arXiv subjects

Jorge J. Garcés

Publications and source records attributed to Jorge J. Garcés.

At least 19 recordsLinked to original sources

arXiv2026

Applications of compact multipliers to algebrability of $(\ell_{\infty}\setminus c_0)\cup\{0\}$ and $(B(\ell_2(\mathbb{N}))\setminus K(\ell_2(\mathbb{N}))\cup \{ 0\}.$

We introduce a generator-counting refinement of algebrability for abelian $C^*$-algebras and related Banach algebras. Given an abelian $C^*$-algebra $A$, we define $(C^*)$-genalgebrability in terms of the minimal possible cardinality of a generating set, encoded by the invariants $gen_{C^*}(A)$ and $gen(A)$. Using compact multipliers and the ideal $K(A)$ of compact elements, we develop embedding results into $\ell_\infty$ whose ranges avoid $c_0$ (except for the zero vector), and we obtain a universal $(\ell_\infty\setminus c_0)$-embeddability phenomenon under the assumption $K(A)=\{0\}$. As an application, we construct a $^*$-isomorphic copy of $\ell_\infty$ inside $(\ell_\infty\setminus c_0)\cup\{0\}$ and transfer the results to Calkin-type settings such as $(B(\ell_2)\setminus K(\ell_2))\cup\{0\}$ and their unitizations. We also establish a generator-counting theorem for abelian $C^*$-algebras: $gen_{C^*}(A)$ equals the smallest cardinal $n$ for which the spectrum $Δ(A)$ embeds into $\mathbb R^n$, and we derive topological formulas for $gen_{C^*}(A)$ in the non-finitely generated case. Finally, we provide a complete classification of the pairs $(d,κ)$ for which $(\ell_\infty\setminus c_0)\cup\{0\}$ is $(d,κ)$-$(C^*)$-genalgebrable, and we discuss the connection with classical algebrability.

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arXiv2025

Bilinear maps having Jordan product property

We study symmetric continuous bilinear maps $V$ on a C$^*$-algebra $A$ that have the Jordan product property at a fixed element $z\in A$. We show that, whenever $A$ is a finite direct sum or a $c_0$-sum of infinite simple von Neumann algebras, such a map $V$ has the square-zero property. Then, it is proved that $V(a,b)=T(a\circ b)$ for some bounded linear map $T$ on $A$. As a consequence, Jordan homomorphisms and derivations at $z\in A$ are characterized.

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arXiv2024

Maps preserving the truncation of triple products on Cartan factors

Let $\{C_i\}_{i\in Γ_1},$ and $\{D_j\}_{j\in Γ_2},$ be two families of Cartan factors such that all of them have dimension at least $2$, and consider the atomic JBW$^*$-triples $A=\bigoplus\limits_{i\in Γ_1}^{\ell_{\infty}} C_i$ and $B=\bigoplus\limits_{j\in Γ_2}^{\ell_{\infty}} D_j$. Let $Δ:A \to B$ be a {\rm(}non-necessarily linear nor continuous{\rm)} bijection preserving the truncation of triple products in both directions, that is, $$\begin{aligned} \boxed{a \mbox{ is a truncation of } \{b,c,b\}} \Leftrightarrow \boxed{Δ(a) \mbox{ is a truncation of } \{Δ(b),Δ(c),Δ(b)\}} \end{aligned}$$ Assume additionally that the restriction of $Δ$ to each rank-one Cartan factor in $A$, if any, is a continuous mapping. Then we show that $Δ$ is an isometric real linear triple isomorphism. We also study some general properties of bijections preserving the truncation of triple products in both directions between general JB$^*$-triples.

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arXiv2024

Invertibility preservers of finitary incidence algebras

Let $X$ be an arbitrary poset and $K$ an arbitrary field. We describe linear unital invertibility preservers of the finitary incidence algebra $FI(X,K)$ in terms of certain maps of the power set algebra $\mathcal{P}(X)$ and linear maps $FI(X,K)\to J(FI(X,K))$.

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arXiv2024

Complete orthogonality preservers between C$^*$-algebras

We introduce $n$-orthogonality (and completely orthogonality) preserving operators between C$^*$-algebras. Our main theorem states that every completely orthogonality preserving bounded linear mapping between C$^*$-algebras is a weighted TRO homomorphism. We also give several characterisations of TRO homomorphisms among triple homomorphisms.

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arXiv2024

A Kaplansky Theorem for JB*-triples

Let $T:E\rightarrow F$ be a non-necessarily continuous triple homomorphism from a (complex) JB$^*$-triple (respectively, a (real) J$^*$B-triple) to a normed Jordan triple. The following statements hold: (1) $T$ has closed range whenever $T$ is continuous (2) $T$ has closed range whenever $T$ is continuous This result generalises classical theorems of I. Kaplansky and S.B. Cleveland in the setting of C$^*$-algebras and of A. Bensebah and J.Pérez, L. Rico and A. Rodr'\iguez Palacios in the setting of JB$^*$-algebras.

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arXiv2024

Automatic continuity of biorthogonality preservers between weakly compact JB$^$-triples and atomic JBW$^$-triples

We prove that every biorthogonality preserving linear surjection from a weakly compact JB$^*$triple containing no infinite dimensional rank-one summands onto another JB$^*$-triple is automatically continuous. We also show that every biorthogonality preserving linear surjection between atomic JBW$^*$triples containing no infinite dimensional rank-one summands is automatically continuous. Consequently, two atomic JBW$^*$-triples containing no rank-one summands are isomorphic if, and only if, there exists a (non necessarily continuous) biorthogonality preserving linear surjection between them.

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arXiv2024

Generalised triple homomorphisms and Derivations

We introduce generalised triple homomorphism between Jordan Banach triple systems as a concept which extends the notion of generalised homomorphism between Banach algebras given by Jarosz and Johnson in 1985 and 1987, respectively. We prove that every generalised triple homomorphism between JB$^*$-triples is automatically continuous. When particularised to C$^*$-algebras, we rediscover one of the main theorems established by Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module, proving that every generalised triple derivation from a JB$^*$-triple $E$ into $E^*$ is automatically continuous.

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arXiv2023

Bilinear maps on C$^*$-algebras that have product property at a compact element

We study bounded bilinear maps on a C$^*$-algebra $A$ having product property at $c\in A$. This leads us to the question of when a C$^*$-algebra is determined by products at $c.$ In the first part of our paper, we investigate this question for compact C$^*$-algebras, and in the second part, we deal with von Neumann algebras having non-trivial atomic part. Our results are applicable to descriptions of homomorphism-like and derivation-like maps at a fixed point on such algebras.

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arXiv2022

On the strict topology of the multipliers of a JB$^*$-algebra

We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonality preserving operators between JB$^*$-algebras admit J-strict continuous extensions to the corresponding type of operators between the multipliers algebras. We characterize J-strict continuous functionals on the multipliers algebra of a JB$^*$-algebra $\mathfrak{A}$, and we establish that the dual of $M(\mathfrak{A})$ with respect to the J-strict topology is isometrically isomorphic to $\mathfrak{A}^*$. We also present a first applications of the J-strict topology of the multipliers algebra, by showing that under the extra hypothesis that $\mathfrak{A}$ and $\mathfrak{B}$ are $σ$-unital JB$^*$-algebras, every surjective Jordan $^*$-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $\mathfrak{A}$ onto $\mathfrak{B}$ admits an extension to a surjective J-strict continuous Jordan $^*$-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $M(\mathfrak{A})$ onto $M(\mathfrak{B})$.

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arXiv2022

Linear maps preserving products equal to primitive idempotents of an incidence algebra

Let $A$, $B$ be algebras and $a\in A$, $b\in B$ a fixed pair of elements. We say that a map $φ:A\to B$ preserves products equal to $a$ and $b$ if for all $a_1,a_2\in A$ the equality $a_1a_2=a$ implies $φ(a_1)φ(a_2)=b$. In this paper we study bijective linear maps $φ:I(X,F)\to I(X,F)$ preserving products equal to primitive idempotents of $I(X,F)$, where $I(X,F)$ is the incidence algebra of a finite connected poset $X$ over a field $F$. We fully characterize the situation, when such a map $φ$ exists, and whenever it does, $φ$ is either an automorphism of $I(X,F)$ or the negative of an automorphism of $I(X,F)$.

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arXiv2022

A projection--less approach to Rickart Jordan structures

The main goal of this paper is to introduce and explore an appropriate notion of weakly Rickart JB$^*$-triples. We introduce weakly order Rickart JB$^*$-triples, and we show that a C$^*$-algebra $A$ is a weakly (order) Rickart JB$^*$-triple precisely when it is a weakly Rickart C$^*$-algebra. We also prove that the Peirce-2 subspace associated with a tripotent in a weakly order Rickart JB$^*$-triple is a Rickart JB$^*$-algebra in the sense of Ayupov and Arzikulov. By extending a classical property of Rickart C$^*$-algebras, we prove that every weakly order Rickart JB$^*$-triple is generated by its tripotents.

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arXiv2021

Potent preservers of incidence algebras

Let $X$ be a finite connected poset, $F$ a field and $I(X,F)$ the incidence algebra of $X$ over $F$. We describe the bijective linear idempotent preservers $φ:I(X,F)\to I(X,F)$. Namely, we prove that, whenever $\mathrm{char}(F)\ne 2$, $φ$ is either an automorphism or an anti-automorphism of $I(X,F)$. If $\mathrm{char}(F)=2$ and $|F|>2$, then $φ$ is a (in general, non-proper) Lie automorphism of $I(X,F)$. Finally, if $F=\mathbb{Z}_2$, then $φ$ is the composition of a bijective shift map and a Lie automorphism of $I(X,F)$. Under certain restrictions on the characteristic of $F$ we also obtain descriptions of the bijective linear maps which preserve tripotents and, more generally, $k$-potents of $I(X,F)$ for $k\ge 3$.

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arXiv2020

One-parameter groups of orthogonality preservers on C$^*$-algebras

We establish a more precise description of those surjective or bijective continuous linear operators preserving orthogonality between C$^*$-algebras. The new description is applied to determine all uniformly continuous one-parameter semigroups of orthogonality preserving operators on an arbitrary C$^*$-algebra. We prove that given a family $\{T_t: t\in \mathbb{R}_0^{+}\}$ of orthogonality preserving bounded linear bijections on a general C$^*$-algebra $A$ with $T_0=Id$, if for each $t\geq 0,$ we set $h_t = T_t^{**} (1)$ and we write $r_t$ for the range partial isometry of $h_t$ in $A^{**},$ and $S_t$ stands for the triple isomorphism on $A$ associated with $T_t$ satisfying $h_t^* S_t(x)$ $= S_t(x^*)^* h_t$, $h_t S_t(x^*)^* =$ $ S_t(x) h_t^*$, $h_t r_t^* S_t(x) =$ $S_t(x) r_t^* h_t$, and $T_t(x) = h_t r_t^* S_t(x) = S_t(x) r_t^* h_t, \hbox{ for all } x\in A,$ the following statements are equivalent: $(a)$ $\{T_t: t\in \mathbb{R}_0^{+}\}$ is a uniformly continuous one-parameter semigroup of orthogonality preserving operators on $A$; $(b)$ $\{S_t: t\in \mathbb{R}_0^{+}\}$ is a uniformly continuous one-parameter semigroup of surjective linear isometries (i.e. triple isomorphisms) on $A$ (and hence there exists a triple derivation $δ$ on $A$ such that $S_t = e^{t δ}$ for all $t\in \mathbb{R}$), the mapping $t\mapsto h_t $ is continuous at zero, and the identity $ h_{t+s} = h_t r_t^* S_t^{**} (h_s),$ holds for all $s,t\in \mathbb{R}.$

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arXiv2020

One-parameter groups of orthogonality preservers on JB$^*$-algebras

In a first objective we improve our understanding about surjective and bijective bounded linear operators preserving orthogonality from a JB$^*$-algebra $\mathcal{A}$ into a JB$^*$-triple $E$. Among many other conclusions, it is shown that a bounded linear bijection $T: \mathcal{A}\to E$ is orthogonality preserving if, and only if, it is biorthogonality preserving if, and only if, it preserves zero-triple-products in both directions (i.e., $\{a,b,c\}=0 \Leftrightarrow \{T(a),T(b),T(c)\}=0$). In the second main result we establish a complete characterization of all one-parameter groups of orthogonality preserving operators on a JB$^*$-algebra.

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arXiv2017

Tingley's problem for spaces of trace class operators

We prove that every surjective isometry between the unit spheres of two trace class spaces admits a unique extension to a surjective complex linear or conjugate linear isometry between the spaces. This provides a positive solution to Tingley's problem in a new class of operator algebras.

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arXiv2014

A Kowalski-Słodkowski theorem for 2-local $^*$-homomorphisms on von Neumann algebras

It is established that every (not necessarily linear) 2-local $^*$-homomorphism from a von Neumann algebra into a C$^*$-algebra is linear and a $^*$-homomorphism. In the setting of (not necessarily linear) 2-local $^*$-homomorphism from a compact C$^*$-algebra we prove that the same conclusion remains valid. We also prove that every 2-local Jordan $^*$-homomorphism from a JBW$^*$-algebra into a JB$^*$-algebra is linear and a Jordan $^*$-homomorphism.

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arXiv2013

Orthogonally additive, orthogonality preserving, holomorphic mappings between C*-algebras

We study holomorphic maps between C$^*$-algebras $A$ and $B$. When $f:B_A (0,\varrho) \longrightarrow B$ is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball $U=B_{A}(0,δ)$ and we assume that $f$ is orthogonality preserving on $A_{sa}\cap U$, orthogonally additive on $U$ and $f(U)$ contains an invertible element in $B$, then there exist a sequence $(h_n)$ in $B^{**}$ and Jordan $^*$-homomorphisms $Θ, \widetildeΘ : M(A) \to B^{**}$ such that $$ f(x) = \sum_{n=1}^\infty h_n \widetildeΘ (a^n)= \sum_{n=1}^\infty Θ (a^n) h_n,$$ uniformly in $a\in U$. When $B$ is abelian the hypothesis of $B$ being unital and $f(U)\cap \hbox{inv} (B) \neq \emptyset$ can be relaxed to get the same statement.

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