  We review an approach to the index theory of operator algebras associated
with Lie groups of quantized canonical transformations. Main points are an
ellipticity condition ensuring the Fredholm property, the definition of
localized algebraic and analytic indices and the proof of their equality. This
framework encompasses many well-known index problems, such as the classical
theory on closed manifold, the Atiyah-Weinstein problem and the index theory
for operators with shifts.
