A Bott periodicity proof for real graded $C^\ast$-algebras

TL;DR

This paper proves Bott periodicity for real graded C*-algebras using K-theory and E-theory, extending previous complex cases to the real setting and establishing 8-fold periodicity.

Contribution

It extends Bott periodicity proof from complex to real graded C*-algebras, demonstrating 8-fold periodicity in E-theory with explicit inverse maps.

Findings

Established 8-fold periodicity in E-theory for real graded C*-algebras

Constructed explicit inverse maps demonstrating periodicity

Extended complex case results to the real setting

Abstract

We give a proof of Bott periodicity for real graded $C^\ast$-algebras in terms of K- theory and E-theory. Guentner and Higson proved a similar result in the complex graded case but we extend this to cover all graded $C^\ast$-algebras. We obtain the 8-fold periodicity in E-theory by constructing two maps that are inverse to each other.