Topological and analytical indices in C∗-algebras

A notion of topological index for the continuous symbol functions of generalized Toeplitz operators is introduced. This generalizes the winding number of functions on the circle and the average winding number of almost periodic functions on the real line and makes fundamental use of the quantized differential calculus of Alain Connes. The analytic index of a generalized Toeplitz operator—defined in terms of a trace—is shown to be equal to minus the topological index of the symbol function of the operator, a result that extends some well-known index theorems.