Cohomological Formula for the Λ-Index

Abstract

In Chap. 7, we obtained a K-theoretical index formula

$$ ind_\Lambda \left( D \right) = p_! \left[ {\sigma \left( D \right)} \right], $$

for nonlocal elliptic operators, where

• 1. Λ = C*(Г) is the group C*-algebra of group Г.

• 2. ind_Λ(D) ∈ K₀(Λ) is the Λ-Fredholm index of an operator in Hilbert Λ-modules associated with symbol σ(D) (see Sec. 5.2).

• 3.
  $$ p_! :K_0 \left( {C_0 \left( {T^ * M} \right) \rtimes \Gamma } \right) \to K_0 \left( \Lambda \right) $$

  is the direct image mapping corresponding to the projection p: M → {pt} of the manifold M into the one-point space.

However, the cohomological formula was obtained only for the usual Fredholm index, which can be extracted from the Λ-Fredholm index by the formula

$$ indD = \alpha _ * ind_\Lambda \left( D \right), $$

where α: Λ → ℂ is the “forgetful” homomorphism associated with the trivial representation of Г in ℂ (see Proposition 5.2).