2022/03/31 by B. V. Rajarama Bhat, Bhat, B V Rajarama, Soumyashant Nayak +3 · 1 citation
Mathematics · #46L10 #47C15 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2204.00009
openalex publication_date 2022/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathscrR be a type II1 von Neumann algebra. We show that every unitary in \mathscrR may be decomposed as the product of six symmetries (that is, self-adjoint unitaries) in \mathscrR, and every unitary in \mathscrR with finite spectrum may be decomposed as the product of four symmetries in \mathscrR. Consequently, the set of products of four symmetries in \mathscrR is norm-dense in the unitary group of \mathscrR. Furthermore, we show that the set of products of three symmetries in a von Neumann algebra \mathscrM is not norm-dense in the unitary group of \mathscrM. This strengthens a result of Halmos-Kakutani which asserts that the set of products of three symmetries in B(\mathscrH), the ring of bounded operators on a Hilbert space \mathscrH, is not the full unitary group of B(\mathscrH).