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Akhlaq A. Siddiqui

Publications and source records attributed to Akhlaq A. Siddiqui.

5 recordsLinked to original sources
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}

math.OA↗
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.

math.OA↗
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.

math.OA↗
arXiv2014

Jordan weak amenability and orthogonal forms on JB*-algebras

We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$.

math.OA↗
arXiv2014

Approximation and convex decomposition by extremals and the $λ$-function in JBW*-triples

We establish new estimates to compute the $λ$-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 $$\hbox{dist} (a, \mathfrak{E} (E_1)) = \max \left\{ 1- m_q (a) , \|a\|-1\right\},$$ where $\mathfrak{E} (E_1)$ denotes the set of extreme points of the closed unit ball $E_1$ of $E$. It is proved that $λ(a) = \frac{1+m_q (a)}{2},$ for every Brown-Pedersen quasi-invertible element $a$ in $E_1$, where $m_q (a)$ is the square root of the quadratic conorm of $a$. For an element $a$ in $E_1$ which is not Brown-Pedersen quasi-invertible we can only estimate that $λ(a)\leq \frac12 (1-α_q (a)).$ A complete description of the $λ$-function on the closed unit ball of every JBW$^*$-triple is also provided, and as a consequence, we prove that every JBW$^*$-triple satisfies the uniform $λ$-property.

math.OA↗