Searcharxiv⌕ Search

arXiv subjects

Matej Brešar

Publications and source records attributed to Matej Brešar.

At least 19 recordsLinked to original sources
arXiv2025

Products of commutators in simple algebras

Let $A$ be a finite-dimensional simple algebra that is not a field. We show that every $a\in A$ can be written as $a=(bc-cb)(de-ed)$ for some $b,c,d,e\in A$. This is not always true for infinite-dimensional simple algebras. In fact, for any $m\in \mathbb N$ we provide an example of an infinite-dimensional simple unital $C^*$-algebra $A$ in which $1$ cannot be written as $\sum_{i=1}^m x_i(a_ib_i-b_ia_i)y_i$ for some $x_i,a_i,b_i,y_i\in A$.

math.RA↗
arXiv2025

Jordan homomorphisms and T-ideals

Let $A$ and $B$ be associative algebras over a field $F$ with {\rm char}$(F)\ne 2$. Our first main result states that if $A$ is unital and equal to its commutator ideal, then every Jordan epimorphism $φ:A\to B$ is the sum of a homomorphism and an antihomomorphism. Our second main result concerns (not necessarily surjective) Jordan homomorphisms from $H(A,*)$ to $B$, where $*$ is an involution on $A$ and $H(A,*)=\{a\in A\,|\, a^*=a\}$. We show that there exists a ${\rm T}$-ideal $G$ having the following two properties: (1) the Jordan homomorphism $φ:H(G(A),*)\to B$ can be extended to an (associative) homomorphism, subject to the condition that the subalgebra generated by $φ(H(A,*))$ has trivial annihilator, and (2) every element of the ${\rm T}$-ideal of identities of the algebra of $2\times 2$ matrices is nilpotent modulo $G$. A similar statement is true for Jordan homomorphisms from $A$ to $B$. A counter-example shows that the assumption on trivial annihilator cannot be removed.

math.RA↗
arXiv2025

Matrix evaluations of noncommutative rational functions and Waring problems

Let $r$ be a nonconstant noncommutative rational function in $m$ variables over an algebraically closed field $K$ of characteristic 0. We show that for $n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$ distinct and nonzero eigenvalues. This result is used to study the linear and multiplicative Waring problems for matrix algebras. Concerning the linear problem, we show that for $n$ large enough, every matrix in $sl_n(K)$ can be written as $r(Y)-r(Z)$ for some $Y,Z\in M_n(K)^m$. We also discuss variations of this result for the case where $r$ is a noncommutative polynomial. Concerning the multiplicative problem, we show, among other results, that if $f$ and $g$ are nonconstant polynomials, then, for $n$ large enough, every nonscalar matrix in $GL_n(K)$ can be written as $f(Y)g(Z)$ for some $Y,Z\in M_n(K)^m$.

math.RA↗
arXiv2024

Splittable Jordan homomorphisms and commutator ideals

We define a Jordan homomorphism $φ$ from a ring $R$ to a ring $R'$ to be splittable if the ideal (of the subring generated by the image of $φ$) generated by all $φ(xy)-φ(x)φ(y)$, $x,y\in R$, has trivial intersection with the ideal generated by all $φ(xy)-φ(y)φ(x)$, $x,y\in R$. Our main result states that a splittable Jordan homomorphism is the sum of a homomorphism and an antihomomorphism on the commutator ideal. As applications, we obtain results that give new insight into the question of the structure of Jordan homomorphisms on some classes of rings.

math.RA↗
arXiv2024

The Wedderburn-Artin Theorem

The celebrated Wedderburn-Artin theorem states that a simple left artinian ring is isomorphic to the ring of matrices over a division ring. We give a short and self-contained proof which avoids the use of modules.

math.RA↗
arXiv2024

Frobenius' Theorem on Division Algebras

Frobenius' Theorem states that the only finite-dimensional real division algebras are the algebra of real numbers $\mathbb R$, the algebra of complex numbers $\mathbb C$, and the algebra of quaternions $\mathbb H$. We present a short proof which uses only standard undergraduate mathematics.

math.RA↗
arXiv2024

Products of commutators in matrix rings

Let $R$ be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when $R$ is commutative, is provided. If, however, $R$ has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if $R$ is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If $R$ is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of $a$ is greater than $2$.

math.RA↗
arXiv2023

The Waring Problem for Matrix Algebras, II

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $α_1,α_2,α_3$ are nonzero elements from $F$ such that $α_1+α_2+α_3=0$, then every trace zero $n\times n$ matrix over $F$ can be written as $α_1 A_1+α_2A_2+α_3A_3$ for some $A_i$ in the image of $f$ in $M_n(F)$.

math.RA↗
arXiv2022

Two-sided zero product determined algebras

An algebra $A$ is said to be two-sided zero product determined if every bilinear functional $φ:A\times A\to F$ satisfying $ φ(x,y)=0$ whenever $xy=yx=0$ is of the form $φ(x,y)=τ_1(xy) + τ_2(yx)$ for some linear functionals $τ_1,τ_2$ on $A$. We present some basic properties and equivalent definitions, examine connections with some properties of derivations, and as the main result prove that a finite-dimensional simple algebra that is not a division algebra is two-sided zero product determined if and only if it is separable.

math.RA↗
arXiv2022

Jordan maps and zero Lie product determined algebras

Let $A$ be an algebra over a field $F$ with {\rm char}$(F)\ne 2$. If $A$ is generated as an algebra by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $Φ:A\times A\to X$, where $X$ is an arbitrary vector space over $F$, the condition that $Φ(x^2,x)=0 $ for all $x\in A$ implies that $Φ(xy,z) +Φ(zx,y) + Φ(yz,x)=0$ for all $x,y,z\in A$. This is applicable to the question of whether $A$ is zero Lie product determined, and is also used in proving that a Jordan homomorphism from $A$ onto a semiprime algebra $B$ is the sum of a homomorphism and an antihomomorphism.

math.RA↗
arXiv2021

Weighted Jordan homomorphisms

Let $A$ and $B$ be unital rings. An additive map $T:A\to B$ is called a weighted Jordan homomorphism if $c=T(1)$ is an invertible central element and $cT(x^2) = T(x)^2$ for all $x\in A$. We provide assumptions, which are in particular fulfilled when $A=B=M_n(R)$ with $n\ge 2$ and $R$ any unital ring with $\frac{1}{2}$, under which every surjective additive map $T:A\to B$ with the property that $T(x)T(y)+T(y)T(x)=0$ whenever $xy=yx=0$ is a weighted Jordan homomorphism. Further, we show that if $A$ is a prime ring with char$(A)\ne 2,3,5$, then a bijective additive map $T:A\to A$ is a weighted Jordan homomorphism provided that there exists an additive map $S:A\to A$ such that $S(x^2)=T(x)^2$ for all $x\in A$.

math.RA↗
arXiv2021

Automorphisms and derivations of finite-dimensional algebras

Let $A$ be a finite-dimensional algebra over a field $F$ with char$(F)\ne 2$. We show that a linear map $D:A\to A$ satisfying $xD(x)x\in [A,A]$ for every $x\in A$ is the sum of an inner derivation and a linear map whose image lies in the radical of $A$. Assuming additionally that $A$ is semisimple and char$(F)\ne 3$, we show that a linear map $T:A\to A$ satisfies $T(x)^3- x^3 \in [A,A]$ for every $x\in A$ if and only if there exist a Jordan automorphism $J$ of $A$ lying in the multiplication algebra of $A$ and a central element $α$ satisfying $α^3=1$ such that $T(x)=αJ(x)$ for all $x\in A$. These two results are applied to the study of local derivations and local (Jordan) automorphisms. In particular, the second result is used to prove that every local Jordan automorphism of a finite-dimensional simple algebra $A$ (over a field $F$ with char$(F)\ne 2,3$) is a Jordan automorphism.

math.RA↗
arXiv2020

Commutators and images of noncommutative polynomials

Let $A$ be an algebra and let $f$ be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between $[A,A]$, the linear span of commutators in $A$, and span$f(A)$, the linear span of the image of $f$ in $A$. In particular, we show that $[A,A]=A$ implies span$f(A)=A$. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if $C$ is a commutative unital algebra over a field $F$ of characteristic $0$, $A$ is the matrix algebra $M_n(C)$, and the polynomial $f$ is neither an identity nor a central polynomial of $M_n(F)$, then every commutator in $A$ can be written as a difference of two elements, each of which is a sum of $7788$ elements from $f(A)$ (if $C=F$ is an algebraically closed field, then $4$ elements suffice). Similar results are obtained for some other algebras, in particular for the algebra $B(H)$ of all bounded linear operators on a Hilbert space $H$.

math.RA↗
arXiv2019

On, Around, and Beyond Frobenius' Theorem on Division Algebras

Frobenius' Theorem states that the algebra of quaternions $\mathbb H$ is, besides the fields of real and complex numbers, the only finite-dimensional real division algebra. We first give a short elementary proof of this theorem, then characterize finite-dimensional real algebras that contain either a copy of $\mathbb C$, a copy of $\mathbb H$, or a pair of anticommuting invertible elements through the dimensions of their (left) ideals, and finally consider the problem of lifting algebraic elements modulo ideals.

math.RA↗
arXiv2019

Functional identities and zero Lie product determined Banach algebras

Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, the second one concerns derivations that preserve commutativity, and the third one concerns bijective commutativity preserving linear maps.

math.RA↗