# Bott Periodicity

Chapter

## Abstract

The Bott periodicity theorem in Of course, a similar isomorphism can also be written out if Here, however, the action of the group Г on

*K*-theory of operator algebras gives the isomorphism of*K*-groups*K*_{0}(*A*) and*K*_{0}(*A*\( \widehat \otimes \)*C*_{0}(ℝ^{2})) and (more generally)*K*_{0}(*A*) and*K*_{0}(*A*\( \widehat \otimes \)*C*_{0}(ℝ^{2n})). In the special case where*A*=*C*(*X*), this isomorphism turns into the isomorphism of*K*-groups^{1}$$
K_0 \left( {C\left( X \right)} \right) \simeq K_0 \left( {C_0 \left( {X \times \mathbb{R}^{2n} } \right)} \right).
$$

(6.1)

*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)

*X*× ℝ^{2n}is the product of its original action on*X*and the*trivial*action on ℝ^{2n}. 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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