Annals of Functional Analysis

Extremally rich JB-triples

Fatmah B. Jamjoom, Antonio M. Peralta, Akhlaq A. Siddiqui, and Haifa M. Tahlawi

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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 e(E1) of all extreme points of the closed unit ball of E. More concretely, we prove that

dist(a,e(E1))=max {1,a1}, for every aE 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)=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 γq() is continuous at a point aE if and only if either a is not von Neumann regular (i.e., γq(a)=0) or a is Brown–Pedersen quasi-invertible.

Article information

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

Received: 7 January 2016
Accepted: 11 April 2016
First available in Project Euclid: 23 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L70: Nonassociative selfadjoint operator algebras [See also 46H70, 46K70]
Secondary: 15A09: Matrix inversion, generalized inverses 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47D25 17C65: Jordan structures on Banach spaces and algebras [See also 46H70, 46L70] 46L05: General theory of $C^*$-algebras

extremally rich JB$^{*}$-triple Brown–Pedersen quasi-invertibility reduced minimum modulus conorm quadratic conorm


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.

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