Related papers: A Random Matrix Approach to the Lack of Projection…

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…

We generalize the recent results on radial projections by Orponen, Shmerkin, Wang using two different methods. In particular, we show that given $X,Y\subset \mathbb{R}^n$ Borel sets and $X\neq \emptyset$. If $\dim Y \in (k,k+1]$ for some…

We give a new proof of the absence of non-trivial idempotents in the group ring of torsion-free cocompact lattices in SL(n,C). It is based on the following procedure. We lift the class of the trace in the cyclic cohomology of the group ring…

We discuss the notion of matrix model, $\pi:C(X)\to M_K(C(T))$, for algebraic submanifolds of the free complex sphere, $X\subset S^{N-1}_{\mathbb C,+}$. When $K\in\mathbb N$ is fixed there is a universal such model, which factorizes as…

We study relatively free associative algebras $F^{(n)}_r$ of ranks $r=2,3$ with the identity $[x_1,\dots, x_n]=0$ of Lie nilpotency of step $n\geqslant 3$ over a field $K$ of characteristic $\neq 2,3$. First we prove a Theorem on the…

The outer automorphism group Out(F_2g) of a free group on 2g generators naturally contains the mapping class group of a punctured surface as a subgroup. We define a subsurface projection of the sphere complex of the connected sum of n…

In this paper two independent and unitarily invariant projection matrices P(N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size $N$ converges to infinity. The…

C*-quantum groups with projection are the noncommutative analogues of semidirect products of groups. Radford's Theorem about Hopf algebras with projection suggests that any C*quantum group with projection decomposes uniquely into an…

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions…

We study the noncommutative topology of the $C^*$-algebras $C(\mathbb{C}P_q^{n})$ of the quantum projective spaces within the framework of Kasparov's bivariant K-theory. In particular, we construct an explicit KK-equivalence with the…

We prove that a simple, separable, nuclear, purely infinite classifiable $C^*$-algebra is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.

We develop a theory of graph C*-algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting a countably infinite set of edges. We show that…

We study $k$-positive linear maps on matrix algebras and address two problems, (i) characterizations of $k$-positivity and (ii) generation of non-decomposable $k$-positive maps. On the characterization side, we derive optimization-based…

Estimating the probability that the Erd\H{o}s-R\'enyi random graph $G(n,m)$ is $H$-free, for a fixed graph $H$, is one of the fundamental problems in random graph theory. If $m$ is such that each edge of $G(n,m)$ belongs to a copy of $H'$…

In this paper we study free actions of groups on separated graphs and their \cstar{}algebras, generalizing previous results involving ordinary (directed) graphs. We prove a version of the Gross-Tucker Theorem for separated graphs yielding a…

Given a ring of ternions $R$, i. e., a ring isomorphic to that of upper triangular $2\times 2$ matrices with entries from an arbitrary commutative field $F$, a complete classification is performed of the vectors from the free left…

Let $F_k=K_1\vee P_{k-1}$ be the fan graph on $k$ vertices. A graph is said to be $F_k$-free if it does not contain $F_k$ as a subgraph. Yu et al. in [arXiv:2404.03423] conjectured that for $k\geq2$ and $m$ sufficiently large, if $G$ is an…

Inspired by the construction of the Cuntz semigroup for a C*-algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic…

For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $\chi(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2\chi(G))$…

The families $\mathcal F_0,\ldots,\mathcal F_s$ of $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ are called cross-union if there is no choice of $F_0\in \mathcal F_0, \ldots, F_s\in \mathcal F_s$ such that $F_0\cup\ldots\cup F_s=[n]$. A…