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Cohomological Formula for the Λ-Index

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

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. 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).

Keywords

Conjugacy Class Curvature Form Free Abelian Group Leibniz Rule Chern Character 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2008

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