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

作者：de Jeu Marcel^*; Tomiyama Jun

来源：Advances in Mathematics, 2016, 301: 79-115.

DOI：10.1016/j.aim.2016.06.008

• Banach algebra
• Algebraically irreducible representation
• Primitive ideal
• Structure space
• Crossed product
• Topological dynamical system

If X is a compact Hausdorff space and sigma is a homeomorphism of X, then a Banach algebra l(1)(Sigma) of crossed product type is naturally associated with this topological dynamical system Sigma = (X, sigma). If X consists of one point, then l(1)(Sigma) is the group algebra of the integers.
We study the algebraically irreducible representations of l(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 l(1)(Sigma) is semisimple. All primitive ideals of l(1)(Sigma) are selfadjoint, and l(1)(Sigma) 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)(Sigma) 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.

• 出版日期2016-10-1