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 R be a type II1 von Neumann algebra. We show that every unitary in R may be decomposed as the product of six symmetries (that is, self-adjoint unitaries) in R, and every unitary in R with finite spectrum may be decomposed as the product of four symmetries in R. Consequently, the set of products of four symmetries in R is norm-dense in the unitary group of R. Furthermore, we show that the set of products of three symmetries in a von Neumann algebra M is {\bf not} norm-dense in the unitary group of M. This strengthens a result of Halmos and Kakutani which asserts that the set of products of three symmetries in B(H), the ring of bounded operators on a Hilbert space H, is not the full unitary group of B(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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