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