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