arXiv subjects

Mateo Tomašević

Publications and source records attributed to Mateo Tomašević.

14 recordsLinked to original sources

arXiv2026

Idempotent words in finite-dimensional simple Jordan algebras

We study finite left-nested Jordan products of idempotents in finite-dimensional simple Jordan algebras over a field $\mathbb{F}$ of characteristic different from $2$. Over an algebraically closed field, the structure theorem reduces the problem to the scalar algebra, spin factors, the full matrix Jordan algebras $M_n(\mathbb{F})^+$, the symmetric matrix algebras $H_n(\mathbb{F})$, the symplectic Hermitian algebras $H_{2n}(\mathbb{F},\mathrm{sp})$, and the split Albert algebra. We determine exactly which elements are representable: except in $H_2(\mathbb{F})$, these are all nonscalar elements together with $0$ and $1$, whereas in $H_2(\mathbb{F})$ the nonzero square-zero elements are also not representable. We also study the minimal word length of representable elements. For $M_n(\mathbb{F})^+$ over an arbitrary field, the maximal word length grows logarithmically in $n$, and for $n=2$ we determine the exact length of every nonscalar matrix. The same logarithmic growth holds for $H_{2n}(\mathbb{F},\mathrm{sp})$. The spin-factor and symmetric-matrix results hold over quadratically closed fields, whereas the full matrix, symplectic Hermitian, and split Albert results hold over arbitrary fields of characteristic different from $2$.

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arXiv2026

The structure of $k$-potents and mixed Jordan-power preservers on matrix algebras

Let $M_n(\mathbb{F})$ denote the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic different from $2$. For $n \ge 2$, we classify all maps $ϕ: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ satisfying the mixed Jordan-power identity $$ ϕ(A^{k} \circ B) = ϕ(A)^{k} \circ ϕ(B), \quad \text{for all } A,B \in M_n(\mathbb{F}), $$ where $\circ$ denotes the (normalized) Jordan product $A \circ B := \tfrac{1}{2}(AB + BA)$ and $k \in \mathbb{N}$. We show that every such map is either constant, taking a fixed $(k+1)$-potent value, or there exist an invertible matrix $T \in M_n(\mathbb{F})$, a ring monomorphism $ω: \mathbb{F} \to \mathbb{F}$, and a $k$-th root of unity $\varepsilon \in \mathbb{F}$ such that $ϕ$ takes one of the forms $$ ϕ(X) = \varepsilon\, T\, ω(X)\, T^{-1} \quad \text{ or } \quad ϕ(X) = \varepsilon\, T\, ω(X)^{t}\, T^{-1}, $$ where $ω(X)$ denotes the matrix obtained by applying $ω$ entrywise to $X$, and $(\cdot)^{t}$ denotes matrix transposition. In particular, every nonconstant solution is necessarily additive. The classification relies fundamentally on the preservation of $(k+1)$-potents and their intrinsic structural properties.

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arXiv2026

Jordan rigidity of full matrix algebras

Let $\mathbb{F}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{F})^+$ denote the Jordan algebra of all $n\times n$ matrices over $\mathbb{F}$ with product $X\circ Y:=(XY+YX)/2$. We prove a rigidity theorem for $M_n(\mathbb{F})^+$, $n\ge2$: if $\mathcal{J}$ is any $2$-torsion-free Jordan ring and $ϕ:M_n(\mathbb{F})^+\to\mathcal{J}$ is a Jordan multiplicative (product-preserving) map, then $ϕ(0)$ is an idempotent and $X\mapstoϕ(X)-ϕ(0)$ is either zero or an injective Jordan ring homomorphism. Thus, up to an idempotent constant, preservation of the Jordan product alone forces additivity and the zero-or-injective dichotomy. When specialized to associative codomains, the theorem yields the Jacobson--Rickart decomposition into homomorphic and antihomomorphic parts. In particular, for maps $M_n(\mathbb F)^+\to M_k(\mathbb K)^+$, where $\mathbb K$ is a field of characteristic different from $2$, we also obtain a block normal form governed by finite-dimensional $\mathbb K$-representations of $\mathbb F$, together with a criterion for the existence of nonconstant maps.

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arXiv2026

Automatic additivity for injective Jordan semi-triple maps on structural matrix rings over division rings

Let $\mathbb D$ be a division ring, and let $\mathcal{R}\subseteq M_n(\mathbb{D})$ be a structural matrix ring over $\mathbb{D}$, that is, the subring of $M_n(\mathbb{D})$ supported on the ordered pairs of a preorder on $\{1,\ldots,n\}$. We study injective Jordan semi-triple maps $ϕ:\mathcal{R}\to M_n(\mathbb{D})$, namely injective maps satisfying \[ ϕ(XYX)=ϕ(X)ϕ(Y)ϕ(X), \qquad \text{for all } X,Y\in\mathcal{R}. \] Assuming that the centre of $\mathbb{D}$ has more than two elements, we give a criterion for automatic additivity and show that there are exactly two obstructions. The first one is scalar: it occurs precisely when $\mathcal{R}$ has a direct ring summand isomorphic to $\mathbb{D}$ and $\mathbb{D}$ is isomorphic to neither $\mathbb{F}_3$ nor $\mathbb{F}_4$. The second one is order-theoretic: it occurs when a nonsymmetric comparable pair $i\preceq j$, $j\not\preceq i$, admits no third index $k\notin\{i,j\}$ comparable with both $i$ and $j$. If neither obstruction occurs, all injective Jordan semi-triple maps are additive. The centre-size hypothesis is sharp: for $n\ge3$, the upper-triangular ring $T_n(\mathbb{F}_2)$ has neither obstruction but nevertheless admits nonadditive injective Jordan semi-triple maps. Finally, in the additive case, we describe the maps componentwise, in terms of endomorphisms, anti-endomorphisms, and transitive multipliers.

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arXiv2026

The Jordan multiplication semigroup of matrix algebras is the full endomorphism semigroup

Let $\mathbb{K}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{K})$ be the algebra of all $n\times n$ matrices over $\mathbb{K}$. We consider the corresponding special Jordan algebra $\mathcal{A}:=M_n(\mathbb{K})^+$ with symmetrized product $A\circ B:=(AB+BA)/2$, and write $\mathcal{A}_{\mathrm v}:=M_n(\mathbb{K})$ for the underlying $\mathbb{K}$-vector space of $\mathcal{A}$. For $A\in\mathcal{A}$, let $\mathrm{L}_A(X):=A\circ X$ be the multiplication operator. We consider the Jordan multiplication semigroup generated by all multiplication operators, \[ \mathrm{JMS}(\mathcal{A}):=\langle \mathrm{L}_A:A\in\mathcal{A}\rangle\subseteq \mathrm{End}_{\mathbb{K}}(\mathcal{A}_{\mathrm v}). \] We prove that $\mathrm{JMS}(\mathcal{A})=\mathrm{End}_{\mathbb{K}}(\mathcal{A}_{\mathrm v})$. Equivalently, every $\mathbb{K}$-linear endomorphism of $\mathcal{A}_{\mathrm v}$ is a composition of multiplication operators. The proof is primarily linear-algebraic. The main step is to show that $\mathrm{SL}(\mathcal{A}_{\mathrm v})\subseteq \mathrm{JMS}(\mathcal{A})$ by constructing elementary transvections inside the semigroup. We then prove determinant surjectivity on the unit group of $\mathrm{JMS}(\mathcal{A})$ and combine it with the existence of a singular element of rank $n^2-1$ to obtain the full endomorphism semigroup. In the finite-field case, the determinant-surjectivity step is established via Jacobi-sum estimates.

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arXiv2025

Classification of Jordan multiplicative maps on matrix algebras

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $ϕ: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative, i.e.\ it satisfies the functional equation $$ ϕ(XY+YX)=ϕ(X)ϕ(Y)+ϕ(Y)ϕ(X), \quad \text{for all } X,Y \in M_n(\mathbb{F}) $$ if and only if one of the following holds: either $ϕ$ is constant, equal to $P/2$ for some idempotent $P \in M_n(\mathbb{F})$, or there exists an invertible matrix $T \in M_n(\mathbb{F})$ and a ring monomorphism $ω: \mathbb{F} \to \mathbb{F}$ such that $$ ϕ(X)=Tω(X)T^{-1} \quad \text{ or } \quad ϕ(X)=Tω(X)^tT^{-1}, \quad \text{for all } X \in M_n(\mathbb{F}), $$ where $ω(X)$ denotes the matrix obtained by applying $ω$ entrywise to $X$. In particular, any Jordan multiplicative map $ϕ: M_n(\mathbb{F}) \to M_n(\mathbb{F})$ with $ϕ(0)=0$ is automatically additive. The analogous characterization fails when $\mathbb{F}$ has characteristic $2$.

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arXiv2025

Multiplicative and Jordan multiplicative maps on structural matrix algebras

Let $M_n$ denote the algebra of $n \times n$ complex matrices and let $\mathcal{A}\subseteq M_n$ be an arbitrary structural matrix algebra, i.e. a subalgebra of $M_n$ that contains all diagonal matrices. We consider injective maps $ϕ: \mathcal{A}\to M_n$ that satisfy the condition $$ ϕ(X \bullet Y) = ϕ(X) \bullet ϕ(Y), \quad \text{for all } X,Y \in \mathcal{A}, $$ where $\bullet$ is either the standard matrix multiplication $(X,Y)\mapsto XY$, the Jordan product $(X,Y) \mapsto XY+YX$, or the normalized Jordan product $(X,Y) \mapsto \frac{1}{2}(XY+YX)$. We show that all such maps $ϕ$ are automatically additive if and only if $\mathcal{A}$ does not contain a central rank-one idempotent. Moreover, in this case, we fully characterize the form of these maps.

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arXiv2025

Continuous spectrum-shrinking maps between finite-dimensional algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital finite-dimensional complex algebras, each equipped with the unique Hausdorff vector topology. Denote by $\mathrm{Max}(\mathcal{A})=\{\mathcal{M}_1, \ldots, \mathcal{M}_p\}$ and $\mathrm{Max}(\mathcal{B})=\{\mathcal{N}_1, \ldots, \mathcal{N}_q\}$ the sets of all maximal ideals of $\mathcal{A}$ and $\mathcal{B}$, respectively. For each $1 \leq i \leq p$ and $1 \leq j \leq q$ define the quantities $$ k_i:=\sqrt{\dim(\mathcal{A}/\mathcal{M}_i)} \quad \text{ and } \quad m_j:=\sqrt{\dim(\mathcal{B}/\mathcal{N}_j)}, $$ which are positive integers by Wedderburn's structure theorem. We show that there exists a continuous spectrum-shrinking map $ϕ: \mathcal{A} \to \mathcal{B}$ (i.e. $\mathrm{sp}(ϕ(a))\subseteq \mathrm{sp}(a)$ for all $a \in \mathcal{A}$) if and only if for each $1\leq j \leq q$ the linear Diophantine equation $$ k_1x_{1}^{j} + \cdots + k_px_{p}^j = m_j $$ has a non-negative integer solution $(x_{1}^j,\ldots,x_{p}^j)\in \mathbb{N}_{0}^p$. In a similar manner we also characterize the existence of continuous spectrum-preserving maps $ϕ: \mathcal{A} \to \mathcal{B}$ (i.e. $\mathrm{sp}(ϕ(a))= \mathrm{sp}(a)$ for all $a \in \mathcal{A}$). Finally, we analyze conditions under which all continuous spectrum-shrinking maps $ϕ: \mathcal{A} \to \mathcal{B}$ are automatically spectrum-preserving.

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arXiv2025

A variant of Šemrl's preserver theorem for singular matrices

For positive integers $1 \leq k \leq n$ let $M_n$ be the algebra of all $n \times n$ complex matrices and $M_n^{\le k}$ its subset consisting of all matrices of rank at most $k$. We first show that whenever $k>\frac{n}{2}$, any continuous spectrum-shrinking map $ϕ: M_n^{\le k} \to M_n$ (i.e. $\mathrm{sp}(ϕ(X)) \subseteq \mathrm{sp}(X)$ for all $X \in M_n^{\le k}$) either preserves characteristic polynomials or takes only nilpotent values. Moreover, for any $k$ there exists a real analytic embedding of $M_n^{\le k}$ into the space of $n\times n$ nilpotent matrices for all sufficiently large $n$. This phenomenon cannot occur when $ϕ$ is injective and either $k > n - \sqrt{n}$ or the image of $ϕ$ is contained in $M_n^{\le k}$. We then establish a main result of the paper -- a variant of Šemrl's preserver theorem for $M_n^{\le k}$: if $n \geq 3$, any injective continuous map $ϕ:M_n^{\le k} \to M_n^{\le k}$ that preserves commutativity and shrinks spectrum is of the form $ϕ(\cdot)=T(\cdot)T^{-1}$ or $ϕ(\cdot)=T(\cdot)^tT^{-1}$, for some invertible matrix $T\in M_n$. Moreover, when $k=n-1$, which corresponds to the set of singular $n\times n$ matrices, this result extends to maps $ϕ$ which take values in $M_n$. Finally, we discuss the indispensability of assumptions in our main result.

math.SP↗

arXiv2025

Continuous spectrum-shrinking maps and applications to preserver problems

For a positive integer $n$ let $\mathcal{X}_n$ be either the algebra $M_n$ of $n \times n$ complex matrices, the set $N_n$ of all $n \times n$ normal matrices, or any of the matrix Lie groups $\mathrm{GL}(n)$, $\mathrm{SL}(n)$ and $\mathrm{U}(n)$. We first give a short and elementary argument that for two positive integers $m$ and $n$ there exists a continuous spectrum-shrinking map $ϕ: \mathcal{X}_n \to M_m$ (i.e.\ $\mathrm{sp}(ϕ(X))\subseteq \mathrm{sp}(X)$ for all $X \in \mathcal{X}_n$) if and only if $n$ divides $m$. Moreover, in that case we have the equality of characteristic polynomials $k_{ϕ(X)}(\cdot) = k_{X}(\cdot)^\frac{m}{n}$ for all $X \in \mathcal{X}_n$, which in particular shows that $ϕ$ preserves spectra. Using this we show that whenever $n \geq 3$, any continuous commutativity preserving and spectrum-shrinking map $ϕ: \mathcal{X}_n \to M_n$ is of the form $ϕ(\cdot)=T(\cdot)T^{-1}$ or $ϕ(\cdot)=T(\cdot)^tT^{-1}$, for some $T\in \mathrm{GL}(n)$. The analogous results fail for the special unitary group $\mathrm{SU}(n)$ but hold for the spaces of semisimple elements in either $\mathrm{GL}(n)$ or $\mathrm{SL}(n)$. As a consequence, we also recover (a strengthened version of) Šemrl's influential characterization of Jordan automorphisms of $M_n$ via preserving properties.

math.SP↗

arXiv2025

An extension of Petek-Šemrl preserver theorems for Jordan embeddings of structural matrix algebras

Let $M_n$ be the algebra of $n \times n$ complex matrices and $\mathcal{T}_n \subseteq M_n$ the corresponding upper-triangular subalgebra. In their influential work, Petek and Šemrl characterize Jordan automorphisms of $M_n$ and $\mathcal{T}_n$, when $n \geq 3$, as (injective in the case of $\mathcal{T}_n$) continuous commutativity and spectrum preserving maps $ϕ: M_n \to M_n$ and $ϕ: \mathcal{T}_n \to \mathcal{T}_n$. Recently, in a joint work with Petek, the authors extended this characterization to the maps $ϕ: \mathcal{A} \to M_n$, where $\mathcal{A}$ is an arbitrary subalgebra of $M_n$ that contains $\mathcal{T}_n$. In particular, any such map $ϕ$ is a Jordan embedding and hence of the form $ϕ(X)=TXT^{-1}$ or $ϕ(X)=TX^tT^{-1}$, for some invertible matrix $T\in M_n$. In this paper we further extend the aforementioned results in the context of structural matrix algebras (SMAs), i.e. subalgebras $\mathcal{A}$ of $M_n$ that contain all diagonal matrices. More precisely, we provide both a necessary and sufficient condition for an SMA $\mathcal{A}\subseteq M_n$ such that any injective continuous commutativity and spectrum preserving map $ϕ: \mathcal{A} \to M_n$ is necessarily a Jordan embedding. In contrast to the previous cases, such maps $ϕ$ no longer need to be multiplicative/antimultiplicative, nor rank-one preservers.

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arXiv2024

Jordan embeddings and linear rank preservers of structural matrix algebras

We consider subalgebras $\mathcal{A}$ of the algebra $M_n$ of $n \times n$ complex matrices that contain all diagonal matrices, known in the literature as the structural matrix algebras (SMAs). Let $\mathcal{A} \subseteq M_n$ be an arbitrary SMA. We first show that any commuting family of diagonalizable matrices in $\mathcal{A}$ can be intrinsically simultaneously diagonalized (i.e. the corresponding similarity can be chosen from $\mathcal{A}$). Using this, we then characterize when one SMA Jordan-embeds into another and in that case we describe the form of such Jordan embeddings. As a consequence, we obtain a description of Jordan automorphisms of SMAs, generalizing Coelho's result on their algebra automorphisms. Next, motivated by the results of Marcus-Moyls and Molnar-Šemrl, connecting the linear rank-one preservers with Jordan embeddings $M_n \to M_n$ and $\mathcal{T}_n \to M_n$ (where $\mathcal{T}_n$ is the algebra of $n \times n$ upper-triangular matrices) respectively, we show that any linear unital rank-one preserver $\mathcal{A} \to M_n$ is necessarily a Jordan embedding. As the converse fails in general, we also provide a necessary and sufficient condition for when it does hold true. Finally, we obtain a complete description of linear rank preservers $\mathcal{A} \to M_n$, as maps of the form $X\mapsto S\left(PX + (I-P)X^t\right)T$, for some invertible matrices $S,T \in M_n$ and a central idempotent $P\in\mathcal{A}$.

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arXiv2024

Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties

Let $M_n$ be the algebra of $n \times n$ complex matrices. We consider arbitrary subalgebras $\mathcal{A}$ of $M_n$ which contain the algebra of all upper-triangular matrices (i.e.\ block upper-triangular subalgebras), and their Jordan embeddings. We first describe Jordan embeddings $ϕ: \mathcal{A} \to M_n$ as maps of the form $ϕ(X)=TXT^{-1}$ or $ϕ(X)=TX^tT^{-1}$, where $T\in M_n$ is an invertible matrix, and then we obtain a simple criteria of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings $ϕ: \mathcal{A} \to M_n$ (when $n\geq 3$) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless $\mathcal{A} = M_n$ when injectivity is superfluous).

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arXiv2022

A Variant of the Kaplansky Problem for Maps on Positive Matrices

We prove that all injective maps on positive complex matrices which preserve order and shrink spectrum are implemented by unitary or antiunitary conjugations. We show by counterexamples that all assumptions are indispensable. The result easily generalizes to maps on hermitian matrices.

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