Real $K$-Theory for $C^*$-Algebras: Just the Facts

TL;DR

This paper reviews the fundamental aspects of real K-theory ($KO$-theory) for $C^*$-algebras, clarifying its connections with complex K-theory, $KR$-theory, and applications in physics, aiming to serve as a foundational reference.

Contribution

It provides a comprehensive overview of $KO$-theory for real $C^*$-algebras, highlighting its properties, relationships with other theories, and potential applications in physics.

Findings

Clarifies the relationship between $KO$-theory and complex $K$-theory.

Explains the connection of $KO$-theory with the Ten-Fold Way in physics.

Highlights the role of $KO$-theory in operator algebras and mathematical physics.

Abstract

This work is intended to present the basic properties of $KO$-theory for real $C^*$-algebras and to explain its relationship with complex $K$-theory and with $KR$- theory. Whenever possible we will rely upon proofs in printed literature, particularly the work of Karoubi, Wood, Schr\"oder, and more recent work of Boersema and J. M. Rosenberg. In addition, we shall explain how $KO$-theory is related to the Ten-Fold Way in physics and point out how some deeper features of $KO$-theory for operator algebras may provide powerful new tools there. Commutative real $C^*$-algebras not of the form $C(X, R)$ will play a special role. Unfortunately, there is no single reference for $KO$-theory for operator algebras that begins to compare with Blackadar's wonderful exposition of complex $K$-theory. This work is intended to provide a platform upon which mathematicians and mathematical physicists can rely in order to use these new tools in their research. As we are writing for a diverse audience of functional analysts, topologists, and physicists, we often present material well-known to one group of people and unfamiliar to another.