Tingley's problem through the facial structure of operator algebras

Abstract

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this paper, we give an affirmative answer to Tingley's problem when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras or the spaces of self-adjoint normal functionals on von Neumann algebras. We also show that every surjective isometry between the unit spheres of unital C${}^{\ast }$-algebras restricts to a bijection between their unitary groups. In addition, we show that every surjective isometry between the normal state spaces or the normal quasi-state spaces of two von Neumann algebras extends to a linear surjective isometry.

19 pages

References in corpus (2)

• Noncommutative Lp structure encodes exactly Jordan structure
• Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras

Cited by in corpus (9)

• A reflection on Tingley's problem and some applications
• Extension of isometries from the unit sphere of a rank-2 Cartan factor
• Isometries between projection lattices of von Neumann algebras
• Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras
• On the extension of isometries between the unit spheres of a JBW${}^{\ast }$-triple and a Banach space
• Metric characterisation of unitaries in JB${}^{\ast }$-algebras
• The Mazur-Ulam property for commutative von Neumann algebras
• Tingley's problem on uniform algebras
• Linearity of isometries between convex Jordan curves