Denis Potapov

Unbounded Fredholm modules and double operator integrals

Abstract In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators D given by ƒ(D) = D(1 + D 2)−1/2. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods work in the setting of semifinite noncommutative geometry where one has D as an unbounded self adjoint linear operator affiliated with a semi-finite von Neumann algebra ℳ. More precisely we show that for a pair D, D 0 of such operators with D – D 0 a bounded self-adjoint linear operator from ℳ and , where 𝓔 is a noncommutative symmetric space associated with ℳ, then . This result is further used to show continuous differentiability of the mapping between an odd 𝓔-summable spectral triple and its bounded counterpart.

On the Witten index in terms of spectral shift functions

AbstractWe study the model operator DA = (d/dt) + A in L2(R;H) associated with the operator path {A(t)}t=−∞∞, where (Af)(t) = A(t)f(t) for a.e. t ∈ R, and appropriate f ∈ L2(R;H) (with H a separable, complex Hilbert space). Denoting by A± the norm resolvent limits of A(t) as t → ±∞, our setup permits A(t) in H to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator A-, and no discrete spectrum assumptions are made on A±. Introducing H1 = DA*DA, H2 = DADA*, the resolvent and semigroup regularized Witten indices of DA, denoted by Wr(DA) and Ws(DA), are defined by

Trace formulas for a class of non-Fredholm operators: A review

\begin{gathered} {W_r}\left( {{D_A}} \right) = \mathop {lim}\limits_{\lambda \uparrow 0} \left( { - \lambda } \right)t{r_{{L^2}\left( {\mathbb{R};H} \right)}}\left( {{{\left( {{H_1} - \lambda I} \right)}^{ - 1}} - {{\left( {{H_2} - \lambda I} \right)}^{ - 1}}} \right), \hfill \\ {W_s}\left( {{D_A}} \right) = \mathop {lim}\limits_{t \uparrow \infty } t{r_{{L^2}\left( {\mathbb{R};H} \right)}}\left( {{e^{ - t{H_1}}} - {e^{ - t{H_2}}}} \right), \hfill \\ \end{gathered}

Double operator integral methods applied to continuity of spectral shift functions

Wr(DA)=limλ↑0(−λ)trL2(ℝ;H)((H1−λI)−1−(H2−λI)−1),Ws(DA)=limt↑∞trL2(ℝ;H)(e−tH1−e−tH2), whenever these limits exist. These regularized indices coincide with the Fredholm index of DA whenever the latter is Fredholm.In situations where DA ceases to be a Fredholm operator in L2(R;H) we compute its resolvent (resp., semigroup) regularizedWitten index in terms of the spectral shift function ξ(•; A+, A-) associated with the pair (A+, A-) as follows: Assuming 0 to be a right and a left Lebesgue point of ξ(•; A+, A-), denoted by ξL(0+; A+, A-) and ξL(0-; A+, A-), we prove that 0 is also a right Lebesgue point of ξ(•; H2, H1), denoted by ξL(0+; H2, H1), and that

On the Global Limiting Absorption Principle for Massless Dirac Operators

\begin{array}{*{20}{c}} {{W_r}\left( {{D_A}} \right)}&\begin{gathered} = {W_s}\left( {{D_A}} \right) \hfill \\ = {\xi _L}\left( {{0_ + };{H_2},{H_1}} \right) \hfill \\ = {{\left[ {{\xi _L}\left( {{0_ + };{A_ + },{A_ - }} \right) + {\xi _L}\left( {{0_ - };{A_ + },{A_ - }} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {{\xi _L}\left( {{0_ + };{A_ + },{A_ - }} \right) + {\xi _L}\left( {{0_ - };{A_ + },{A_ - }} \right)} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}, \hfill \\ \end{gathered} \end{array}

Operator-Lipschitz functions in Schatten–von Neumann classes

Wr(DA)=Ws(DA)=ξL(0+;H2,H1)=[ξL(0+;A+,A−)+ξL(0−;A+,A−)]/2, the principal result of this paper.In the special case where dim(H) < ∞, we prove that the Witten indices of DA are either integer, or half-integer-valued.