ON PRODUCTS OF SYMMETRIES IN VON NEUMANN ALGEBRAS

Article Type

Research Article

Publication Title

Journal of Operator Theory

Abstract

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 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 ).

First Page

579

Last Page

596

DOI

10.7900/jot.2022nov23.2411

Publication Date

1-1-2024

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