Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47C15

Retrieve articles in all journals with MSC (2010): 47C15


Additional Information

Junhao Shen
Affiliation: Department of Mathematics & Statistics, University of New Hampshire, Durham, New Hampshire 03824
Email: Junhao.Shen@unh.edu

Rui Shi
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China
Email: ruishi@dlut.edu.cn, ruishi.math@gmail.com

DOI: https://doi.org/10.1090/proc/14910
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