Related papers: Unique prime factorization and bicentralizer probl…
We obtain a complete classification of a large class of non almost periodic free Araki-Woods factors $\Gamma(\mu,m)"$ up to isomorphism. We do this by showing that free Araki-Woods factors $\Gamma(\mu, m)"$ arising from finite symmetric…
We consider the second order Cauchy problem $$u''+\m{u}Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is…
The category of $C^*$-algebras is blessed with many different tensor products. In contrast, virtually the only tensor product ever used in the category of von Neumann algebras is the normal spatial tensor product. We propose a definition of…
In 1960 Schwinger [J. Schwinger, Proc.Natl.Acad.Sci. 46 (1960) 570- 579] proposed the algorithm for factorization of unitary operators in the finite M dimensional Hilbert space according to a coprime decomposition of M. Using a special…
We prove a generalisation of Vinogradov's theorem by finding for $m\geqslant 3$ and fixed positive integers $c_1, \dots ,c_m, r_1, \dots , r_m$ the asymptotics of the number of sequences $(n_1, \dots ,n_m) \in \mathbf{N}^{m}$ such that…
We consider a joint ordered multifactorisation for a given positive integer $n\geq 2$ into $m$ parts, where $n=n_1~\times~\ldots~\times~n_m$, and each part $n_j$ is split into one or more component factors. Our central result gives an…
Given $k, \ell \in {\bf N}^+$, let $x_{i,j}$ be, for $1 \le i \le k$ and $0 \le j \le \ell$, some fixed integers, and define, for every $n \in {\bf N}^+$, $s_n := \sum_{i=1}^k \prod_{j=0}^\ell x_{i,j}^{n^j}$. We prove that the following are…
We study upgraded free independence phenomena for unitary elements $u_1$, $u_2$, \dots representing the large-$n$ limit of Haar random unitaries, showing that free independence extends to several larger algebras containing $u_j$ in the…
Irrespective of whether n is prime, prime power with exponent >1, or composite, the group U_n of units of Z_n can sometimes be obtained as the direct product of cyclic groups generated by x, x+k and x+2k, for x, k in Z_n. Indeed, for many…
We study conjugacy orbits of certain types of subalgebras in tracial von Neumann algebras. For any separable II$_1$ factor $N_0$ we construct a highly indecomposable non Gamma II$_1$ factor $N$ such that $N_0 \subset N$ and moreover every…
Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th…
The three-way table problem is to decide if there exists an l x m x n table satisfying given line sums, and find a table if there is one. It is NP-complete already for l=3 and every bounded integer program can be isomorphically represented…
The structure of the multiplicative group $M_n = ({\mathbb Z}/n{\mathbb Z})^\times$ encodes a great deal of arithmetic information about the integer $n$ (examples include $\phi(n)$, the Carmichael function $\lambda(n)$, and the number…
Starting from a (small) rigid C$^*$-tensor category $\mathscr{C}$ with simple unit, we construct von Neumann algebras associated to each of its objects. These algebras are factors and can be either semifinite (of type II$_1$ or II$_\infty$,…
If $\mathcal{C}$ is a category of algebras closed under finite direct products, and $M_\mathcal{C}$ the commutative monoid of isomorphism classes of members of $\mathcal{C},$ with operation induced by direct product, A.Tarski defined a…
We define hypersurfaces $f\colon M^n\to \mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ of a product of two space forms as those that have flat normal bundle when regarded as submanifolds of the underlying flat…
A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each…
We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime…
In this paper we introduce a new class of non-amenable groups denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras. This means that for every $\Gamma \in {\bf NC}_1 \cap {\bf…
Recently, Brannan and Vergnioux showed that the free orthogonal quantum group factors $\mathcal{L}\mathbb{F}O_M$ have Jung's strong 1-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for…