Abstract

Let \(\mathcal{M}\) be an atomless semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ (or else, an atomic von Neumann algebra with all atoms having the same trace) acting on a separable Hilbert space \(\mathcal{H}\). Let \(E(\mathcal{M},\tau )\) be a separable symmetric space of τ-measurable operators, whose norm is not proportional to the Hilbert norm ||⋅||₂ on \({{L}_{2}}(\mathcal{M},\tau )\). We provide a description of all bounded Hermitian operators on \(E(\mathcal{M},\tau )\) and all surjective linear isometries of this space.

中文翻译：

---

非交换对称空间上的等价性

摘要

让\（\ mathcal {M} \）是semifinite配备有忠实正常semifinite跟踪τ（否则，原子冯·诺依曼代数与具有相同的迹线的所有原子）作用于可分离的Hilbert空间冯·诺依曼代数的atomless \（ \ mathcal {H} \）。令\（E（\ mathcal {M}，\ tau）\）是τ个可测算子的可分离对称空间，其范数与希尔伯特范数不成比例||⋅|| ₂在\（{{L} _ {2}}（\ mathcal {M}，\ tau）\）上。我们提供\（E（\ mathcal {M}，\ tau）\）上所有有界Hermitian算子的描述以及该空间的所有射影线性等式。