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

Abstract

Abstract. If X is a compact Hausdorff space and σ is a homeomorphism of X, then a Banach algebra ℓ 1 (Σ) of crossed product type is naturally associated with this topological dynamical system Σ = (X, σ). If X consists of one point, then ℓ 1 (Σ) is the group algebra of the integers.We study the algebraically irreducible representations of ℓ 1 (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 ℓ 1 (Σ) is semisimple. All primitive ideals of ℓ 1 (Σ) are selfadjoint, and ℓ 1 (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 ℓ 1 (Σ) is conditionally shown to be homeomorphic to the product of a space of finite orbits and 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/Z and T. For rational rotations of T, this implies that the structure space is homeomorphic to T 2 .

Topologically irreducible representations of the Banach -algebra associated with a dynamical system

“…If µ is completely non-atomic then this follows from Theorem 3.4. If µ is atomic, then L 2 (X, µ) ∼ = ℓ 2 (Z) and this is proved in [5].…”

“…We naturally expect that a topologically irreducible, infinite-dimensional, representation of ℓ 1 (Σ) is not algebraically irreducible. Indeed this is shown for irreducible representations induced from aperiodic orbits in X ( [5]). There must be other properties which exhibit a stark difference between these two objects, deserving thorough investigation but beyond the scope of our present research.…”

“…Thus we are here confined to the problem of irreducible representations. (See [2,3,4,5,8,9,10] for the ideal structures and some irreducible representations).…”

Amenable Crossed Product Banach Algebras Associated with a Class of $$\varvec{{\mathrm C}^*}$$ C ∗ -Dynamical Systems

“…and n ∈ Z. The algebra ℓ 1 (Z, C(X); α) has been studied in [9], [10], and [11], and the question whether this algebra is amenable was what originally led to this paper. According to Theorem 2.4, the answer is affirmative.…”