Journal of Operator Theory

Volume 92, Issue 2, Autumn 2024 pp. 579-596.

On products of symmetries in von Neumann algebras

Authors: B.V. Rajarama Bhat (1), Soumyashant Nayak (2), and P. Shankar (3)
Author institution: (1) Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bengaluru, Karnataka - 560059, India
(2) Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bengaluru, Karnataka - 560059, India
(3) Department of Mathematics, Cochin University of Science and Technology, Kochi, Kerala - 682022, India

Summary: Let $\mathscr{R}$ be a type II$_1$ von Neumann algebra. We show that every unitary in $\mathscr{R}$ may be decomposed as the product of six symmetries (that is, self-adjoint unitaries) in $\mathscr{R}$, and every unitary in $\mathscr{R}$ with finite spectrum may be decomposed as the product of four symmetries in $\mathscr{R}$. Consequently, the set of products of four symmetries in $\mathscr{R}$ is norm-dense in the unitary group of $\mathscr{R}$. Furthermore, we show that the set of products of three symmetries in a von Neumann algebra $\mathscr{M}$ is {\bf not} norm-dense in the unitary group of $\mathscr{M}$. This strengthens a result of Halmos and Kakutani which asserts that the set of products of three symmetries in $\mathcal{B}(\mathscr{H})$, the ring of bounded operators on a Hilbert space $\mathscr{H}$, is not the full unitary group of $\mathcal{B}(\mathscr{H})$.

DOI: http://dx.doi.org/10.7900/jot.2022nov23.2411
Keywords: Reflections in von Neumann algebras, products of unitaries, Halmos--Kakutani theorem, Cartan-Hadamard theorem

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