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

TL;DR

This paper investigates the algebraic irreducible representations, primitive ideals, and structure space of a Banach algebra associated with a topological dynamical system, revealing conditions for semisimplicity and topological descriptions of the structure space.

Contribution

It provides a detailed classification of algebraically irreducible representations and primitive ideals of the algebra, and describes the structure space in various cases, including finite and periodic point scenarios.

Findings

Finite dimensional irreducible representations are classified up to algebraic equivalence.

The algebra is shown to be semisimple with all primitive ideals being selfadjoint.

The structure space is topologically described as a product of orbit spaces and tori, depending on the 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 $\mathbb T^2$.