Journal of Operator Theory

Volume 80, Issue 2,  Fall  2018  pp. 429-452.

Metric preserving bijections between positive spherical shells of non-commutative $L^p$-spaces

Authors:  Chi-Wai Leung (1), Chi-Keung Ng (2), and Ngai-Ching Wong (3)
Author institution: (1) Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
(2) Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
(3) Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

Summary:  Let $L^p(M)$ be the non-commutative $L^p$-space associated to a von Neumann algebra $M$ with the canonical positive cone $L^p_+(M)$. Consider $$L^p_+(M)^1_{1-\varepsilon} := \{T\in L^p_+(M): 1-\varepsilon \leqslant \|T\| \leqslant 1\} \quad (0< \varepsilon < 1),$$ the positive spherical shell of $L^p_+(M)$. If $N$ is another von Neumann algebra, $p\in [1,\infty]$ and $\Phi: L^p_+(M)^1_{1-\varepsilon} \to L^p_+(N)^1_{1-\varepsilon}$ is a metric preserving bijection, then $M, N$ are isomorphic as Jordan $*$-algebras. Assume further that $M\ncong \mathbb{C}$ is approximately semifinite and $1 DOI: http://dx.doi.org/10.7900/jot.2017oct30.2199 Keywords: non-commutative$L^p$-spaces, Jordan$*\$-isomorphisms, metric bijections