On the trace formula for elliptic pseudodifferential operators over \(C^*\)-algebras (Q1072204)

For a \(C^*\) algebra A, the author considers pseudo-differential operators which act on smooth functions with values in \(A^ n\), the direct product of n copies of A. For such operators he proves a number of results well known for specific \(C^*\) algebras. These include: the existence of an adjoint, the existence of an inverse of an elliptic pseudo-differential operator, the existence of powers of a suitable positive operator and finally the existence of a trace of such powers and identification of its properties. He concludes by remarking that his framework allows one to extend the Atiyah-Bott index formula to such elliptic pseudo-differential operators for \(C^*\) algebras.