Integration on locally compact noncommutative spaces

Abstract:

Dixmier traces on semifinite von Neumann algebras play an important role in noncommutative geometry (spectral characterization of the dimension of a manifold, construction of Yang-Mills type actions and etc.). One of the principal difficulty is computation of such non-normal traces. Generalizing a result of Connes in the B(H) case, Carey, Philips and Sukochev proved that the Dixmier traces of a positive Dixmier traceable operator G may be computed by mean of a generalized residue of the associated zeta function z → T(Gz). In collaboration with Carey, Rennie and Sukochev, we extend this result for product of operators a and G, none of them being Dixmier traceable but only the product aG is, by mean of generalized residues of a zeta type function z → T(aGz).