arXiv subjects

M. Brešar

Publications and source records attributed to M. Brešar.

12 recordsLinked to original sources

arXiv2026

Commutativity preserving mappings in Banach algebras

Let $A$ and $B$ be unital complex Banach algebras having no quotients isomorphic to $\mathbb{C}$ or $M_2(\mathbb{C})$. Assume additionally that $B$ is semisimple. If a surjective additive mapping $Φ\colon A\to B$ satisfies $[Φ(x^2),Φ(x)] = 0$ for all $x\in A$, then there exist a surjective direct sum of an additive homomorphism and an additive anti-homomorphism $Ψ\colon A\to B$, an invertible element $λ\in\mathcal{Z}(B)$, and an additive mapping $ζ\colon A\to\mathcal{Z}(B)$ such that $Φ(x)=λΨ(x)+ζ(x)$ for all $x\in A$.

math.RA↗

arXiv2023

f-zpd algebras and a multilinear Nullstellensatz

Let $f=f(x_1,\dots,x_m)$ be a multilinear polynomial over a field $F$. An $F$-algebra $A$ is said to be $f$-zpd ($f$-zero product determined) if every $m$-linear functional $φ\colon A^{m}\rightarrow F$ which preserves zeros of $f$ is of the form $φ(a_1,\dots,a_m)=τ(f(a_1,\dots,a_m))$ for some linear functional $τ$ on $A$. We are primarily interested in the question whether the matrix algebra $M_d(F)$ is $f$-zpd. While the answer is negative in general, we provide several families of polynomials for which it is positive. We also consider a related problem on the form of a multilinear polynomial $g=g(x_1,\dots,x_m)$ with the property that every zero of $f$ in $M_d(F)^{m}$ is a zero of $g$. Under the assumption that $m<2d-3$, we show that $g$ and $f$ are linearly dependent.

math.RA↗

arXiv2022

Derivations and homomorphisms in commutator-simple algebras

We call an algebra $A$ commutator-simple if $[A,A]$ does not contain nonzero ideals of $A$. After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation $D\colon L^1(G)\to L^1(G)$, where $G$ is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras.

math.FA↗

arXiv2021

Maps preserving two-sided zero products on Banach algebras

Let $A$ and $B$ be Banach algebras with bounded approximate identities and let $Φ:A\to B$ be a surjective continuous linear map which preserves two-sided zero products (i.e., $Φ(a)Φ(b)=Φ(b)Φ(a)=0$ whenever $ab=ba=0$). We show that $Φ$ is a weighted Jordan homomorphism provided that $A$ is zero product determined and weakly amenable. These conditions are in particular fulfilled when $A$ is the group algebra $L^1(G)$ with $G$ any locally compact group. We also study a more general type of continuous linear maps $Φ:A\to B$ that satisfy $Φ(a)Φ(b)+Φ(b)Φ(a)=0$ whenever $ab=ba=0$. We show in particular that if $Φ$ is surjective and $A$ is a $C^*$-algebra, then $Φ$ is a weighted Jordan homomorphism.

math.FA↗

arXiv2019

Zero Jordan product determined Banach algebras

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $φ\colon A\times A\to X$, where $X$ is an arbitrary Banach space, which satisfies $φ(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $φ(a,b)=σ(ab+ba)$ for some continuous linear map $σ$. We show that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally compact groups have this property, and also discuss some applications.

math.FA↗

arXiv2016

Zero Lie product determined Banach algebras

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $φ\colon A\times A\to \mathbb{C}$ with the property that $φ(a,b)=0$ whenever $a$ and $b$ commute is of the form $φ(a,b)=τ(ab-ba)$ for some $τ\in A^*$. In the first part of the paper we give some general remarks on this class of algebras. In the second part we consider amenable Banach algebras and show that all group algebras $L^1(G)$ with $G$ an amenable locally compact group are zero Lie product determined.

math.FA↗

arXiv2013

Derivations preserving quasinilpotent elements

We consider a Banach algebra $A$ with the property that, roughly speaking, sufficiently many irreducible representations of $A$ on nontrivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property turns out to be quite large -- it includes $C^*$-algebras, group algebras on arbitrary locally compact groups, commutative algebras, $L(X)$ for any Banach space $X$, and various other examples. Our main result states that every derivation of $A$ that preserves the set of quasinilpotent elements has its range in the radical of $A$.

math.OA↗

arXiv2012

Lie Superautomorphisms on Associative Algebras, II

Lie superautomorphisms of prime associative superalgebras are considered. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.

math.RA↗

arXiv2012

Maps preserving zeros of a polynomial

Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $ϕ:\A\to \A$ that preserve zeros of $f$. Under certain technical restrictions we solve the problem for general polynomials $f$ in the case where $\A=M_n(F)$. We also consider quite general algebras $\A$, but only for specific polynomials $f$.

math.RA↗

arXiv2012

On Lie and associative algebras containing inner derivations

We describe subalgebras of the Lie algebra $\mf{gl}(n^2)$ that contain all inner derivations of $A=M_n(F)$ (where $n\ge 5$ and $F$ is an algebraically closed field of characteristic 0). In a more general context where $A$ is a prime algebra satisfying certain technical restrictions, we establish a density theorem for the associative algebra generated by all inner derivations of $A$.

math.RA↗

arXiv2012

Determining elements in Banach algebras through spectral properties

Let $A$ be a Banach algebra. By $σ(x)$ and $r(x)$ we denote the spectrum and the spectral radius of $x\in A$, respectively. We consider the relationship between elements $a,b\in A$ that satisfy one of the following two conditions: (1) $σ(ax) = σ(bx)$ for all $x\in A$, (2) $r(ax) \le r(bx)$ for all $x\in A$. In particular we show that (1) implies $a=b$ if $A$ is a $C^*$-algebra, and (2) implies $a\in \mathbb C b$ if $A$ is a prime $C^*$-algebra. As an application of the results concerning the conditions (1) and (2) we obtain some spectral characterizations of multiplicative maps.

math.OA↗

arXiv2012

Identifying derivations through the spectra of their values

We consider the relationship between derivations $d$ and $g$ of a Banach algebra $B$ that satisfy $\s(g(x)) \subseteq \s(d(x))$ for every $x\in B$, where $\s(\, . \,)$ stands for the spectrum. It turns out that in some basic situations, say if $B=B(X)$, the only possibilities are that $g=d$, $g=0$, and, if $d$ is an inner derivation implemented by an algebraic element of degree 2, also $g=-d$. The conclusions in more complex classes of algebras are not so simple, but are of a similar spirit. A rather definitive result is obtained for von Neumann algebras. In general $C^*$-algebras we have to make some adjustments, in particular we restrict our attention to inner derivations implemented by selfadjoint elements. We also consider a related condition $\|[b,x]\|\leq M\|[a,x]\|$ for all selfadjoint elements $x$ from a $C^*$-algebra $B$, where $a,b\in B$ and $a$ is normal.

math.OA↗