Bott Periodicity

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


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
$$ K_0 \left( {C\left( X \right)} \right) \simeq K_0 \left( {C_0 \left( {X \times \mathbb{R}^{2n} } \right)} \right). $$
Of course, a similar isomorphism can also be written out if C(X) is replaced by the crossed product C(X)Г:
$$ K_0 \left( {C\left( X \right)_\Gamma } \right) \simeq K_0 \left( {C_0 \left( {X \times \mathbb{R}^{2n} } \right)_\Gamma } \right). $$
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).


Fredholm Operator Homogeneous Element Exterior Product Hilbert Module Euler Operator 
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© Birkhäuser Verlag AG 2008

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