Trace Formulas

This volume introduces trace formulas for singular traces on a separable Hilbert space. In general terms, trace formulas compute the value of a singular trace by integration over the principal part of the symbol of a bounded operator. Formulas and symbols are described from several applications in Alain Connes' noncommutative geometry. Singular traces are shown to compute the noncommutative residue in differential geometry using the principal symbol of a pseudodifferential operator. An extension of the symbol map and computation of the noncommutative residue using a trace formula are shown for the noncommutative torus, noncommutative Euclidean space, and the C*-algebra of SU(2)-valued continuous functions. Several other trace formulas from noncommutative geometry are computed using the same principle, including integration of functions over quantum densities, Connes' character formula concerning the Hochschild cohomology class of the Chern character, and extension of the density of states measure in solid state physics.