A Banach *-algebra associated with the action of a group on a topological space

Given a discrete and countable infinite group $G$ acting via homeomorphism on a compact and Hausdorff space $X$, we consider $\Sigma$ the dynamical system associated to the action. One can naturally associate to $\Sigma$ the crossed product type Banach *-algebra $\ell^1(\Sigma)$. We will study how different dynamical properties of $\Sigma$ are characterized as analytic-algebraic properties of $\ell^1(\Sigma)$. The dynamical properties of $\Sigma$ that we will study are topological freeness, minimality, topological transitivity and residual topological freeness. We will also see how, under some assumptions on $G$, one can detect if $\Sigma$ is free by detecting closed non-self-adjoint ideals of $\ell^1(\Sigma)$.