[Paper Review] On the extension of isometries between the unit spheres of von Neumann algebras

Antonio M. Peralta, Francisco J. Fernández-Polo|arXiv (Cornell University)|Sep 25, 2017

Advanced Topics in Algebra5 citations

TL;DR

This paper establishes a complete positive solution to Tingley's problem for von Neumann algebras by proving that every surjective isometry between the unit spheres of two von Neumann algebras extends uniquely to a surjective real linear isometry. The key result shows that such isometries preserve unitary elements and are induced by Jordan *-isomorphisms composed with multiplication by a unitary in the target algebra.

ABSTRACT

We prove that every surjective isometry between the unit spheres of two von Neumann algebras admits a unique extension to a surjective real linear isometry between these two algebras.

Motivation & Objective

• To resolve Tingley's problem in the context of general von Neumann algebras, where the extension of isometries from the unit sphere to the entire algebra remains open.
• To extend known results on isometry extensions from C*-algebras and JBW*-triples to the broader class of von Neumann algebras.
• To establish that surjective isometries between unit spheres of von Neumann algebras preserve key algebraic and geometric structures, such as unitary elements and norm-closed faces.
• To prove that such isometries arise from Jordan *-isomorphisms composed with unitary multiplication, yielding a canonical extension to real linear isometries.
• To overcome technical challenges in non-atomic von Neumann algebras by leveraging convex combinations of unitaries and facial structure preservation.

Proposed method

• Utilize the geometric property that surjective isometries between unit spheres preserve maximal convex sets and norm-closed faces of the unit ball.
• Prove that a surjective isometry between unit spheres of C*-algebras maps non-zero partial isometries to partial isometries and preserves associated norm-closed faces.
• Establish that such isometries preserve unitary elements, generalizing a result from finite von Neumann algebras to arbitrary von Neumann algebras.
• Apply Hatori and Molnár’s result that every surjective isometry between unitary groups of von Neumann algebras extends to a real linear isometry.
• Use the theory of convex combinations of unitary operators in von Neumann algebras (Olsen and Pedersen) to approximate partial isometries and extend the isometry to the entire sphere.
• Construct the extension $ T(x) = f(1)ig(pJ(x) + (1-p)J(x)^*ig) $, where $ J $ is a Jordan *-isomorphism and $ p $ is a central projection in the target algebra, to define the real linear extension.

Experimental results

Research questions

• RQ1 Does every surjective isometry between the unit spheres of two von Neumann algebras extend to a surjective real linear isometry on the entire algebras?
• RQ2 How do isometries between unit spheres interact with the facial structure of the unit ball in von Neumann algebras?
• RQ3 Can the preservation of unitary elements by such isometries be established in general von Neumann algebras, not just finite or atomic ones?
• RQ4 What is the precise form of the linear extension of a sphere isometry in the context of von Neumann algebras?
• RQ5 To what extent do partial isometries and their associated norm-closed faces behave under isometric sphere mappings in non-atomic von Neumann algebras?

Key findings

• Every surjective isometry between the unit spheres of two von Neumann algebras extends uniquely to a surjective real linear isometry on the entire algebras.
• The extension is given by $ T(x) = f(1)ig(pJ(x) + (1-p)J(x)^*ig) $, where $ J $ is a Jordan *-isomorphism and $ p $ is a central projection in the target algebra.
• Surjective isometries between unit spheres preserve unitary elements, generalizing a result from finite von Neumann algebras.
• The isometry maps norm-closed faces of the unit ball associated with non-zero partial isometries to corresponding faces in the target algebra.
• Every element in the unit sphere of a von Neumann algebra can be approximated by convex combinations of unitaries, enabling the extension of isometries via real affine structure.
• In the case of a factor, the isometry is either $ f(x) = f(1)J(x) $ or $ f(x) = f(1)J(x^*) $, where $ J $ is a Jordan *-isomorphism, showing a clean classification of the extension.