertible elements of the suspended algebra
to the additive real line. The deformation properties of this extended eta
homomorphism are discussed and a related `divisor flow' is shown to label
the components of the set of invertible elements within each component of the
elliptic set.

*Richard B. Melrose *

Sat Mar 30 08:04:27 EST 1996