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Antonio M. Peralta

Publications and source records attributed to Antonio M. Peralta.

At least 91 records · Page 5Linked to original sources
arXiv2016

Low rank compact operators and Tingley's problem

Let $E$ and $B$ be arbitrary weakly compact JB$^*$-triples whose unit spheres are denoted by $S(E)$ and $S(B)$, respectively. We prove that every surjective isometry $f: S(E) \to S(B)$ admits an extension to a surjective real linear isometry $T: E\to B$. This is a complete solution to Tingley's problem in the setting of weakly compact JB$^*$-triples. Among the consequences, we show that if $K(H,K)$ denotes the space of compact operators between arbitrary complex Hilbert spaces $H$ and $K$, then every surjective isometry $f: S(K(H,K)) \to S(K(H,K))$ admits an extension to a surjective real linear isometry $T: K(H,K)\to K(H,K)$.

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arXiv2016

A solution to Tingley's problem for isometries between the unit spheres of compact C$^*$-algebras and JB$^*$-triples

Let $f: S(E) \to S(B)$ be a surjective isometry between the unit spheres of two weakly compact JB$^*$-triples not containing direct summands of rank smaller than or equal to 3. Suppose $E$ has rank greater than or equal to 5. Applying techniques developed in JB$^*$-triple theory, we prove that $f$ admits an extension to a surjective real linear isometry $T: E\to B$. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C$^*$-algebras $A$ and $B$ (and in particular when $A=K(H)$ and $B=K(H')$) extends to a surjective real linear isometry from $A$ into $B$. These results provide new examples of infinite dimensional Banach spaces where Tingley's problem admits a positive answer.

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arXiv2016

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Let $A$ and $B$ be C$^*$-algebras. A linear map $T:A\to B$ is said to be a $^*$-homomorphism at an element $z\in A$ if $a b^*=z$ in $A$ implies $T (a b^*) =T (a) T (b)^* =T(z)$, and $ c^* d=z$ in $A$ gives $T (c^* d) =T (c)^* T (d) =T(z).$ Assuming that $A$ is unital, we prove that every linear map $T: A\to B$ which is a $^*$-homomorphism at the unit of $A$ is a Jordan $^*$-homomorphism. If $A$ is simple and infinite, then we establish that a linear map $T: A\to B$ is a $^*$-homomorphism if and only if $T$ is a $^*$-homomorphism at the unit of $A$. For a general unital C$^*$-algebra $A$ and a linear map $T:A\to B$, we prove that $T$ is a $^*$-homomorphism if, and only if, $T$ is a $^*$-homomorphism at $0$ and at $1$. Actually if $p$ is a non-zero projection in $A$, and $T$ is a $^*$-homomorphism at $p$ and at $1-p$, then we prove that $T$ is a Jordan $^*$-homomorphism. We also study bounded linear maps that are $^*$-homomorphisms at a unitary element in $A$.

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arXiv2016

Preduals of JBW$^*$-triples are 1-Plichko spaces

We prove that the predual, $M_*$, of a JBW$^*$-triple $M$ is a 1-Plichko space (i.e. it admits a countably 1-norming Markushevich basis or, equivalently, it has a commutative 1-projectional skeleton), and obtain a natural description of the $Σ$-subspace of $M$. This generalizes and improves similar results for von Neumann algebras and JBW$^*$-algebras. Consequently, dual spaces of JB$^*$-triples also are 1-Plichko spaces. We also show that $M_*$ is weakly Lindelöf determined if and only if $M$ is $σ$-finite if and only if $M_*$ is weakly compactly generated. Moreover, contrary to the proof for JBW$^*$-algebras, our proof dispenses with the use of elementary submodels theory.

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arXiv2016

Stability of derivations under weak-2-local continuous perturbations

Let $Ω$ be a compact Hausdorff space and let $A$ be a C$^*$-algebra. We prove that if every weak-2-local derivation on $A$ is a linear derivation and every derivation on $C(Ω,A)$ is inner, then every weak-2-local derivation $Δ:C(Ω,A)\to C(Ω,A)$ is a {\rm(}linear{\rm)} derivation. As a consequence we derive that, for every complex Hilbert space $H$, every weak-2-local derivation $Δ: C(Ω,B(H)) \to C(Ω,B(H))$ is a (linear) derivation. We actually show that the same conclusion remains true when $B(H)$ is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if $B$ denotes a compact C$^*$-algebra (in particular, when $B=K(H)$), then every weak-2-local derivation on $C(Ω,B)$ is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra $M$ and every compact Hausdorff space $Ω$, every 2-local derivation on $C(Ω,M)$ is a (linear) derivation.

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arXiv2016

Geometric implications of the M(r,s)-properties and the uniform Kadec-Klee property in JB*-triples

We explore new implications of the $M(r,s)$ and $M^*(r,s)$ properties for Banach spaces. We show that a Banach space $X$ satisfying property $M(1,s)$ for some $0<s\leq 1$, admitting a point $x_{0}$ in its unit sphere at which the relative weak and norm topologies agree, satisfies the generalized Gossez-Lami Dozo property. We establish sufficient conditions, in terms of the $(r, s)$-Lipschitz weak$^*$ Kadec-Klee property on a Banach space $X$, to guarantee that its dual space satisfies the UKK$^*$ property. We determine appropriate conditions to assure that a Banach space $X$ satisfies the $(r, s)$-Lipschitz weak$^*$ Kadec-Klee property. These results are applied to prove that every spin factor satisfies the UKK property, and consequently, the KKP and the UKK properties are equivalent for real and complex JB$^*$-triples.

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arXiv2015

Orthogonal forms and orthogonality preservers on real function algebras revisited

In 2014, we determine the precise form of a continuous orthogonal form on a commutative real C$^*$-algebra. We also describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between commutative unital real C$^*$-algebras. Among the consequences, we show that every orthogonality preserving linear bijection between commutative unital real C$^*$-algebras is continuous. In this note we revisit these results and their proofs with the idea of filling a gap in the arguments, and to extend the original conclusions.

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arXiv2015

Weak-2-local symmetric maps on C*-algebras

We introduce and study weak-2-local symmetric maps between C$^*$-algebras $A$ and $B$ as non necessarily linear nor continuous maps $Δ: A\to B$ such that for each $a,b\in A$ and $ϕ\in B^{*}$, there exists a symmetric linear map $T_{a,b,ϕ}: A\to B$, depending on $a$, $b$ and $ϕ$, satisfying $ϕΔ(a) = ϕT_{a,b,ϕ}(a)$ and $ϕΔ(b) = ϕT_{a,b,ϕ}(b)$. We prove that every weak-2-local symmetric map between C$^*$-algebras is a linear map. Among the consequences we show that every weak-2-local $^*$-derivation on a general C$^*$-algebra is a (linear) $^*$-derivation. We also establish a 2-local version of the Kowalski-Słodkowski theorem for general C$^*$-algebras by proving that every 2-local $^*$-homomorphism between C$^*$-algebras is a (linear) $^*$-homomorphism.

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arXiv2015

Inner ideals, compact tripotents and Čebyšëv subtriples of JB$^{*}$-triples and C$^*$-algebras

The aim of this note is to study Čebyšëv JB$^*$-subtriples of general JB$^*$-triples. It is established that if $F$ is a non-zero Čebyšëv JB$^*$-subtriple of a JB$^*$-triple $E$, then exactly one of the following statements holds:\begin{enumerate}\item $F$ is a rank one JBW$^*$-triple with dim$(F)\geq 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, $F$ may be a closed subspace of arbitrary dimension and $E$ may have arbitrary rank, \item $F= \mathbb{C} e$, where $e$ is a complete tripotent in $E$, \item $E$ and $F$ are rank two JBW$^*$-triples, but $F$ may have arbitrary dimension, \item $F$ has rank greater or equal than three and $E=F$. \end{enumerate}

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arXiv2015

Linear maps between C*-algebras preserving extreme points and strongly linear preservers

We study new classes of linear preservers between C$^*$-algebras and JB$^*$-triples. Let $E$ and $F$ be JB$^*$-triples with $\partial_{e} (E_1)$. We prove that every linear map $T:E\to F$ strongly preserving Brown-Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital C$^*$-algebras $A$ and $B,$ for each linear map $T$ strongly preserving Brown-Pedersen quasi-invertible elements, then there exists a Jordan $^*$-homomorphism $S: A\to B$ satisfying $T(x) = T(1) S(x)$, for every $x\in A$. We also study the connections between linear maps strongly preserving Brown-Pedersen quasi-invertibility and other clases of linear preservers between C$^*$-algebras like Bergmann-zero pairs preservers, Brown-Pedersen quasi-invertibility preservers and extreme points preservers.

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arXiv2015

Cebysev subspaces of JBW*-triples

We describe the one-dimensional Čebyšëv subspaces of a JBW$^*$-triple $M,$ by showing that for a non-zero element $x$ in $M$, $\mathbb{C}x$ is a Čebyšëv subspace of $M$ if, and only if, $x$ is a Brown-Pedersen quasi-invertible element in ${M}$. We study the Čebyšëv JBW$^*$-subtriples of a JBW$^*$-triple $M$. We prove that, for each non-zero Čebyšëv JBW$^*$-subtriple $N$ of $M$, then exactly one of the following statements holds: $(a)$ $N$ is a rank one JBW$^*$-triple with dim$(N)\geq 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, $N$ may be a closed subspace of arbitrary dimension and $M$ may have arbitrary rank; $(b)$ $N= \mathbb{C} e$, where $e$ is a complete tripotent in $M$; $(c)$ $N$ and $M$ have rank two, but $N$ may have arbitrary dimension; $(d)$ $N$ has rank greater or equal than three and $N=M$. We also provide new examples of Čebyšëv subspaces of classic Banach spaces in connection with ternary rings of operators.

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arXiv2015

Weak Banach-Saks property and Komlos theorem for preduals of JBW$^*$-triples

We show that the predual of a JBW$^*$-triple has the weak Banach-Saks property, that is, reflexive subspaces of a JBW$^*$-triple predual are super-reflexive. We also prove that JBW$^*$-triple preduals satisfy the Komlós property (which can be considered an abstract version of the weak law of large numbers). The results rely on two previous papers from which we infer the fact that, like in the classical case of $L^1$, a subspace of a JBW$^*$-triple predual contains $\ell_1$ as soon as it contains uniform copies of $\ell_1^n$.

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arXiv2015

Quadratic Conorm and extremally rich JB*-triples

We introduce and study the class of extremally rich JB$^*$-triples. We establish new results to determine the distance from an element $a$ in an extremally rich JB$^*$-triple $E$ to the set $\partial_{e} (E_1)$ of all extreme points of the closed unit ball of $E$. More concretely, we prove that $$\hbox{dist} (a,\partial_e (E_1)) =\max \{ 1, \|a\|-1\},$$ for every $a\in E$ which is not Brown-Pedersen quasi-invertible. As a consequence, we determine the form of the $λ$-function of Aron and Lohman on the open unit ball of an extremally rich JB$^*$-triple $E$, by showing that $λ(a)= \frac12$ for every non-BP quasi-invertible element $a$ in the open unit ball of $E$. We also prove that for an extremally rich JB$^*$-triple $E$, the quadratic connorm $γ^{q}(.)$ is continuous at a point $a\in E$ if, and only if, either $a$ is not von Neumann regular {\rm(}i.e. $γ^{q}(a)=0${\rm)} or $a$ is Brown-Pedersen quasi-invertible.

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arXiv2015

Weak 2-local derivations on $\mathbb{M}_n$

We introduce the notion of weak-2-local derivation (respectively, $^*$-derivation) on a C$^*$-algebra $A$ as a (non-necessarily linear) map $Δ: A\to A$ satisfying that for every $a,b\in A$ and $ϕ\in A^*$ there exists a derivation (respectively, a $^*$-derivation) $D_{a,b,ϕ}: A\to A$, depending on $a$, $b$ and $ϕ$, such that $ϕΔ(a) = ϕD_{a,b,ϕ} (a)$ and $ϕΔ(b) = ϕD_{a,b,ϕ} (b)$. We prove that every weak-2-local $^*$-derivation on $M_n$ is a linear derivation. We also show that the same conclusion remains true for weak-2-local $^*$-derivations on finite dimensional C$^*$-algebras.

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arXiv2014

Bilocal *-automorphisms of B(H) satisfying the 3-local property

We prove that, for a complex Hilbert space $H$ with dimension bigger or equal than three, every linear mapping $T: B(H)\to B(H)$ satisfying the 3-local property is a $^*$-monomorphism, that is, every linear mapping $T: B(H) \to B(H)$ satisfying that for every $a$ in $B(H)$ and every $ξ,η$ in $H$, there exists a $^*$-automorphism $π_{a,ξ,η}: B(H)\to B(H)$, depending on $a$, $ξ$, and $η$, such that $$T(a) (ξ) = π_{a,ξ,η} (a) (ξ), \hbox{ and } T(a) (η) = π_{a,ξ,η} (a) (η),$$ is a $^*$-monomorphism. This solves a question posed by L. Molnár in [\emph{Arch. Math.} \textbf{102}, 83-89 (2014)].

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