2015/01/26 by Sara Azzali, Cyril Lévy, Azzali, Sara +5
Mathematics · Physics and Astronomy · #19K56 #35K08 #58J42 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Primary 47G30 #Secondary 58B34 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.OA #math.SP #msc:19K56 #msc:35K08 #msc:47G30 #msc:58B34 #msc:58J42
paper · pdf · doi:10.48550/arxiv.1501.06338
to appear in Geometric Methods in Physics, XXXIII Worskhop, Bialoweza, Poland, June 29-July 5, 2014, Birkhäuser
arxiv created 2015/01/26 · openalex publication_date 2015/01/26 · arxiv updated 2015/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks; the noncommutative torus and Hilbert modules. These traces are meromorphic functions whose residues at the poles as well as the constant term of the Laurent expansion at zero (the latter when the family at zero is a differential operator) can be expressed in terms of Wodzicki residues and extended Wodzicki residues involving logarithmic operators. They are therefore local and contain geometric information. For holomorphic families leading to zeta regularised traces, they relate to the heat-kernel asymptotic coefficients via an inverse Mellin mapping theorem. We revisit Atiyah's L2-index theorem by means of the (extended) Wodzicki residue and interpret the scalar curvature on the noncommutative two torus as an (extended) Wodzicki residue.