复Hilbert空间上的实算子代数

李炳仁

数学学报 ›› 2009, Vol. 52 ›› Issue (6) : 1041-1046.

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PDF(401 KB)
数学学报 ›› 2009, Vol. 52 ›› Issue (6) : 1041-1046. DOI: 10.12386/A2009sxxb0131
论文

复Hilbert空间上的实算子代数

    李炳仁
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Real Operator Algebras on a Complex Hilbert Space

    Bing Ren LI
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摘要

本文研究在一个复Hilbert空间H上的实算子代数.从H可以得到一个实Hilbert空间Hr,进而又有一个复Hilbert空间 Hrc=Hr Hr.通过这个过程, 证明了如下结果.如果A, M分别是H上一致闭的,弱闭的实*算子代数, 则它们的复 扩张A+iA, M+iM分别是H上的(复) C*-代数, (复) von Neumann代数.这里, 不需要条件A∩iA={0}, M∩iM={0}. 因此, 我们的结果是Stormer结果的推广.  

Abstract

We study real operator algebras on a complex Hilbert space H. From H, we can get a real Hilbert space Hr. Further, we have a complex Hilbert space Hrc=Hr iHr. Through this process, we prove the following. If A and M are uniformly closed and weakly closed real * operator algebras on H respectively, then their complex span A+iA and M+iM are (complex) C*-algebra and (complex) von Neumann algebra on H, respectively. Here, we don't need the condition: A∩iA={0}, M∩iM={0}. So our result is a generalization of Stormer's result.  

关键词

复Hilbert空间的实化 / 实Hilbert空间的复化 / 实算子代数 / 实von Neumann代数

Key words

realification of complex Hilbert space / complexification of real Hilbert space / real operator algebra / real von Neumann algebra

引用本文

导出引用
李炳仁. 复Hilbert空间上的实算子代数. 数学学报, 2009, 52(6): 1041-1046 https://doi.org/10.12386/A2009sxxb0131
Bing Ren LI. Real Operator Algebras on a Complex Hilbert Space. Acta Mathematica Sinica, Chinese Series, 2009, 52(6): 1041-1046 https://doi.org/10.12386/A2009sxxb0131

参考文献



[1] Li B. R., Real Operator Algebras, Singapore: World Scientific, 2003.



[2] Stormer E., Real structure in the Hyperfinite Factor, Duke Math. J., 1980, 47(1): 145--153.



[3] Stormer E., Irreducible jordan algebras of Self-Adjoint operators, Trans. AMS, 1968, 130(1): 153--166.



[4] Li B. R., Introduction to Operator Algebras, Singapore: World Scientific, 1992.
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