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 ${\mathbb{T}}^2$.

32 pages. Editorial improvements from the first version, and a few remarks added. Final version, to appear in Advances in Mathematics

References in corpus (3)

• Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces
• Simplicity of UHF and Cuntz algebras on $L^p$~spaces
• Topologically irreducible representations of the Banach *-algebra associated with a dynamical system

Cited by in corpus (4)

• Amenable crossed product Banach algebras associated with a class of ${\mathrm{C}}^{\ast }$-dynamical systems
• Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces
• Topologically irreducible representations of the Banach *-algebra associated with a dynamical system
• The closure of ideals of ${\mathbf{\ell }}^1(\mathbf{\Sigma })$ in its enveloping ${\mathrm{C}}^{\ast }$-algebra