Noncommutative \(L^p\) structure encodes exactly Jordan structure (Q1772694)

This paper studies the structure of surjective isometries between noncommutative \(L^p\)-spaces. Let \(\mathcal M\) and \(\mathcal N\) be two von Neumann algebras and \(1\leq p\neq2 \leq\infty\). The main result of the paper then asserts that \(L^p(\mathcal M)\) and \(L^p(\mathcal N)\) are isometrically isomorphic as Banach spaces iff \(\mathcal M\) and \(\mathcal N\) are Jordan *-isomorphic. In the case that both \(\mathcal M\) and \(\mathcal N\) are semifinite, this result is due to \textit{F. J. Yeadon} [Math. Proc. Camb. Philos. Soc. 90, 41--50 (1981; Zbl 0483.46041)]. On the other hand, it extends Kadison's classical theorem on Jordan isomorphisms of von Neumann algebras (corresponding to the case \(p=\infty\) in the previous statement) to \(L^p\)-spaces. Like in Yeadon's paper, the arguments here depend heavily on the equality case of the noncommutative Clarkson inequality.