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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sum of irreducible operators in von Neumann factors
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by Junhao Shen and Rui Shi PDF
Proc. Amer. Math. Soc. 148 (2020), 2901-2908 Request permission

Abstract:

Let $\mathcal {M}$ be a factor acting on a complex, separable Hilbert space $\mathcal {H}$. An operator $a\in \mathcal {M}$ is said to be irreducible in $\mathcal {M}$ if $W^*(a)$, the von Neumann subalgebra generated by $a$ in $\mathcal M$, is an irreducible subfactor of $\mathcal {M}$, i.e., $W^*(a)’\cap \mathcal {M}=\mathbb {C} I$. In this note, we prove that each operator $a\in \mathcal {M}$ is a sum of two irreducible operators in $\mathcal {M}$, which can be viewed as a natural generalization of a theorem in [Proc. Amer. Math. Soc. 21 (1969), pp. 251–252], with a completely different proof.
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Additional Information
  • Junhao Shen
  • Affiliation: Department of Mathematics & Statistics, University of New Hampshire, Durham, New Hampshire 03824
  • MR Author ID: 626774
  • Email: Junhao.Shen@unh.edu
  • Rui Shi
  • Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China
  • MR Author ID: 856944
  • Email: ruishi@dlut.edu.cn, ruishi.math@gmail.com
  • Received by editor(s): August 6, 2019
  • Received by editor(s) in revised form: October 19, 2019
  • Published electronically: February 4, 2020
  • Additional Notes: The second author was partly supported by NSFC (Grant No.11871130) and the Fundamental Research Funds for the Central Universities (Grant No.DUT18LK23)
    The second author is the corresponding author
  • Communicated by: Stephan Ramon Garcia
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 2901-2908
  • MSC (2010): Primary 47C15
  • DOI: https://doi.org/10.1090/proc/14910
  • MathSciNet review: 4099778