Hilbert丛的线性正交变换
Linear orthogonality preservers of Hilbert bundles
摘要：
由于希尔伯特空间的相应事实，很自然地会问希尔伯特$C^*$ -模的线性和正交结构是否决定了它的$C^*$ -代数值内积。我们在$C^*$ -代数是交换的情况下验证了这一点（或者等价地，我们考虑局部紧Hausdorff空间上的Hilbert丛）。更精确地说，两个希尔伯特$C^*$ -模之间的$\mathbb{C}$ -线性映射$θ$（不假定是有界的）被称为“保持正交性的”，如果$\left<θ(x),θ(y)\right> =0$无论何时$\left<x,y\right> =0$ .我们证明了如果$θ$是从一个满Hilbert $C_0(Ω)$ -模$E$到另一个满足较弱的$C_0(Ω)$ -线性性概念的Hilbert $C_0(Ω)$ -模$F$的保正交映射（称为“局部性”），则$θ$有界且存在$ϕ\in C_b(Ω)_+$使得$$ \left<θ(x),θ(y)\right>\ =\ ϕ\cdot\left<x,y\right>, \quad \forall x,y \in E. $$另一方面，如果$F$是另一个交换$C^*$ -代数$C_0(Δ)$上的全Hilbert $C^*$ -模，我们证明了具有某种“局部型性质”的“保双正交”双射映射$θ$是有界的，并且满足$$ \left<θ(x),θ(y)\right>\ =\ ϕ\cdot\left<x,y\right>\circσ, \quad \forall x,y \in E $$，其中$ϕ\in C_b(Ω)_+$和$σ: Δ\rightarrow Ω$是同胚.
英文摘要：
Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $θ$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if $\left<θ(x),θ(y)\right> =0$ whenever $\left<x,y\right> =0$. We prove that if $θ$ is an orthogonality preserving map from a full Hilbert $C_0(Ω)$-module $E$ into another Hilbert $C_0(Ω)$-module $F$ that satisfies a weaker notion of $C_0(Ω)$-linearity (known as "localness"), then $θ$ is bounded and there exists $ϕ\in C_b(Ω)_+$ such that $$ \left<θ(x),θ(y)\right>\ =\ ϕ\cdot\left<x,y\right>, \quad \forall x,y \in E. $$ On the other hand, if $F$ is a full Hilbert $C^*$-module over another commutative $C^*$-algebra $C_0(Δ)$, we show that a "bi-orthogonality preserving" bijective map $θ$ with some "local-type property" will be bounded and satisfy $$ \left<θ(x),θ(y)\right>\ =\ ϕ\cdot\left<x,y\right>\circσ, \quad \forall x,y \in E $$ where $ϕ\in C_b(Ω)_+$ and $σ: Δ\rightarrow Ω$ is a homeomorphism.
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