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Bott Periodicity

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

Abstract

The Bott periodicity theorem in K-theory of operator algebras gives the isomorphism of K-groups K0(A) and K0(A\( \widehat \otimes \)C0(ℝ2)) and (more generally) K0(A) and K0(A\( \widehat \otimes \)C0(ℝ2n)). In the special case where A = C(X), this isomorphism turns into the isomorphism of K-groups1
$$ K_0 \left( {C\left( X \right)} \right) \simeq K_0 \left( {C_0 \left( {X \times \mathbb{R}^{2n} } \right)} \right). $$
(6.1)
Of course, a similar isomorphism can also be written out if C(X) is replaced by the crossed product C(X)Г:
$$ K_0 \left( {C\left( X \right)_\Gamma } \right) \simeq K_0 \left( {C_0 \left( {X \times \mathbb{R}^{2n} } \right)_\Gamma } \right). $$
(6.2)
Here, however, the action of the group Г on X × ℝ2n is the product of its original action on X and the trivial action on ℝ2n. From this point of view, the isomorphism (6.2) cannot be considered as a full-fledged analog of the isomorphism (6.1).

Keywords

Fredholm Operator Homogeneous Element Exterior Product Hilbert Module Euler Operator 
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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