Bott Periodicity

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).

We assume that X is compact.