Related papers: Application of Operator Theory for the Collatz Con…

This work is motivated by Radulescu's result on the comparison of C*-tensor norms on C*(F_n) x C*(F_n). For unital C*-algebras A and B, there are natural inclusions of A and B into their unital free product, their maximal tensor product and…

We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to…

In this paper a new conjecture equivalent to Collatz conjecture is presented. In particural, showing that (all) the solution(s) of newly introduced iterative functional equation(s) have a given property is equivalent to prove Collatz…

The Collatz conjecture (or ``Syracuse problem'') considers recursively-defined sequences of positive integers where $n$ is succeeded by $\tfrac{n}{2}$, if $n$ is even, or $\tfrac{3n+1}{2}$, if $n$ is odd. The conjecture states that for all…

In this article, we reduce the unsolved problem of convergence of Collatz sequences to convergence of Collatz sequences of odd numbers that are divisible by 3. We give an elementary proof of the fact that a Collatz sequence does not…

We show how the Gelfand spectrum of certain commutative operator algebras can be studied based on the theorem of Stone and von Neumann. The method presented is a natural addition to the tools of quantum spectral synthesis, which were…

This article introduces a new kind of number systems on $p$-adic integers which is inspired by the well-known $3n+1$ conjecture of Lothar Collatz. A $p$-adic system is a piecewise function on $\mathbb{Z}_p$ which has branches for all…

Let $E$ be a finite directed graph with no sources or sinks and write $X_E$ for the graph correspondence. We study the $C^*$-algebra $C^*(E,Z):=\mathcal{T}(X_E,Z)/\mathcal{K}$ where $\mathcal{T}(X_E,Z)$ is the $C^*$-algebra generated by…

In this paper we consider the operator system $\cl{S}_n$ generated by $n$ Cuntz isometries, i.e. the span of the generators of the Cuntz algebra $\cl{O}_n$ together with their adjoints and the identity. We define an operator subsystem…

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\Gamma)$ for $\Gamma$ a finitely generated group with solvable word problem, $C^*(\Gamma)$ for $\Gamma$ a finitely…

In this paper we are shown the following facts: The probability of increased $ A_{k}=P(T^{k} (x_{0})>T^{k-1} (x_{0})) $, and the probability of decrease $B_{k}=P(T^{k} (x_{0})<T^{k-1} (x_{0}))$ in step $ k $ of a Collataz procedure…

We find first structural background information about the reasons that for any C*-algebra $A$ and any two Hilbert $A$-modules $M \subseteq N$ with $M^\perp=\{0\}$, every bounded $A$-linear map $N \to A$ (or $N \to N)$ vanishing on $M$ might…

This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive…

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…

The $C^{\ast}$-algebra $\mathcal{U}_{nc}(n)$ is the universal $C^{\ast}$-algebra generated by $n^2$ generators $u_{ij}$ that make up a unitary matrix. We prove that Kirchberg's formulation of Connes' embedding problem has a positive answer…

We show that the representations of the Cuntz C$^\ast$-algebras $O_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this…

Let $T$ be a bounded linear operator on a Hilbert space $H$ such that \[ \alpha[T^*,T]:=\sum_{n=0}^\infty \alpha_n T^{*n}T^n\ge 0. \] where $\alpha(t)=\sum_{n=0}^\infty \alpha_n t^n$ is a suitable analytic function in the unit disc…

We prove that if an analytic map $f:=(f_1,\ldots ,f_n):U\subset \mathbb{C}^n\rightarrow \mathbb{C}^n$ admits an algebraic addition theorem then there exists a meromorphic map $g:=(g_1,\ldots ,g_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$…

Two conjectures are presented. The first, Conjecture 1, is that the pushforward of a geometric distribution on the integers under $n$ Collatz iterates, modulo $2^p$, is usefully close to uniform distribution on the integers modulo $2^p$, if…

We construct a family of purely infinite $C^*$-algebras, $\mathcal{Q}^\lambda$ for $\lambda\in (0,1)$ that are classified by their $K$-groups. There is an action of the circle $\T$ with a unique ${\rm KMS}$ state $\psi$ on each…