满等距映射的刻画：实情形

中文摘要 AI 辅助

设$(\boldsymbol{\Omega},\mu)$与$(\boldsymbol{\Lambda},\lambda)$是完备无原子可局部化半有限测度空间，$E(\Omega,\mu)$与$F(\Lambda,\lambda)$是具有序连续范数的实重排不变巴拿赫函数空间（巴拿赫格意义下），且两范数均不与$L_2$-范数成比例。每个满实线性等距映射$U:E(\Omega,\mu)\longrightarrow F(\Lambda,\lambda)$均可表示为$Uf=w\Phi(f)$，其中$w$具有全支撑，$\Phi$由完全测度类布尔同构诱导，且两个因子由$U$唯一确定。设$(\mathcal{M},\tau)$与$(\mathcal{N},\nu)$是无原子半有限冯·诺依曼代数，$E(\mathcal{M},\tau)$与$F(\mathcal{N},\nu)$是满足相同范数假设的对称算子空间，每个满实线性等距映射$V:E(\mathcal{M},\tau)_{\mathrm{sa}}\longrightarrow F(\mathcal{N},\nu)_{\mathrm{sa}}$均可表示为$V(x)=hJ(x)$，其中$J:\mathcal{M}\longrightarrow\mathcal{N}$是正规满约旦$*$-同构，$h\in LS(\mathcal{Z}(\mathcal{N}))_{\mathrm{sa}}$是具有全支撑的中心元，两个因子同样唯一。我们还确定了实自伴部分上的有界斜厄米算子，并推导了Mityagin问题的交换与非交换等距形式。

英文摘要

Let $(Ω,μ)$ and $(Λ,λ)$ be complete atomless localizable semifinite measure spaces. Suppose that $E(Ω,μ)$ and $F(Λ,λ)$ are real rearrangement-invariant Banach function spaces with order-continuous norms, in the Banach-lattice sense, and that neither norm is proportional to the $L_2$-norm. Every surjective real-linear isometry $U:E(Ω,μ)\longrightarrow F(Λ,λ)$ has the form $Uf=wΦ(f)$, where $w$ has full support and $Φ$ is induced by a complete measure-class Boolean isomorphism. Both factors are uniquely determined by $U$. Let $(\mathcal{M},τ)$ and $(\mathcal{N},ν)$ be atomless semifinite von Neumann algebras, and let $E(\mathcal{M},τ)$ and $F(\mathcal{N},ν)$ be symmetric operator spaces satisfying the same assumptions on their norms. Every surjective real-linear isometry $V:E(\mathcal{M},τ)_{\mathrm{sa}}\longrightarrow F(\mathcal{N},ν)_{\mathrm{sa}}$ has the form $V(x)=hJ(x)$, where $J:\mathcal{M}\longrightarrow\mathcal{N}$ is a normal surjective Jordan $*$-isomorphism and $h\in LS(\mathcal{Z}(\mathcal{N}))_{\mathrm{sa}}$ is central with full support; again, the two factors are unique. We also identify the bounded skew-Hermitian operators on the real self-adjoint part and derive commutative and noncommutative isometric forms of Mityagin's question.

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preliminary version