Application of Operator Theory for the Collatz Conjecture

TL;DR

This paper explores the Collatz conjecture through the lens of $C^{*}$-algebra theory, establishing new equivalences and conditions that relate algebraic properties to the conjecture's truth.

Contribution

It introduces novel formulations of the Collatz conjecture using $C^{*}$-algebras and proves equivalences between algebraic properties and the conjecture.

Findings

Proves that certain $C^{*}$-algebra conditions imply the Collatz conjecture.

Establishes equivalences between algebraic properties and the conjecture for specific operator frameworks.

Generalizes the connection between the conjecture and algebraic irreducibility for similar maps.

Abstract

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.