Hilbert $C^*$-模上的共轭
Conjugations on Hilbert $C^*$-modules
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- Ferdowsi University of Mashhad（马什哈德 Ferdowsi 大学）
- Farhangian University（Farhangian 大学）
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中文总结 AI 辅助
本文在Hilbert $C^*$-模上系统建立共轭理论，引入$\sharp$-共轭并刻画其结构，推广Garcia--Putinar极分解定理，证明$C$-对称酉算子可分解为两个共轭之积。
中文摘要 AI 辅助
本文在Hilbert $C^*$-模的框架内系统开展了共轭理论的研究，将经典的复对称算子理论从Hilbert空间情形推广到更一般的设定。我们首先探讨了$C^*$-代数上$*$-共轭自同构$\sharp$的基础结构，并通过与反代数之间的同构刻画了其存在性。我们证明了Hilbert $C^*$-模$\mathscr{E}$上所有可伴共轭线性算子构成的集合$\mathbb{L}_c(\mathscr{E})$具有Hilbert $\mathbb{L}(\mathscr{E})$-模的结构。我们还引入了Hilbert $C^*$-模上$\sharp$-共轭的关键概念，并利用实形式和实$C^*$-代数完全刻画了此类模。此外，我们研究了$\sharp$-共轭对偶的几何性质。一个核心成果是发展了作用在配备$\sharp$-共轭$C$的Hilbert $C^*$-模上的半正则线性（共轭线性）算子的极分解理论，从而推广了著名的Garcia--Putinar定理[Trans. Amer. Math. Soc. 359 (2007), 3913--3931]。作为推论，我们证明了任何$C$-对称酉算子可以分解为两个$\sharp$-共轭的乘积。对替代定义的比较分析表明，要求朴素伴随条件$\langle Cx,y\rangle = \langle x,Cy\rangle^*$会迫使底层$C^*$-代数本质上是交换的，从而证明了所采用方法的合理性。我们提供了各种例子和反例来广泛说明我们的结果。
英文摘要
This work initiates a systematic development of conjugation theory within the framework of Hilbert $C^*$-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of $*$-conjugate-automorphisms $\sharp$ on $C^*$-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set $\mathbb{L}_c(\mathscr{E})$ of all adjointable conjugate-linear operators on a Hilbert $C^*$-module $\mathscr{E}$ possesses the structure of a Hilbert $\mathbb{L}(\mathscr{E})$-module. We also introduce the key notion of $\sharp$-conjugation on a Hilbert $C^*$-module and fully characterize such modules in terms of real forms and real $C^*$-algebras. Moreover, we investigate the geometric properties of $\sharp$-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert $C^*$-module equipped with a $\sharp$-conjugation $C$, thereby extending the celebrated Garcia--Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913--3931]. As a consequence, we show that any $C$-symmetric unitary operator factors as a product of two $\sharp$-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition $\langle Cx,y\rangle = \langle x,Cy\rangle^*$ would force the underlying $C^*$-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.
