K-Theory for Real C*-algebras via Unitary Elements with Symmetries

TL;DR

This paper demonstrates that all KO groups for real C*-algebras can be constructed from symmetric unitary matrices, providing explicit formulas for boundary maps similar to the complex case.

Contribution

It introduces a new approach to KO groups using symmetric unitaries and computes boundary maps explicitly for real C*-algebras.

Findings

KO groups derived from symmetric unitaries

Boundary maps computed as exponential or index maps

Formulas closely resemble the complex case

Abstract

We prove that all eight KO groups for a real C*-algebra can be constructed from homotopy classes of unitary matrices that respect a variety of symmetries. In this manifestation of the KO groups, all eight boundary maps in the 24-term exact sequence associated to an ideal in a real C*-algebra can be computed as exponential or index maps with formulas that are nearly identical to the complex case.