arXiv subjects

Antonio M. Peralta

Publications and source records attributed to Antonio M. Peralta.

At least 37 records · Page 2Linked to original sources

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.

math.OA↗

arXiv2024

$M$-ideals, yet again: the case of real JB$^*$-triples

We prove that a subspace of a real JBW$^*$-triple is an $M$-summand if and only if it is a weak$^*$-closed triple ideal. As a consequence, $M$-ideals of real JB$^*$-triples correspond to norm-closed triple ideals. As in the setting of complex JB$^*$-triples, a geometric property is characterized in purely algebraic terms. This is a newfangled treatment of the classical notion of $M$-ideal in the real setting by a fully new approach due to the unfeasibility of the known arguments in the setting of complex C$^*$-algebras and JB$^*$-triples. The results in this note also provide a full characterization of all $M$-ideals in real C$^*$-algebras, real JB$^*$-algebras and real TROs.

math.OA↗

arXiv2023

Metric invariants in Banach and Jordan--Banach algebras

In this note we collect some significant contributions on metric invariants for complex Banach algebras and Jordan--Banach algebras established during the last fifteen years. This note is mainly expository, but it also contains complete proofs and arguments, which in many cases are new or have been simplified. We have also included several new results. The common goal in the results is to seek for "natural" subsets, $\mathfrak{S}_{A},$ associated with each complex Banach or Jordan--Banach algebra $A$, sets which when equipped with a certain metric, $d_{A}$, enjoys the property that each surjective isometry from $(\mathfrak{S}_{A},d_A)$ to a similar set, $(\mathfrak{S}_{B},d_B),$ associated with another Banach or Jordan--Banach algebra $B$, extends to a surjective real-linear isometry from $A$ onto $B$. In case of a positive answer to this question, the problem of discussing whether in such a case the algebras $A$ and $B$ are in fact isomorphic or Jordan isomorphic is the subsequent question. The main results presented here will cover the cases in which the sets $(\mathfrak{S}_{A},d_A)$ and $(\mathfrak{S}_{B},d_B)$ are in one of the following situations: $(\checkmark)$ Subsets of the set of invertible elements in a unital complex Banach algebra or in a unital complex Jordan--Banach algebra with the metric induced by the norm. Specially in the cases of unital C$^*$- and JB$^*$-algebras. $(\checkmark)$ The sets of positive invertible elements in unital C$^*$- or JB$^*$-algebras with respect to the metric induced by the norm and with respect to the Thompson's metric. $(\checkmark)$ Subsets of the set of unitary elements in unital C$^*$- and JB$^*$-algebras.

math.FA↗

arXiv2023

Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition

In 2022, Hatori gave a sufficient condition for complex Banach spaces to have the complex Mazur--Ulam property. In this paper, we introduce a class of complex Banach spaces $B$ that do not satisfy the condition but enjoy the property that every surjective isometry on the unit sphere of such $B$ admits an extension to a surjective real linear isometry on the whole space $B$. Typical examples of Banach spaces studied in this note are the spaces ${\rm Lip}([0,1])$ of all Lipschitz complex-valued functions on $[0,1]$ and $C^1([0,1])$ of all continuously differentiable complex-valued functions on $[0,1]$ equipped with the norm $|f(0)|+\|f'\|_\infty$.

math.FA↗

arXiv2023

Lie--Trotter formulae in Jordan--Banach algebras with applications to the study of spectral-valued multiplicative functionals

We establish some Lie--Trotter formulae for unital complex Jordan--Banach algebras, showing that for each couple of elements $a,b$ in a unital complex Jordan--Banach algebra $\mathfrak{A}$ the identities $$ \lim_{n\to \infty} \left(e^{\frac{a}{n}}\circ e^{\frac{b}{n}} \right)^{n} = e^{a+b},\ \lim_{n\to \infty} \left(U_{e^{\frac{a}{n}}} \left( e^{\frac{b}{n}}\right) \right)^{n} = e^{2 a+b}, \hbox{ and }$$ $$ \lim_{n\to \infty} \left(U_{e^{\frac{a}{n}},e^{\frac{c}{n}}} \left( e^{\frac{b}{n}}\right) \right)^{n} = e^{a+b + c}$$ hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in $\mathfrak{A}$. These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals $f:\mathfrak{A}\to \mathbb{C}$ satisfying $f(U_x (y))=U_{f(x)}f(y),$ for all $x,y\in \mathfrak{A}$. We prove that for any such a functional $f,$ there exists a unique continuous (Jordan-)multiplicative linear functional $ψ\colon \mathfrak{A}\to\mathbb{C}$ such that $ f(x)=ψ(x),$ for every $x$ in the connected component of set of all invertible elements of $\mathfrak{A}$ containing the unit element. If we additionally assume that $\mathfrak{A}$ is a JB$^*$-algebra and $f$ is continuous, then $f$ is a linear multiplicative functional on $\mathfrak{A}$. The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Shulz, and Tour{é}.

math.FA↗

arXiv2023

Estimations of the numerical index of a JB$^*$-triple

We prove that every commutative JB$^*$-triple has numerical index one. We also revisit the notion of commutativity in JB$^*$-triples to show that a JBW$^*$-triple $M$ has numerical index one precisely when it is commutative, while $e^{-1}\leq n(M) \leq 2^{-1}$ otherwise. Consequently, a JB$^*$-triple $E$ is commutative if and only if $n(E^*) =1$ (equivalently, $n(E^{**}) =1$). In the general setting we prove that the numerical index of each JB$^*$-triple $E$ admitting a non-commutative element also satisfies $e^{-1}\leq n(M) \leq 2^{-1}$, and the same holds when the bidual of $E$ contains a Cartan factor of rank $\geq 2$ in its atomic part.

math.OA↗

arXiv2023

A strengthened Kadison's transitivity theorem for unital JB$^*$-algebras with applications to the Mazur--Ulam property

The principal result in this note is a strengthened version of Kadison's transitivity theorem for unital JB$^*$-algebras, showing that for each minimal tripotent $e$ in the bidual, $\mathfrak{A}^{**}$, of a unital JB$^*$-algebra $\mathfrak{A}$, there exists a self-adjoint element $h$ in $\mathfrak{A}$ satisfying $e\leq \exp(ih)$, that is, $e$ is bounded by a unitary in the principal connected component of the unitary elements in $\mathfrak{A}$. This new result opens the way to attack new geometric results, for example, a Russo--Dye type theorem for maximal norm closed proper faces of the closed unit ball of $\mathfrak{A}$ asserting that each such face $F$ of $\mathfrak{A}$ coincides with the norm closed convex hull of the unitaries of $\mathfrak{A}$ which lie in $F$. Another geometric property derived from our results proves that every surjective isometry from the unit sphere of a unital JB$^*$-algebra $\mathfrak{A}$ onto the unit sphere of any other Banach space is affine on every maximal proper face. As a final application we show that every unital JB$^*$-algebra $\mathfrak{A}$ satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $\mathfrak{A}$ onto the unit sphere of any other Banach space $Y$ admits an extension to a surjective real linear isometry from $\mathfrak{A}$ onto $Y$. This extends a result of M. Mori and N. Ozawa who have proved the same for unital C$^*$-algebras.

math.OA↗

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 $M(\mathfrak{B})$.

math.OA↗

arXiv2022

Preservers of triple transition pseudo-probabilities in connection with orthogonality preservers and surjective isometries

We prove that every bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW$^*$-triples automatically preserves orthogonality in both directions. Consequently, each bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW$^*$-triples is precisely the restriction of a (complex-)linear triple isomorphism between the corresponding JBW$^*$-triples. This result can be regarded as triple version of the celebrated Wigner theorem for Wigner symmetries on the posets of minimal projections in $B(H)$. We also present a Tingley type theorem by proving that every surjective isometry between the sets of minimal tripotents in two atomic JBW$^*$-triples admits an extension to a real linear surjective isometry between these two JBW$^*$-triples. We also show that the class of surjective isometries between the sets of minimal tripotents in two atomic JBW$^*$-triples is, in general, strictly wider than the set of bijections preserving triple transition pseudo-probabilities.

math.FA↗

arXiv2022

The Daugavet equation for polynomials on C$^$-algebras and JB$^$-triples

We prove that every JB$^*$-triple $E$ (in particular, every $C^*$-algebra) satisfying the Daugavet property also satisfies the stronger polynomial Daugavet property, that is, every weakly compact polynomial $P\colon E \longrightarrow E$ satisfies the Daugavet equation $\|\hbox{id}_{E} + P\| = 1+\|P\|$. The analogous conclusion also holds for the alternative Daugavet property.

math.OA↗

arXiv2022

Linear orthogonality preservers between function spaces associated with commutative JB$^*$-triples

It is known, by Gelfand theory, that every commutative JB$^*$-triple admits a representation as a space of continuous functions of the form $$C_0^{\mathbb{T}}(L) = \{ a\in C_0(L) : a(λt ) = λa(t), \ \forall λ\in \mathbb{T}, t\in L\},$$ where $L$ is a principal $\mathbb{T}$-bundle and $\mathbb{T}$ denotes the unit circle in $\mathbb{C}.$ We provide a description of all orthogonality preserving (non-necessarily continuous) linear maps between commutative JB$^*$-triples. We show that each linear orthogonality preserver $T: C_{0}^{\mathbb{T}} (L_1)\to C_{0}^{\mathbb{T}} (L_2)$ decomposes in three main parts on its image, on the first part as a positive-weighted composition operator, on the second part the points in $L_2$ where the image of $T$ vanishes, and a third part formed by those points $s$ in $L_2$ such that the evaluation mapping $δ_s\circ T$ is non-continuous. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB$^*$-triples is automatically continuous and biorthogonality preserving.

math.FA↗

arXiv2022

Maps preserving triple transition pseudo-probabilities

Let $e$ and $v$ be minimal tripotents in a JBW$^*$-triple $M$. We introduce the notion of triple transition pseudo-probability from $e$ to $v$ as the complex number $TTP(e,v)= φ_v(e),$ where $φ_v$ is the unique extreme point of the closed unit ball of $M_*$ at which $v$ attains its norm. In the case of two minimal projections in a von Neumann algebra, this correspond to the usual transition probability. We prove that every bijective transformation $Φ$ preserving triple transition pseudo-probabilities between the lattices of tripotents of two atomic JBW$^*$-triples $M$ and $N$ admits an extension to a bijective {\rm(}complex{\rm)} linear mapping between the socles of these JBW$^*$-triples. If we additionally assume that $Φ$ preserves orthogonality, then $Φ$ can be extended to a surjective (complex-)linear {\rm(}isometric{\rm)} triple isomorphism from $M$ onto $N$. In case that $M$ and $N$ are two spin factors or two type 1 Cartan factors we show, via techniques and results on preservers, that every bijection preserving triple transition pseudo-probabilities between the lattices of tripotents of $M$ and $N$ automatically preserves orthogonality, and hence admits an extension to a triple isomorphism from $M$ onto $N$.

math.OA↗

arXiv2022

Determinants in Jordan matrix algebras

We introduce a natural notion of determinant in matrix JB$^*$-algebras, i.e., for hermitian matrices of biquaternions and for hermitian $3\times 3$ matrices of complex octonions. We establish several properties of these determinants which are useful to understand the structure of the Cartan factor of type $6$. As a tool we provide an explicit description of minimal projections in the Cartan factor of type $6$ and a variety of its automorphisms.

math.OA↗

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.

math.OA↗

arXiv2022

Every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property

We prove that every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property. Thanks to the representation theory, we can identify commutative JB$^*$-triples as spaces of complex-valued continuous functions on a principal $\mathbb{T}$-bundle $L$ in the form $$C_0^\mathbb{T}(L):=\{a\in C_0(L):a(λt)=λa(t)\text{ for every } (λ,t)\in\mathbb{T}\times L\}.$$ We prove that every surjective isometry from the unit sphere of $C_0^\mathbb{T}(L)$ onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.

math.FA↗

arXiv2021

Order type relations on the set of tripotents in a JB$^*$-triple

We introduce, investigate and compare several order type relations on the set of tripotents in a JB$^*$-triple. The main two relations we address are $\le_h$ and $\le_n$. We say that $u\le_h e$ (or $u\le_n e$) if $u$ is a self-adjoint (or normal) element of the Peirce-2 subspace associated to $e$ considered as a unital JB$^*$-algebra with unit $e$. It turns out that these relations need not be transitive, so we consider their transitive hulls as well. Properties of these transitive hulls appear to be closely connected with types of von Neumann algebras, with the results on products of symmetries, with determinants in finite-dimensional Cartan factors, with finiteness and other structural properties of JBW$^*$-triples.

math.OA↗

arXiv2021

Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors

There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states $\mathbf{P}$ -- i.e., the set of one-dimensional projections on a complex Hilbert space $H$ -- and the orthomodular lattice $\mathbf{L}$ of closed subspaces of $H$. These six groups are isomorphic when the dimension of $H$ is $\geq 3$. Despite of the difficulties caused by $M_2(\mathbb{C})$, rank two algebras are used for quantum mechanics description of the spin state of spin-$\frac12$ particles, there is a counterexample for Uhlhorn's version of Wigner's theorem for such state space. In this note we prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and orthogonality, but we also need to preserve the partial order set of all tripotents and orthogonality among them (a set which strictly enlarges the lattice of projections). Concretely, let $M$ and $N$ be two atomic JBW$^*$-triples not containing rank-one Cartan factors, and let $\mathcal{U} (M)$ and $\mathcal{U} (N)$ denote the set of all tripotents in $M$ and $N$, respectively. We show that each bijection $Φ: \mathcal{U} (M)\to \mathcal{U} (N)$, preserving the partial ordering in both directions, orthogonality in one direction and satisfying some mild continuity hypothesis can be extended to a real linear triple automorphism. This, in particular, extends a result of Moln{á}r to the wider setting of atomic JBW$^*$-triples not containing rank-one Cartan factors, and provides new models to present quantum behavior.

quant-ph↗

arXiv2021

Exploring new solutions to Tingley's problem for function algebras

In this note we present two new positive answers to Tingley's problem in certain subspaces of function algebras. In the first result we prove that every surjective isometry between the unit spheres, $S(A)$ and $S(B)$, of two uniformly closed function algebras $A$ and $B$ on locally compact Hausdorff spaces can be extended to a surjective real linear isometry from $A$ onto $B$. In a second goal we study surjective isometries between the unit spheres of two abelian JB$^*$-triples represented as spaces of continuous functions of the form $$C^{\mathbb{T}}_0 (X) := \{ a \in C_0(X) : a (λt) = λa(t) \hbox{ for every } (λ, t) \in \mathbb{T}\times X\},$$ where $X$ is a (locally compact Hausdorff) principal $\mathbb{T}$-bundle. We establish that every surjective isometry $Δ: S(C_0^{\mathbb{T}}(X))\to S(C_0^{\mathbb{T}}(Y))$ admits an extension to a surjective real linear isometry between these two abelian JB$^*$-triples.

math.FA↗