动力系统与巴拿赫代数通过$\mathbb{Z}$的交叉积之间的联系
Connections between dynamical systems and crossed products of Banach algebras by $\mathbb{Z}$
AI总结：
本文研究巴拿赫代数与整数群交叉积的性质，证明理想与子代数相交、极大交换性及非周期点稠密性之间的等价关系，并刻画交叉积的单纯性与素性对应的动力系统条件。
摘要：
从一个复交换半单正则巴拿赫代数 $A$ 及其自同构 $σ$ 出发，我们构造 $A$ 与整数群的交叉积，其中整数群通过 $σ$ 的迭代作用于 $A$。该自同构以自然方式在 $A$ 的特征空间 $Δ(A)$ 上诱导出一个拓扑动力系统。我们证明了以下性质之间的等价性：交叉积中每个非零理想与子代数 $A$ 的交集非零、$A$ 在交叉积中的极大交换性、以及诱导系统在特征空间上非周期点的稠密性。我们还证明了交叉积中每个非平凡理想总是与 $A$ 的交换子非平凡地相交。此外，在假设 $A$ 是单位的且 $Δ(A)$ 包含无穷多个点的条件下，我们证明了交叉积的单纯性与诱导系统的极小性之间的等价性，以及交叉积的素性与系统的拓扑传递性之间的等价性。
英文摘要：
Starting with a complex commutative semi-simple regular Banach algebra $A$ and an automorphism $σ$ of $A$, we form the crossed product of $A$ with the integers, where the latter act on $A$ via iterations of $σ$. The automorphism induces a topological dynamical system on the character space $Δ(A)$ of $A$ in a natural way. We prove an equivalence between the property that every non-zero ideal in the crossed product has non-zero intersection with the subalgebra $A$, maximal commutativity of $A$ in the crossed product, and density of the non-periodic points of the induced system on the character space. We also prove that every non-trivial ideal in the crossed product always intersects the commutant of $A$ non-trivially. Furthermore, under the assumption that $A$ is unital and such that $Δ(A)$ consists of infinitely many points, we show equivalence between simplicity of the crossed product and minimality of the induced system, and between primeness of the crossed product and topological transitivity of the system.
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