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
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