Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system

Abstract

If $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a Banach algebra $\ell^1(\Sigma)$ of crossed product type is naturally associated with this topological dynamical system $\Sigma=(X,\sigma)$. If $X$ consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the integers. We study the algebraically irreducible representations of $\ell^1(\Sigma)$ on complex vector spaces, its primitive ideals and its structure space. The finite dimensional algebraically irreducible representations are determined up to algebraic equivalence, and a sufficiently rich family of infinite dimensional algebraically irreducible representations is constructed to be able to conclude that $\ell^1(\Sigma)$ is semisimple. All primitive ideals of $\ell^1(\Sigma)$ are selfadjoint, and $\ell^1(\Sigma)$ is Hermitian if there are only periodic points in $X$. If $X$ is metrisable or all points are periodic, then all primitive ideals arise as in our construction. A part of the structure space of $\ell^1(\Sigma)$ is conditionally shown to be homeomorphic to the product of a space of finite orbits and $\mathbb T$. If $X$ is a finite set, then the structure space is the topological disjoint union of a number of tori, one for each orbit in $X$. If all points of $X$ have the same finite period, then it is the product of the orbit space $X/\mathbb Z$ and $\mathbb T$. For rational rotations of $\mathbb T$, this implies that the structure space is homeomorphic to $\mathbb T^2$.