论文标题
von Neumann代数中的对称性产品
On products of symmetries in von Neumann algebras
论文作者
论文摘要
令$ \ mathscr {r} $为类型$ II_1 $ von Neumann代数。我们表明，$ \ Mathscr {r} $中的每个单一都可以被分解为$ \ Mathscr {r} $的六个对称性（即自行接个性单位）的产物，并且每个单位均以有限的频谱分解为$ nsymetries $ nsumpries $ ns $ nsyscr。因此，在$ \ mathscr {r} $的统一组中，$ \ mathscr {r} $中四个对称的产品集是标准密集的。此外，我们表明，在$ \ Mathscr {M} $的单一组中，von Neumann代数$ \ Mathscr {M} $中三个对称的产品集并非规范。这加强了Halmos-Kakutani的结果，该结果断言$ \ Mathcal {b}（\ Mathscr {h}）$的三个对称产品集，Hilbert Space $ \ Mathscr {h} $上的有限运营商的环，不是$ \ nathccr $ \ nathcal $ \ b}（Hilbert space $ \ mathscr {h} $）（H）（H）。
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 not norm-dense in the unitary group of $\mathscr{M}$. This strengthens a result of Halmos-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 $\mathcal{B}(\mathscr{H})$.
