[Paper Review] The Mazur-Ulam property for commutative von Neumann algebras

Antonio M. Peralta, María Cueto-Avellaneda|arXiv (Cornell University)|Mar 1, 2018

Advanced Operator Algebra Research29 references3 citations

TL;DR

This paper establishes that commutative von Neumann algebras, specifically $L^∞(\Omega,\mu)$ and $C(K)$ for Stonean spaces $K$, satisfy the Mazur-Ulam property: every surjective isometry between their unit spheres extends to a surjective real linear isometry on the entire space. The result is derived via a general extension theorem for isometries on $C(K)$ with $K$ Stonean, leveraging the structure of projections and monotone completeness in these C*-algebras.

ABSTRACT

Let $(Ω,μ)$ be a $σ$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (Ω,μ)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $Δ: S(L^{\infty} (Ω,μ)) o S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (Ω,μ) o X$. This conclusion is derived from a more general statement which assures that every surjective isometry $Δ: S(C(K)) o S(X),$ where $K$ is a Stonean space, admits an extension to a surjective real linear isometry from $C(K)$ onto $X$.

Motivation & Objective

• To resolve the open problem of whether commutative von Neumann algebras satisfy the Mazur-Ulam property, particularly in the complex setting.
• To extend known results on isometry extensions from real Banach spaces to complex ones, especially for $L^\infty(\Omega,\mu)$ and $C(K)$ with $K$ Stonean.
• To provide a general framework for extending surjective isometries from the unit sphere of $C(K)$ to real linear isometries on the whole space when $K$ is Stonean.
• To overcome the limitations of previous real-variable methods by developing a complex-analytic and C*-algebraic approach tailored to complex Banach spaces.
• To clarify the role of Stonean spaces and monotone completeness in enabling the extension of isometries on unit spheres.

Proposed method

• The authors prove that for a Stonean space $K$, every surjective isometry $\Delta: S(C(K)) \to S(X)$ extends to a surjective real linear isometry $T: C(K) \to X$, using the fact that $C(K)$ is monotone complete and elements can be uniformly approximated by finite linear combinations of projections.
• They exploit the structure of $C(K)$ as a dual Banach space when $K$ is hyper-Stonean, but focus on the more general Stonean case to ensure applicability to $L^\infty(\Omega,\mu)$ via representing it as a $C(K)$ space.
• The proof relies on the existence of sufficiently many projections and the use of slices in the unit sphere to control the behavior of isometries, adapting techniques from Tingley’s problem and the Mazur-Ulam theorem.
• The authors use the fact that in a Stonean space, every bounded increasing net of real-valued continuous functions has a supremum in $C(K)$, ensuring monotone completeness and enabling the extension of isometries.
• They apply a general extension result for isometries on the unit sphere of $C(K)$ to the specific case of $L^\infty(\Omega,\mu)$, which is isometrically isomorphic to $C(K)$ for some Stonean $K$, thus transferring the result to $L^\infty$.
• The method avoids reliance on real-variable techniques from prior works (e.g., Tan [42]) by constructing a complex-adapted framework based on the algebraic and topological properties of $C(K)$ for Stonean $K$.

Experimental results

Research questions

• RQ1 Does the space $L^\infty(\Omega,\mu)$ of complex-valued essentially bounded functions satisfy the Mazur-Ulam property for complex Banach spaces?
• RQ2 Can every surjective isometry between the unit spheres of $C(K)$ and another complex Banach space $X$ be extended to a real linear isometry when $K$ is a Stonean space?
• RQ3 What structural properties of $C(K)$ for Stonean $K$ enable the extension of isometries from the unit sphere to the whole space?
• RQ4 How does the Mazur-Ulam property behave in the context of complex commutative von Neumann algebras, given the challenges posed by complex linear and conjugate-linear isometries?
• RQ5 To what extent can the results on real Banach spaces be generalized to complex Banach spaces in the setting of Tingley’s problem?

Key findings

• The space $L^\infty(\Omega,\mu)$ satisfies the Mazur-Ulam property for complex Banach spaces, meaning every surjective isometry between its unit sphere and that of any complex Banach space $X$ extends to a surjective real linear isometry from $L^\infty(\Omega,\mu)$ onto $X$.
• For any Stonean space $K$, every surjective isometry $\Delta: S(C(K)) \to S(X)$ extends to a surjective real linear isometry $T: C(K) \to X$, establishing a general extension principle.
• The extension result holds due to the monotone completeness and projection structure of $C(K)$ when $K$ is Stonean, which allows control over the isometric image of the unit sphere.
• The result generalizes previous positive solutions of Tingley’s problem to complex commutative von Neumann algebras, filling a key gap in the literature.
• The authors show that the complex version of Tan’s real $L^\infty$ result cannot be extended directly, necessitating a new approach based on C*-algebraic and topological properties of Stonean spaces.
• The study confirms that $C(K)$ for Stonean $K$ is a natural setting for the Mazur-Ulam property in the complex category, extending known results from real Banach spaces.