K-theoretic classification for certain inductive limit $Z_2$ actions on real rank zero C$^*$-algebras

In this paper a K-theoretic classification is given of the C$^*$-algebra dynamical systems $(A, \alpha , Z_2)= \lim \limits_\to (A_n, {\alpha }_n, Z_2)$ where $A$ is of real rank zero, each $A_n$ is a finite direct sum of matrix algebras over finite connected graphs, and each $\alpha _n$ is induced by an action on each component of the spectrum of $A_n$. Corresponding to the trivial actions is the K-theoretic classification for real rank zero C$^*$-algebras that can be expressed as finite direct sums of matrix algebras over finite graphs obtained in Mem. Amer. Math. Soc. no. 547, vol. 114.