Bounded transform of a spectral triple - Imported Knowlpedia Content

Definition

Let $(\mathcal A,H,D)$ be a spectral triple, with $D$ densely defined and self-adjoint. Its bounded transform is the bounded self-adjoint contraction

F_D=D(1+D^2)^{-1/2},

defined by functional calculus. Compact resolvent gives $F_D^2-I=-(1+D^2)^{-1}$ compact, and the bounded-commutator axiom implies that $[F_D,a]$ is compact for every $a\in\mathcal A$ . Thus $(H,\pi,F_D)$ is a Fredholm module in the compact-defect convention. If the spectral triple is even, its grading anticommutes with $F_D$ ; otherwise the transform is odd.

Why the transform is bounded

The scalar function

f(x)=\frac{x}{\sqrt{1+x^2}}

is real-valued and bounded by $1$ . The self-adjoint functional calculus therefore turns the generally unbounded $D$ into a bounded self-adjoint operator. Its square tends to $1$ at spectral infinity, while

I-F_D^2=(1+D^2)^{-1}

retains the compactness information. The transform preserves the sign of every nonzero spectral value but sends large magnitudes toward $\pm1$ .

The compactness of $[F_D,a]$ follows from the resolvent integral for $(1+D^2)^{-1/2}$ and the boundedness of $[D,a]$ . This is the bounded-transform theorem of Baaj and Julg; Connes, Appendix A, Theorem 15 states the Hilbert-module version and the integral argument.

K-homological role

The bounded transform forgets metric scale but retains the stable Fredholm data needed for analytic K-homology. Homotopies satisfying the appropriate uniform unbounded-cycle hypotheses descend to operator homotopies of bounded cycles. The resulting class can then be paired with $K$ -theory by the even or odd Fredholm index.

For a nonunital unbounded Kasparov cycle, compact resolvent is replaced by local compactness $a(1+D^2)^{-1}\in K(H)$ . The same formula then gives the local compact-defect conditions $a(F_D^2-I)\in K(H)$ . Connes, Appendix A, Theorem 15.

Example and distinction from the phase

For the spin Dirac operator on a closed Riemannian spin manifold,

D(1+D^2)^{-1/2}

is an order-zero bounded operator and defines the manifold's analytic K-homology class. Its commutators with smooth functions are compact.

The bounded transform is not literally the phase $\operatorname{sgn}(D)$ . On $\ker D$ , both may be assigned $0$ , but on nonzero finite spectral values $x/\sqrt{1+x^2}\neq\operatorname{sgn}(x)$ . The two yield related normalized cycles only after an additional compact perturbation or kernel convention.

References