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

If X is a compact Hausdorff space and σ is a homeomorphism of X  , then a Banach algebra ℓ1(Σ)ℓ1(Σ) of crossed product type is naturally associated with this topological dynamical system Σ=(X,σ)Σ=(X,σ). If X   consists of one point, then ℓ1(Σ)ℓ1(Σ) is the group algebra of the integers.We study the algebraically irreducible representations of ℓ1(Σ)ℓ1(Σ) 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 ℓ1(Σ)ℓ1(Σ) is semisimple. All primitive ideals of ℓ1(Σ)ℓ1(Σ) are selfadjoint, and ℓ1(Σ)ℓ1(Σ) is Hermitian if there are only periodic points in X. If X   is metrizable or all points are periodic, then all primitive ideals arise as in our construction. A part of the structure space of ℓ1(Σ)ℓ1(Σ) is conditionally shown to be homeomorphic to the product of a space of finite orbits and TT. 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/ZX/Z and TT. For rational rotations of TT, this implies that the structure space is homeomorphic to T2T2.