[Paper Review] Unbounded Fredholm modules and double operator integrals

Denis Potapov, Fedor Sukochev|ArXiv.org|Aug 21, 2008

Advanced Operator Algebra Research14 references3 citations

TL;DR

This paper establishes commutator estimates and continuous differentiability for the map $ D \mapsto F_D = D(1+D^2)^{-1/2} $ in semifinite noncommutative geometry, proving that if $ (1+D_0^2)^{-1/2} \in \mathcal{E} $ and $ D - D_0 \in \mathcal{M} $, then $ \|f(D) - f(D_0)\|_{\mathcal{E}} \leq C \|D - D_0\|_{\mathcal{M}} $. This result ensures the bounded Fredholm module $ F_D $ inherits $ \mathcal{E} $-summability and the mapping is continuously differentiable, removing prior technical assumptions in spectral flow theory.

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 $f(D)=D(1+D^2)^{-1/2}$. 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 $\aM$. 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 $\aM$ and $ ({\bf 1}+D_0^2)^{-1/2}\in \sE$, where $\sE$ is a noncommutative symmetric space associated with $\aM$, then $$ \Vert f(D) - f (D_0) \Vert_{\sE} \leq C\cdot \Vert D-D_0\Vert_{\aM}. $$ This result is further used to show continuous differentiability of the mapping between an odd $\sE$-summable spectral triple and its bounded counterpart.

Motivation & Objective

• To establish commutator estimates for the map $ D \mapsto F_D = D(1+D^2)^{-1/2} $ in the context of semifinite von Neumann algebras.
• To show that $ \mathcal{E} $-summability of $ (1+D^2)^{-1/2} $ implies $ \mathcal{E} $-summability of the bounded Fredholm module $ F_D $.
• To prove the mapping $ D \mapsto F_D $ is continuously differentiable on the affine space of bounded self-adjoint perturbations in $ \mathcal{M} $.
• To remove technical assumptions previously required in spectral flow theory by establishing smoothness of the map from unbounded to bounded Fredholm modules.

Proposed method

• The authors analyze the function $ f(t) = t / \sqrt{1 + t^2} $ on unbounded self-adjoint operators $ D $, defining $ F_D = f(D) $.
• They use double operator integrals via the functional calculus and the operator $ T_{\psi_f}(D, D_0) $ to represent the difference $ f(D) - f(D_0) $.
• Key estimates rely on generalized Hölder inequalities in noncommutative symmetric spaces $ \mathcal{E} $, with bounds on $ \|T_{\psi_f}(D, D_0)\|_{B(\mathcal{L}^\infty, \mathcal{E})} \leq c $.
• The proof of differentiability uses the decomposition of the derivative $ \frac{dF_t}{dt} = T_{\psi_f}(D_t, D_t) \frac{dD_t}{dt} $, with convergence in $ \mathcal{E} $-norm.
• Continuity of the derivative is shown by splitting the difference $ \frac{dF_t}{dt}(s) - \frac{dF_t}{dt}(0) $ into three terms, each vanishing in $ \mathcal{E} $ as $ s \to 0 $.
• The analysis is conducted in the framework of semifinite von Neumann algebras with a normal, faithful, tracial weight $ \tau $, and $ \mathcal{E} $ is a noncommutative symmetric space of $ \tau $-measurable operators.

Experimental results

Research questions

• RQ1 Under what conditions does the map $ D \mapsto F_D = D(1+D^2)^{-1/2} $ preserve $ \mathcal{E} $-summability in semifinite noncommutative geometry?
• RQ2 Can commutator estimates $ \|[F_D, a]\|_{\mathcal{E}} \leq c \|[D, a]\| $ be established without restricting to Schatten or Dixmier ideals?
• RQ3 Is the mapping from unbounded to bounded Fredholm modules continuously differentiable under general $ \mathcal{E} $-summability conditions?
• RQ4 How does the operator differentiability of $ f(D) $ relate to the spectral flow along paths of unbounded self-adjoint operators?
• RQ5 Can the technical assumptions on $ D_0 $ and the path $ D_t $ in spectral flow theory be removed using this framework?

Key findings

• The map $ D \mapsto F_D = D(1+D^2)^{-1/2} $ satisfies the estimate $ \|f(D) - f(D_0)\|_{\mathcal{E}} \leq C \|D - D_0\|_{\mathcal{M}} $ for $ D, D_0 $ self-adjoint and $ D - D_0 \in \mathcal{M} $, with $ (1+D_0^2)^{-1/2} \in \mathcal{E} $.
• The bounded Fredholm module $ F_D $ is $ \mathcal{E} $-summable whenever $ (1+D^2)^{-1/2} \in \mathcal{E} $, and the commutator $ [F_D, a] \in \mathcal{E} $ follows from $ [D, a] \in \mathcal{B}(\mathcal{H}) $.
• The mapping $ D \mapsto F_D $ is continuously differentiable on the affine space of bounded self-adjoint perturbations of $ D_0 $, with derivative $ \frac{dF_t}{dt} = T_{\psi_f}(D_t, D_t) \frac{dD_t}{dt} $.
• The derivative $ \frac{dF_t}{dt} $ is continuous in the $ \mathcal{E} $-norm, provided $ \frac{dD_t}{dt} $ is operator norm continuous.
• The result removes prior technical assumptions in spectral flow theory, as the smoothness of the map is now established under general $ \mathcal{E} $-summability conditions.
• The proof relies on double operator integral techniques and generalized Hölder inequalities in noncommutative symmetric spaces, with uniform bounds on the operator $ T_{\psi_f} $.