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ON THE EXTENSION OF ISOMETRIES BETWEEN THE UNIT SPHERES OF A
$\text{JBW}^{\ast }$-TRIPLE AND A BANACH SPACE
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 20 / Issue 1 / January 2021
- Published online by Cambridge University Press:
- 15 April 2019, pp. 277-303
- Print publication:
- January 2021
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- Article
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We prove that if
$M$ is a 
$\text{JBW}^{\ast }$-triple and not a Cartan factor of rank two, then 
$M$ satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of 
$M$ onto the unit sphere of another real Banach space 
$Y$ extends to a surjective real linear isometry from 
$M$ onto 
$Y$.
MAZUR–ULAM PROPERTY OF THE SUM OF TWO STRICTLY CONVEX BANACH SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 3 / June 2016
- Published online by Cambridge University Press:
- 11 November 2015, pp. 473-485
- Print publication:
- June 2016
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- Article
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In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the
$\ell ^{1}$-sum and the 
$\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.
