The $K$-theory of real graph $C*$-algebras

Abstract

In this paper, we will introduce real graph algebras and develop the theory to the point of being able to calculate the $K$-theory of such algebras. The $K$-theory situation is significantly more complicated than in the case for complex graph algebras. To develop the long exact sequence to compute the $K$-theory of a real graph algebra, we need to develop a generalized theory of crossed products for real C*-algebras for groups with involution. We also need to deal with the additional algebraic intricacies related to the period-8 real $K$-theory using united $K$-theory. Ultimately, we prove that the $K$-theory of a real graph algebra is recoverable from the $K$-theory of the corresponding complex graph algebra.