Two normal functionals on a JBW$^*$-triple are known to be orthogonal if and only if they are $L$... more Two normal functionals on a JBW$^*$-triple are known to be orthogonal if and only if they are $L$-orthogonal (meaning that they span an isometric copy of $\ell_1(2)$). This is shown to be stable under small norm perturbations in the following sense: if the linear span of the two functionals is isometric up to $\delta>0$ to $\ell_1(2)$, then the functionals are less far (in norm) than $\eps>0$ from two orthogonal functionals, where $\eps\to0$ as $\delta\to0$. Analogous statements for finitely and even infinitely many functionals hold as well. And so does a corresponding statement for non-normal functionals. Our results have been known for C$^*$-algebras.
We survey the results on linear local and 2-local homomorphisms and zero products preserving oper... more We survey the results on linear local and 2-local homomorphisms and zero products preserving operators between C$^*$-algebras, and we incorporate some new precise observations and results to prove that every bounded linear 2-local homomorphism between C$^*$-algebras is a homomorphism. Consequently, every linear 2-local $^*$-homomorphism between C$^*$-algebras is a $^*$-homomorphism.
ABSTRACT We describe the one-dimensional \v{C}eby\v{s}\"{e}v subspaces of a JBW$^*$-trip... more ABSTRACT 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 quasi-invertible element in ${M}$. We establish a complete description of all \v{C}eby\v{s}\"{e}v JBW$^*$-subtriples of $M$, establishing that a non-zero JBW$^*$-subtriple $N$ of $M$ if a \v{C}eby\v{s}\"{e}v subspace if and only if one of the following statements holds:\begin{enumerate}[$(a)$] $N$ and $M$ have rank one or two; $N= \mathbb{C} e$, where $e$ is a complete tripotent in $M$; $N$ has rank greater or equal than 3 and $N=M$.
In this note we revise and survey some recent results established in (8). We shall show that for ... more In this note we revise and survey some recent results established in (8). We shall show that for each Banach space X, there exists a locally convex topology for X, termed the "Right Topology", such that a linear map T, from X into a Banach space Y, is weakly compact, precisely when T is a continuous map from X, equipped with the "Right" topology, into Y equipped with the norm topology. We provide here a new and shorter proof of this result. We shall also survey the results concerning sequentially Right-to-norm continuous operators.
Two normal functionals on a JBW$^*$-triple are known to be orthogonal if and only if they are $L$... more Two normal functionals on a JBW$^*$-triple are known to be orthogonal if and only if they are $L$-orthogonal (meaning that they span an isometric copy of $\ell_1(2)$). This is shown to be stable under small norm perturbations in the following sense: if the linear span of the two functionals is isometric up to $\delta>0$ to $\ell_1(2)$, then the functionals are less far (in norm) than $\eps>0$ from two orthogonal functionals, where $\eps\to0$ as $\delta\to0$. Analogous statements for finitely and even infinitely many functionals hold as well. And so does a corresponding statement for non-normal functionals. Our results have been known for C$^*$-algebras.
We survey the results on linear local and 2-local homomorphisms and zero products preserving oper... more We survey the results on linear local and 2-local homomorphisms and zero products preserving operators between C$^*$-algebras, and we incorporate some new precise observations and results to prove that every bounded linear 2-local homomorphism between C$^*$-algebras is a homomorphism. Consequently, every linear 2-local $^*$-homomorphism between C$^*$-algebras is a $^*$-homomorphism.
ABSTRACT We describe the one-dimensional \v{C}eby\v{s}\"{e}v subspaces of a JBW$^*$-trip... more ABSTRACT 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 quasi-invertible element in ${M}$. We establish a complete description of all \v{C}eby\v{s}\"{e}v JBW$^*$-subtriples of $M$, establishing that a non-zero JBW$^*$-subtriple $N$ of $M$ if a \v{C}eby\v{s}\"{e}v subspace if and only if one of the following statements holds:\begin{enumerate}[$(a)$] $N$ and $M$ have rank one or two; $N= \mathbb{C} e$, where $e$ is a complete tripotent in $M$; $N$ has rank greater or equal than 3 and $N=M$.
In this note we revise and survey some recent results established in (8). We shall show that for ... more In this note we revise and survey some recent results established in (8). We shall show that for each Banach space X, there exists a locally convex topology for X, termed the "Right Topology", such that a linear map T, from X into a Banach space Y, is weakly compact, precisely when T is a continuous map from X, equipped with the "Right" topology, into Y equipped with the norm topology. We provide here a new and shorter proof of this result. We shall also survey the results concerning sequentially Right-to-norm continuous operators.
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