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

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• 作者：Marcel de Jeu^a ; ^{mdejeu@math.leidenuniv.nl" class="auth_mail" title="E-mail the corresponding author} ; Jun Tomiyama^b ; ^{juntomi@med.email.ne.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 46H15 ; secondary ; 46H10 ; 47L65 ; 54H20
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：301
• 期：Complete
• 页码：79-115
• 全文大小：675 K

文摘

If X is a compact Hausdorff space and σ is a homeomorphism of X  , then a Banach algebra ℓ¹(Σ)${\ell }^1(\mathrm{\Sigma })$ of crossed product type is naturally associated with this topological dynamical system Σ=(X,σ)$\mathrm{\Sigma }=(X,\sigma )$. If X   consists of one point, then ℓ¹(Σ)${\ell }^1(\mathrm{\Sigma })$ is the group algebra of the integers.

We study the algebraically irreducible representations of ℓ¹(Σ)${\ell }^1(\mathrm{\Sigma })$ 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 ℓ¹(Σ)${\ell }^1(\mathrm{\Sigma })$ is semisimple. All primitive ideals of ℓ¹(Σ)${\ell }^1(\mathrm{\Sigma })$ are selfadjoint, and ℓ¹(Σ)${\ell }^1(\mathrm{\Sigma })$ 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 ℓ¹(Σ)${\ell }^1(\mathrm{\Sigma })$ is conditionally shown to be homeomorphic to the product of a space of finite orbits and T$\mathbb{T}$. 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/Z$X/\mathbb{Z}$ and T$\mathbb{T}$. For rational rotations of T$\mathbb{T}$, this implies that the structure space is homeomorphic to T²${\mathbb{T}}^2$.