希尔伯特C*-模中的B样条插值问题
B-spline interpolation problem in Hilbert C*-modules
AI总结：
本文在希尔伯特C*-模中引入B样条插值问题，研究其解的存在性与唯一性，给出充要条件，并推广至W*-模及C*-理想情形，辅以实例说明。
摘要：
我们引入了与希尔伯特C*-模上的C*-值半双线性形式相对应的B样条插值问题，并研究了其基本性质以及解的唯一性。我们首先在希尔伯特C*-模为自对偶的情形下研究该问题。将希尔伯特C*-模上的有界C*-值半双线性形式延拓到其二重对偶上的半双线性形式后，我们给出了B样条插值问题有解的一些必要且充分条件。转而考虑希尔伯特W*-模的设定，我们通过刻画延拓的C*-值半双线性形式到希尔伯特W*-模X的对偶X'上的样条插值问题何时有解，呈现了我们的主要结果。作为推论，我们给出了一个充分条件：自对偶希尔伯特W*-模X的一个正交补子模关于X上的另一个C*-内积也是正交补的。最后，我们广泛讨论了定义在W*-代数的C*-理想上的希尔伯特C*-模的B样条插值问题的解。提供了若干例子来说明该问题解的存在性或缺乏性。
英文摘要：
We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Extending a bounded $C^*$-valued sesquilinear form on a Hilbert $C^*$-module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued sesquilinear to the dual $\mathscr{X}'$ of the Hilbert $W^*$-module $\mathscr{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathscr{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathscr{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem.
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