(Second) Quantised resolvents and regularised traces

Regularised traces on classical pseudodifferential operators are extended to tensor products of classical pseudodifferential operators via a (second) quantisation procedure  . Whereas ordinary ζζ-regularised traces are not generally expected to be local, using techniques borrowed from Connes and Moscovici [A. Connes, H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (2) (1995) 174–243], Higson [N. Higson, The residue index theorem of Connes and Moscovici, in: Clay Mathematics Proceedings, 2004, http://www.math.psu.edu/higson/ResearchPapers.html], we show that if QQ has scalar leading symbol, higher quantised  ζζ-regularised traces   are local since they can be expressed as a finite linear combination of noncommutative residues. Just as ordinary ζζ-regularised traces, they present anomalies (Hochschild coboundary, dependence on the weight QQ), which for quantised ζζ-regularised traces of level nn, are roughly speaking finite linear combinations of quantised regularised traces of level n+1n+1. As a result, anomalies are local for any non negative nn, which yields back as a particular case the fact that ordinary ζζ-regularised traces present local anomalies.1