  We consider elliptic operators associated with discrete groups of quantized
canonical transformations. In order to be able to apply results from algebraic
index theory, we define the localized algebraic index of the complete symbol of
an elliptic operator. With the help of a calculus of semiclassical quantized
canonical transformations, a version of Egorov's theorem and a theorem on trace
asymptotics for semiclassical Fourier integral operators we show that the
localized analytic index and the localized algebraic index coincide. As a
corollary, we express the Fredholm index in terms of the algebraic index for a
wide class of groups, in particular, for finite extensions of Abelian groups.
