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Sum of irreducible operators in von Neumann factors


Authors: Junhao Shen and Rui Shi
Journal: Proc. Amer. Math. Soc. 148 (2020), 2901-2908
MSC (2010): Primary 47C15
DOI: https://doi.org/10.1090/proc/14910
Published electronically: February 4, 2020
MathSciNet review: 4099778
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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

Keywords: Factor von Neumann alegbras, irreducible operators
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
Article copyright: © Copyright 2020 American Mathematical Society