arXiv subjects

Antonio M. Peralta

Publications and source records attributed to Antonio M. Peralta.

At least 55 records · Page 3Linked to original sources

arXiv2021

On optimality of constants in the Little Grothendieck Theorem

We explore the optimality of the constants making valid the recently established Little Grothendieck inequality for JB$^*$-triples and JB$^*$-algebras. In our main result we prove that for each bounded linear operator $T$ from a JB$^*$-algebra $B$ into a complex Hilbert space $H$ and $\varepsilon>0$, there is a norm-one functional $φ\in B^*$ such that $$\|Tx\|\le(\sqrt{2}+\varepsilon)\|T\|\|x\|_φ\quad\mbox{ for }x\in B.$$ The constant appearing in this theorem improves the best value known up to date (even for C$^*$-algebras). We also present an easy example witnessing that the constant cannot be strictly smaller than $\sqrt2$, hence our main theorem is `asymptotically optimal'. For type I JBW$^*$-algebras we establish a canonical decomposition of normal functionals which may be used to prove the main result in this special case and also seems to be of an independent interest. As a tool we prove a measurable version of the Schmidt representation of compact operators on a Hilbert space.

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arXiv2021

Surjective isometries between unitary sets of unital JB$^*$-algebras

This paper is, in a first stage, devoted to establish a topological--algebraic characterization of the principal component, $\mathcal{U}^0 (M)$, of the set of unitary elements, $\mathcal{U} (M)$, in a unital JB$^*$-algebra $M$. We arrive to the conclusion that, as in the case of unital C$^*$-algebras, $$\begin{aligned}\mathcal{U}^0(M) &= M^{-1}_{\textbf{1}}\cap\mathcal{U} (M) =\left\lbrace U_{e^{i h_n}}\cdots U_{e^{i h_1}}(\textbf{1}) \colon \begin{array}{c} n\in \mathbb{N}, \ h_j\in M_{sa} \forall\ 1\leq j \leq n \end{array} \right\rbrace \end{aligned}$$ is analytically arcwise connected. Our second goal is to provide a complete description of the surjective isometries between the principal components of two unital JB$^*$-algebras $M$ and $N$. Contrary to the case of unital C$^*$-algebras, we shall deduce the existence of connected components in $\mathcal{U} (M)$ which are not isometric as metric spaces. We shall also establish necessary and sufficient conditions to guarantee that a surjective isometry $Δ: \mathcal{U}(M)\to \mathcal{U} (N)$ admits an extension to a surjective linear isometry between $M$ and $N$, a conclusion which is not always true. Among the consequences it is proved that $M$ and $N$ are Jordan $^*$-isomorphic if, and only if, their principal components are isometric as metric spaces if, and only if, there exists a surjective isometry $Δ: \mathcal{U}(M)\to \mathcal{U}(N)$ mapping the unit of $M$ to an element in $\mathcal{U}^0(N)$. These results provide an extension to the setting of unital JB$^*$-algebras of the results obtained by O. Hatori for unital C$^*$-algebras.

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arXiv2021

Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Let $M$ and $N$ be unital Jordan-Banach algebras, and let $M^{-1}$ and $N^{-1}$ denote the sets of invertible elements in $M$ and $N$, respectively. Suppose that $\mathfrak{M}\subseteq M^{-1}$ and $\mathfrak{N}\subseteq N^{-1}$ are clopen subsets of $M^{-1}$ and $N^{-1}$, respectively, which are closed for powers, inverses and products of the form $U_{a} (b)$. In this paper we prove that for each surjective isometry $Δ: \mathfrak{M}\to \mathfrak{N}$ there exists a surjective real-linear isometry $T_0: M\to N$ and an element $u_0$ in the McCrimmon radical of $N$ such that $Δ(a) = T_0(a) +u_0$ for all $a\in \mathfrak{M}$.\smallskip Assuming that $M$ and $N$ are unital JB$^*$-algebras we establish that for each surjective isometry $Δ: \mathfrak{M}\to \mathfrak{N}$ the element $Δ(\textbf{1}) =u$ is a unitary element in $N$ and there exist a central projection $p\in M$ and a complex-linear Jordan $^*$-isomorphism $J$ from $M$ onto the $u^*$-homotope $N_{u^*}$ such that $$Δ(a) = J(p\circ a) + J ((\textbf{1}-p) \circ a^*),$$ for all $a\in \mathfrak{M}$. Under the additional hypothesis that there is a unitary element $ω_0$ in $N$ satisfying $U_{ω_0} (Δ(\textbf{1})) = \textbf{1}$, we show the existence of a central projection $p\in M$ and a complex-linear Jordan $^*$-isomorphism $Φ$ from $M$ onto $N$ such that $$Δ(a) = U_{w_0^{*}} \left(Φ(p\circ a) + Φ((\textbf{1}-p) \circ a^*)\right),$$ for all $a\in \mathfrak{M}$.

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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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arXiv2020

A linear preserver problem on maps which are triple derivable at orthogonal pairs

A linear mapping $T$ on a JB$^*$-triple is called triple derivable at orthogonal pairs if for every $a,b,c\in E$ with $a\perp b$ we have $$0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}.$$ We prove that for each bounded linear mapping $T$ on a JB$^*$-algebra $A$ the following assertions are equivalent: $(a)$ $T$ is triple derivable at zero; $(b)$ $T$ is triple derivable at orthogonal elements; $(c)$ There exists a Jordan $^*$-derivation $D:A\to A^{**}$, a central element $ξ\in A^{**}_{sa},$ and an anti-symmetric element $η$ in the multiplier algebra of $A$, such that $$ T(a) = D(a) + ξ\circ a + η\circ a, \hbox{ for all } a\in A;$$ $(d)$ There exist a triple derivation $δ: A\to A^{**}$ and a symmetric element $S$ in the centroid of $A^{**}$ such that $T= δ+S$. The result is new even in the case of C$^*$-algebras. We next establish a new characterization of those linear maps on a JBW$^*$-triple which are triple derivations in terms of a good local behavior on Peirce 2-subspaces. We also prove that assuming some extra conditions on a JBW$^*$-triple $M$, the following statements are equivalent for each bounded linear mapping $T$ on $M$: $(a)$ $T$ is triple derivable at orthogonal pairs; $(b)$ There exists a triple derivation $δ: M\to M$ and an operator $S$ in the centroid of $M$ such that $T = δ+ S$. \end{enumerate}

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arXiv2020

Grothendieck's inequalities for JB$^*$-triples: Proof of the Barton-Friedman conjecture

We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $ψ\in E^*$ satisfying $$\|T(x)\| \leq K \, \|T\| \, \|x\|_ψ,$$ for all $x\in E$. Applying this result we show that, given $G > 8 (1+2\sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product of two JB$^*$-triples $E$ and $B$, there exist norm-one functionals $φ\in E^{*}$ and $ψ\in B^{*}$ satisfying $$|V(x,y)| \leq G \ \|V\| \, \|x\|_φ \, \|y\|_ψ$$ for all $(x,y)\in E \times B$. These results prove a conjecture pursued during almost twenty years.

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arXiv2020

On the extension of surjective isometries whose domain is the unit sphere of a space of compact operators

We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur--Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur--Ulam property.

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arXiv2020

Can one identify two unital JB$^*$-algebras by the metric spaces determined by their sets of unitaries?

Let $M$ and $N$ be two unital JB$^*$-algebras and let $\mathcal{U} (M)$ and $\mathcal{U} (N)$ denote the sets of all unitaries in $M$ and $N$, respectively. We prove that the following statements are equivalent: $(a)$ $M$ and $N$ are isometrically isomorphic as (complex) Banach spaces; $(b)$ $M$ and $N$ are isometrically isomorphic as real Banach spaces; $(c)$ There exists a surjective isometry $Δ: \mathcal{U}(M)\to \mathcal{U}(N).$ We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry $Δ:\mathcal{U} (M) \to \mathcal{U} (N)$ we can find a surjective real linear isometry $Ψ:M\to N$ which coincides with $Δ$ on the subset $e^{i M_{sa}}$. If we assume that $M$ and $N$ are JBW$^*$-algebras, then every surjective isometry $Δ:\mathcal{U} (M) \to \mathcal{U} (N)$ admits a (unique) extension to a surjective real linear isometry from $M$ onto $N$. This is an extension of the Hatori--Moln{á}r theorem to the setting of JB$^*$-algebras.

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arXiv2020

Finite tripotents and finite JBW$^*$-triples

We study two natural preorders on the set of tripotents in a JB$^*$-triple defined in terms of their Peirce decomposition and weaker than the standard partial order. We further introduce and investigate the notion of finiteness for tripotents in JBW$^*$-triples which is a natural generalization of finiteness for projections in von Neumann algebras. We analyze the preorders in detail using the standard representation of JBW$^*$-triples. We also provide a refined version of this representation - in particular a decomposition of any JBW$^*$-triple into its finite and properly infinite parts. Since a JBW$^*$-algebra is finite if and only if the extreme points of its unit ball are just unitaries, our notion of finiteness differs from the concept of modularity widely used in Jordan structures so far. The exact relationship of these two notions is clarified in the last section.

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arXiv2020

Linear maps which are anti-derivable at zero

Let $T:A\to X$ be a bounded linear operator, where $A$ is a C$^*$-algebra, and $X$ denotes an essential Banach $A$-bimodule. We prove that the following statements are equivalent: $(a)$ $T$ is anti-derivable at zero (i.e. $ab =0$ in $A$ implies $T(b) a + b T(a)=0$); $(b)$ There exist an anti-derivation $d:A\to X^{**}$ and an element $ξ\in X^{**}$ satisfying $ξa = a ξ,$ $ξ[a,b]=0,$ $T(a b) = b T(a) + T(b) a - b ξa,$ and $T(a) = d(a) + ξa,$ for all $a,b\in A$. We also prove a similar equivalence when $X$ is replaced with $A^{**}$. This provides a complete characterization of those bounded linear maps from $A$ into $X$ or into $A^{**}$ which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are $^*$-anti-derivable at zero.

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arXiv2019

Measures of weak non-compactness in preduals of von Neumann algebras and JBW$^*$-triples

We prove, among other results, that three standard measures of weak non-compactness coincide in preduals of JBW$^*$-triples. This result is new even for preduals of von Neumann algebras. We further provide a characterization of JBW$^*$-triples with strongly WCG predual and describe the order of seminorms defining the strong$^*$ topology. As a byproduct we improve a characterization of weakly compact subsets of a JBW$^*$-triple predual, providing so a proof for a conjecture, open for almost eighteen years, on weakly compact operators from a JB$^*$-triple into a complex Banach space.

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arXiv2019

Metric characterisation of unitaries in JB$^*$-algebras

Let $M$ be a unital JB$^*$-algebra whose closed unit ball is denoted by $\mathcal{B}_M$. Let $\partial_e(\mathcal{B}_M)$ denote the set of all extreme points of $\mathcal{B}_M$. We prove that an element $u\in \partial_e(\mathcal{B}_M)$ is a unitary if and only if the set $$\mathcal{M}_{u} = \{e\in \partial_e(\mathcal{B}_M) : \|u\pm e\|\leq \sqrt{2} \}$$ contains an isolated point. This is a new geometric characterisation of unitaries in $M$ in terms of the set of extreme points of $\mathcal{B}_M$.

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arXiv2019

Extension of isometries from the unit sphere of a rank-2 Cartan factor

We prove that every surjective isometry from the unit sphere of a rank-2 Cartan factor $C$ onto the unit sphere of a real Banach space $Y$, admits an extension to a surjective real linear isometry from $C$ onto $Y$. The conclusion also covers the case in which $C$ is a spin factor. This result closes an open problem and, combined with the conclusion in a previous paper, allows us to establish that every JBW$^*$-triple $M$ satisfies the Mazur--Ulam property, that is, every surjective isometry from its unit sphere onto the unit sphere of a arbitrary real Banach space $Y$ admits an extension to a surjective real linear isometry from $M$ onto $Y$.

math.FA↗

arXiv2019

Aron-Berner extensions of triple maps with application to the bidual of Jordan Banach triple systems

By extending the notion of Arens regularity of bilinear mappings, we say that a bounded trilinear map on Banach spaces id Aron--Berner regular when all its six Aron-Berner extensions to the bidual spaces coincide. We give some results on the Aron-Berner regularity of certain trilinear maps. We then focus on the bidual, $E^{**},$ of a Jordan banach triple system $(E,π)$, and investigate those conditions under which $E^{**}$ is itself a Jordan Banach triple system under each of the Aron-Berner extensions of the triple product $π.$ We also compare these six triple products with those arising from certain ultrafilters based on the ultrapower formulation of the principle of local reflexivity. In particular, we examine the Aron--Berner triple products on the bidual of a JB$^*$-triple in relation with the so-called Dineen's theorem. Some illuminating examples are included and some questions are also left undecided.

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arXiv2019

On the Mazur--Ulam property for the space of Hilbert-space-valued continuous functions

Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all $H$-valued continuous functions on $K$, equipped with the supremum norm, satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space, every surjective isometry $Δ$ from the unit sphere of $C(K,H)$ onto the unit sphere of $Y$ admits a unique extension to a surjective real linear isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of $C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For this purpose we determine the facial structure of the closed unit ball of a real JB$^*$-triple and its dual space.

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arXiv2019

On the extension of isometries between the unit spheres of a JBW$^*$-triple and a Banach space

We prove that every JBW$^*$-triple $M$ with rank one or rank bigger than or equal to three satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$. We also show that the same conclusion holds if $M$ is not a JBW$^*$-triple factor, or more generally, if the atomic part of $M^{**}$ is not a rank two Cartan factor.

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arXiv2018

Absolutely compatible pairs in a von Neumann algebra

Let $a,b$ be elements in a unital C$^*$-algebra with $0\leq a,b\leq 1$. The element $a$ is absolutely compatible with $b$ if $$\vert a - b \vert + \vert 1 - a - b \vert = 1.$$ In this note we find some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra. These characterizations are applied to measure how close is a pair of absolute compatible positive elements in the closed unit ball from being orthogonal or commutative. In the case of 2 by 2 matrices the results offer a geometric interpretation in terms of an ellipsoid determined by one of the points. The conclusions for 2 by 2 matrices are also applied to describe absolutely compatible pairs of positive elements in the closed unit ball of $\mathbb{M}_n$.

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