K-Theory and Structural Properties of $C^{\ast }$-Algebras Associated with Relative Generalized Boolean Dynamical Systems

We present an explicit formula for the $K$-theory of the $C^{\ast }$-algebra associated with a relative generalized Boolean dynamical system $(\text{CB},\text{CL},\theta ,{\text{CI}}_{\text{af}};\text{CJ})$. In particular, we find concrete generators for the $K_1$-group of $C^{\ast }(\text{CB},\text{CL},\theta ,{\text{CI}}_{\text{af}};\text{CJ})$. We also prove that every gauge-invariant ideal of $C^{\ast }(\text{CB},\text{CL},\theta ,{\text{CI}}_{\text{af}};\text{CJ})$ is Morita equivalent to a $C^{\ast }$-algebra of a relative generalized Boolean dynamical system. As a structural application, we show that if the underlying Boolean dynamical system $(\text{CB},\text{CL},\theta )$ satisfies Condition (K), then the associated $C^{\ast }$-algebra is $K_0$-liftable. Furthermore, we deduce that if $C^{\ast }(\text{CB},\text{CL},\theta ,{\text{CI}}_{\text{af}};\text{CJ})$ is separable and purely infinite, then it has real rank zero.