TY  - JOUR
A1  - Azzali, Sara
A1  - Goette, Sebastian
A1  - Schick, Thomas
T1  - Large time limit and local L-2-index theorems for families
JF  - Journal of noncommutative geometry
N2  - We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L-2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L-2-index formulas.
 As applications, we prove a local L-2-index theorem for families of signature operators and an L-2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L-2-eta forms and L-2-torsion forms as transgression forms.
KW  - Local index theory
KW  - eta forms
KW  - torsion forms
KW  - L-2-invariants
Y1  - 2015
U6  - https://doi.org/10.4171/JNCG/203
SN  - 1661-6952
SN  - 1661-6960
VL  - 9
IS  - 2
SP  - 621
EP  - 664
PB  - EMS Publ.
CY  - Zürich
ER  - 
TY  - JOUR
A1  - Antonini, Paolo
A1  - Azzali, Sara
A1  - Skandalis, Georges
T1  - Bivariant K-theory with R/Z-coefficients and rho classes of unitary representations
JF  - Journal of functional analysis
N2  - We construct equivariant KK-theory with coefficients in and R/Z as suitable inductive limits over II1-factors. We show that the Kasparov product, together with its usual functorial properties, extends to KK-theory with real coefficients. Let Gamma be a group. We define a Gamma-algebra A to be K-theoretically free and proper (KFP) if the group trace tr of Gamma acts as the unit element in KKR Gamma (A, A). We show that free and proper Gamma-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if Gamma is torsion free and satisfies the KK Gamma-form of the Baum-Connes conjecture, then every Gamma-algebra satisfies (KFP). If alpha : Gamma -> U-n is a unitary representation and A satisfies property (KFP), we construct in a canonical way a rho class rho(A)(alpha) is an element of KKR/Z1,Gamma (A A) This construction generalizes the Atiyah-Patodi-Singer K-theory class with R/Z-coefficients associated to alpha. (C) 2015 Elsevier Inc. All rights reserved.
KW  - Operator algebras
KW  - Bivariant K-theory
KW  - Rho invariants
Y1  - 2016
U6  - https://doi.org/10.1016/j.jfa.2015.06.017
SN  - 0022-1236
SN  - 1096-0783
VL  - 270
SP  - 447
EP  - 481
PB  - Elsevier
CY  - San Diego
ER  - 
TY  - JOUR
A1  - Azzali, Sara
A1  - Wahl, Charlotte
T1  - Two-cocycle twists and Atiyah-Patodi-Singer index theory
JF  - Mathematical Proceedings of the Cambridge Philosophical Society
N2  - We construct eta- and rho-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah-Patodi-Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.
Y1  - 2019
U6  - https://doi.org/10.1017/S0305004118000427
SN  - 0305-0041
SN  - 1469-8064
VL  - 167
IS  - 3
SP  - 437
EP  - 487
PB  - Cambridge Univ. Press
CY  - New York
ER  - 
TY  - JOUR
A1  - Azzali, Sara
A1  - Paycha, Sylvie
T1  - Spectral zeta-invariants lifted to coverings
JF  - Transactions of the American Mathematical Society
N2  - The canonical trace and the Wodzicki residue on classical pseudo-differential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral zeta-invariants using lifted defect formulae which express discrepancies of zeta-regularised traces in terms of Wodzicki residues. We derive Atiyah's L-2-index theorem as an instance of the Z(2)-graded generalisation of the canonical lift of spectral zeta-invariants and we show that certain lifted spectral zeta-invariants for geometric operators are integrals of Pontryagin and Chern forms.
Y1  - 2020
U6  - https://doi.org/10.1090/tran/8067
SN  - 0002-9947
SN  - 1088-6850
VL  - 373
IS  - 9
SP  - 6185
EP  - 6226
PB  - American Mathematical Society
CY  - Providence, RI
ER  -