与拓扑动力系统相关的巴拿赫代数的代数不可约表示与结构空间
Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system
摘要：
若$X$是紧Hausdorff空间，且$σ$是$X$的同胚，则交叉积型巴拿赫代数$\ell^1(Σ)$自然地与这个拓扑动力系统$Σ=(X,σ)$相联系.如果$X$由一个点组成，则$\ell^1(Σ)$是整数的群代数。本文研究了$\ell^1(Σ)$在复向量空间、它的本原理想和它的结构空间上的代数不可约表示。有限维代数不可约表示被确定为代数等价，并且构造了一个足够丰富的无限维代数不可约表示族，从而能够得出$\ell^1(Σ)$是半单的结论. $\ell^1(Σ)$的所有本原理想都是自伴的，如果$X$中只有周期点，则$\ell^1(Σ)$是厄米特的.如果$X$是可度量化的或者所有点都是周期的，那么所有的本原理想都像我们的构造一样产生。证明了$\ell^1(Σ)$的部分结构空间与有限轨道空间和$\mathbb T$的乘积同胚.如果$X$是一个有限集，那么结构空间是若干环面的拓扑不交并，$X$中的每个轨道对应一个环面。如果$X$的所有点都有相同的有限周期，则它是轨道空间$X/\mathbb Z$和$\mathbb T$的乘积。对于$\mathbb T$的有理旋转，这意味着结构空间与$\mathbb T^2$同胚。
英文摘要：
If $X$ is a compact Hausdorff space and $σ$ is a homeomorphism of $X$, then a Banach algebra $\ell^1(Σ)$ of crossed product type is naturally associated with this topological dynamical system $Σ=(X,σ)$. If $X$ consists of one point, then $\ell^1(Σ)$ is the group algebra of the integers. We study the algebraically irreducible representations of $\ell^1(Σ)$ 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 $\ell^1(Σ)$ is semisimple. All primitive ideals of $\ell^1(Σ)$ are selfadjoint, and $\ell^1(Σ)$ 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 $\ell^1(Σ)$ 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 $\mathbb T^2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
