## Annals of Functional Analysis

### Extremally rich JB$^{*}$-triples

#### Abstract

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

$$\operatorname{dist}(a,\partial_{e}(E_{1}))=\max \{1,\|a\|-1\},$$ for every $a\in E$ which is not Brown–Pedersen quasi-invertible. As a consequence, we determine the form of the $\lambda$-function of Aron and Lohman on the open unit ball of an extremally rich JB$^{*}$-triple $E$ by showing that $\lambda(a)=1/2$ 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 conorm $\gamma^{q}(\cdot)$ is continuous at a point $a\in E$ if and only if either $a$ is not von Neumann regular (i.e., $\gamma^{q}(a)=0$) or $a$ is Brown–Pedersen quasi-invertible.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 4 (2016), 578-592.

Dates
Accepted: 11 April 2016
First available in Project Euclid: 23 September 2016

https://projecteuclid.org/euclid.afa/1474652183

Digital Object Identifier
doi:10.1215/20088752-3661557

Mathematical Reviews number (MathSciNet)
MR3550937

Zentralblatt MATH identifier
06667755

#### Citation

Jamjoom, Fatmah B.; Peralta, Antonio M.; Siddiqui, Akhlaq A.; Tahlawi, Haifa M. Extremally rich JB $^{*}$ -triples. Ann. Funct. Anal. 7 (2016), no. 4, 578--592. doi:10.1215/20088752-3661557. https://projecteuclid.org/euclid.afa/1474652183

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