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

Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations