On the extension of surjective isometries whose domain is the unit sphere of a space of compact operators

TL;DR

This paper demonstrates that surjective isometries from the unit sphere of compact operators on a Hilbert space can be extended to linear isometries, revealing new properties of non-commutative C*-algebras and JB*-triples.

Contribution

It shows that certain isometries on the sphere of compact operators extend to linear isometries, including a novel example of a non-commutative C*-algebra with this property.

Findings

Surjective isometries extend to linear isometries for compact operators.

First example of a non-commutative C*-algebra with the Mazur–Ulam property.

All compact C*-algebras and weakly compact JB*-triples satisfy the Mazur–Ulam property.

Abstract

We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur--Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur--Ulam property.