Cohomological Formula for the Λ-Index

Part of the Operator Theory: Advances and Applications book series (OT, volume 183)


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. 1.

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

  2. 2.

    indΛ(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. 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).


Conjugacy Class Curvature Form Free Abelian Group Leibniz Rule Chern Character 
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© Birkhäuser Verlag AG 2008

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