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Journal of Operator Theory

Volume 77, Issue 2,  Spring  2017  pp. 421-454.

Complex symmetric generators for operator algebras

Authors:  Junhao Shen (1) and Sen Zhu (2)
Author institution: (1) Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824, U.S.A. (2) Department of Mathematics, Jilin University, Changchun, 130012, P.R. China

Summary:  In this paper we explore the complex symmetric generator problem for operator algebras, that is, the problem of determining which operator algebras can be generated by a single complex symmetric operator. For type I von Neumann algebras, properly infinite von Neumann algebras and a large class of finite von Neumann algebras, we give a complete answer. The complex symmetric generator problem for a large class of $C^*$-algebras, including UHF algebras, AF algebras, irrational rotation algebras and reduced free products, is also studied.

Keywords:  complex symmetric operator, von Neumann algebra, $C^*$-algebra, anti-automorphism, generator, single generation

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