The Mazur-Ulam property for commutative von Neumann algebras

Abstract

Let $(\Omega ,\mu )$ be a $\sigma$-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 $L^{\infty }(\Omega ,\mu )$ 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 $\Delta :S(L^{\infty }(\Omega ,\mu ))\to S(X)$ admits an extension to a surjective real linear isometry $T:L^{\infty }(\Omega ,\mu )\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry $\Delta :S(C(K))\to S(X),$ where $K$ is a Stonean space, admits an extension to a surjective real linear isometry from $C(K)$ onto $X$.

References in corpus (3)

• Tingley's problem through the facial structure of operator algebras
• On the extension of isometries between the unit spheres of a C*-algebra and B(H)
• On the extension of isometries between the unit spheres of von Neumann algebras

Cited by in corpus (2)

• Extension of isometries from the unit sphere of a rank-2 Cartan factor
• The Mazur--Ulam property in ${\ell }_{\infty }$-sum and $c_0$-sum of strictly convex Banach spaces