论文标题
冯·诺伊曼（Von Neumann）代数的统一组几乎且几乎是周期性的功能
Almost and weakly almost periodic functions on the unitary groups of von Neumann algebras
论文作者
论文摘要
令$ M \子集B（\ Mathcal H）$为von Neumann代数，作用于Hilbert Space $ \ Mathcal H $。我们证明$ m $是且仅在M $中的每一个$ x \时才有限的，对于所有向量$ξ，η\ in \ Mathcal h $，系数函数$ \ u \ mapsto \ langle uxu^*exu^*ξ|η\ h \ rangle $几乎是在toopological $ u _ m $ $ u _ m $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $（MON）上的（M $ $ $ $ $）。主设备是$ c^*$ - 代数$ \ permatatorName {wap}（u_m）$的唯一不变式均值，$ u_m $上的薄弱的周期性功能。接下来，我们证明每个系数函数$ u \ mapsto \ langle uxu^*ξ|η\ rangle $几乎是周期性的，并且仅当$ m $是弥漫性的直接总和，阿贝尔von neumann代数和有限量比因素。顺便说一句，我们证明，如果$ m $是弥漫性的von Neumann代数，那么其统一组几乎是周期性的。
Let $M\subset B(\mathcal H)$ be a von Neumann algebra acting on the Hilbert space $\mathcal H$. We prove that $M$ is finite if and only if, for every $x\in M$ and for all vectors $ξ,η\in\mathcal H$, the coefficient function $u\mapsto \langle uxu^*ξ|η\rangle$ is weakly almost periodic on the topological group $U_M$ of unitaries in $M$ (equipped with the weak or strong operator topology). The main device is the unique invariant mean on the $C^*$-algebra $\operatorname{WAP}(U_M)$ of weakly almost periodic functions on $U_M$. Next, we prove that every coefficient function $u\mapsto \langle uxu^*ξ|η\rangle$ is almost periodic if and only if $M$ is a direct sum of a diffuse, abelian von Neumann algebra and finite-dimensional factors. Incidentally, we prove that if $M$ is a diffuse von Neumann algebra, then its unitary group is minimally almost periodic.
