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

Abstract

This paper 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 $C^R(X)$ will play a special role. We also will identify Atiyah's $KR^0(X, \tau ))$ in terms of $KO_0$ of an associated real $C^*$-algebra.