相关论文: Estimations of the numerical index of a JB$^*$-tri…

We prove that a Le Page-type inequality is also valid for metrically characterizing those JB$^*$-triples that are commutative. More precisely, we establish that the following statements are equivalent for any JB$^*$-triple $E$: $(a)$ $E$ is…

The aim of this note is to study \v{C}eby\v{s}\"ev JB$^*$-subtriples of general JB$^*$-triples. It is established that if $F$ is a non-zero \v{C}eby\v{s}\"ev JB$^*$-subtriple of a JB$^*$-triple $E$, then exactly one of the following…

We describe the one-dimensional \v{C}eby\v{s}\"{e}v subspaces of a JBW$^*$-triple $M,$ by showing that for a non-zero element $x$ in $M$, $\mathbb{C}x$ is a \v{C}eby\v{s}\"{e}v subspace of $M$ if, and only if, $x$ is a Brown-Pedersen…

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…

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…

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…

We show that every Hilbert C*-module E is a JB*-triple in a canonical way and establish an explicit expression for the holomorphic automorphisms of the unit ball of E.

The contributions in this note begin with a new characterization of (positive) scalar multiples of minimal tripotents in a general JB$^*$-triple $E$, proving that a non-zero element $a\in E$ is a positive scalar multiple of a minimal…

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 $\psi\in E^*$ satisfying $$\|T(x)\| \leq K \, \|T\| \, \|x\|_{\psi},$$…

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…

We establish new estimates to compute the $\lambda$-function of Aron and Lohman on the unit ball of a JB$^*$-triple. It is established that for every Brown-Pedersen quasi-invertible element $a$ in a JB$^*$-triple $E$ we have $$\hbox{dist}…

Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on…

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…

We prove that the vast majority of JC*-triples satisfy the condition of universal reversibility. Our characterisation is that a JC*-triple is universally reversible if and only if it has no triple homomorphisms onto Hilbert spaces of…

A classical theorem by Jacobson says that a ring in which every element $x$ satisfies the equation $x^n=x$ for some $n>1$ is commutative. According to Birkhoff's Completeness Theorem, if $n$ is fixed, there must be an equational proof of…

We prove that every (not necessarily linear nor continuous) 2-local triple homomorphism from a JBW$^*$-triple into a JB$^*$-triple is linear and a triple homomorphism. Consequently, every 2-local triple homomorphism from a von Neumann…

Motivated by advances in categorical probability, we introduce non-commutative almost everywhere (a.e.) equivalence and disintegrations in the setting of C*-algebras. We show that C*-algebras (resp. W*-algebras) and a.e. equivalence classes…

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…

We show that every closed (resp., weak$^*$-closed) inner ideal $I$ of a real JB$^*$-triple (resp. a real JBW$^*$-triple) $E$ is Hahn--Banach smooth (resp., weak$^*$-Hahn--Banach smooth). Contrary to what is known for complex JB$^*$-triples,…

We call an operator algebra A {\em reversible} if A with reversed multiplication is also an abstract operator algebra (in the modern operator space sense). This class of operator algebras is intimately related to the {\em symmetric operator…