算子理论在Collatz猜想中的应用
Application of Operator Theory for the Collatz Conjecture
AI总结：
本研究通过单算子、双算子和Cuntz algebra三种$C^{*}$-代数表述Collatz猜想，证明其无非平凡约化子空间条件分别蕴含或等价于该猜想，并推广了相关代数不可约性与猜想的联系。
摘要：
Collatz映射（或$3n{+}1$-映射）$f$定义在正整数上，当$n$为奇数时$f(n)$等于$3n+1$，当$n$为偶数时等于$n/2$。Collatz猜想断言，从任意正整数$n$出发，$f$的某次迭代取值为1。本研究通过$C^{*}$-代数讨论Collatz猜想的表述，分为以下三种方式：(1) 单算子，(2) 双算子，(3) Cuntz algebra。对于由它们各自生成的$C^{*}$-代数，我们考虑其无非平凡约化子空间的条件。对于(1)，我们证明该条件蕴含Collatz猜想。在(2)和(3)的情况下，我们证明该条件等价于Collatz猜想。对于类似映射，我们引入了由它们定义的等价关系，并推广了Collatz猜想与相关$C^{*}$-代数不可约性之间的联系。
英文摘要：
The Collatz map (or the $3n{+}1$-map) $f$ is defined on positive integers by setting $f(n)$ equal to $3n+1$ when $n$ is odd and $n/2$ when $n$ is even. The Collatz conjecture states that starting from any positive integer $n$, some iterate of $f$ takes value $1$. In this study, we discuss formulations of the Collatz conjecture by $C^{*}$-algebras in the following three ways: (1) single operator, (2) two operators, and (3) Cuntz algebra. For the $C^{*}$-algebra generated by each of these, we consider the condition that it has no non-trivial reducing subspaces. For (1), we prove that the condition implies the Collatz conjecture. In the cases (2) and (3), we prove that the condition is equivalent to the Collatz conjecture. For similar maps, we introduce equivalence relations by them and generalize connections between the Collatz conjecture and irreducibility of associated $C^{*}$-algebras.
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