Let $G$ be a second countable locally compact Hausdorff topological group and $P$ be a closed subsemigroup of $G$ containing the identity element $e\in G$. Assume that the interior of $P$ is dense in $P$. Let $\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be a semigroup of unital normal $\ast$-endomorphisms of a von Neumann algebra $M$ with separable predual satisfying a natural measurability hypothesis. We show that $\unicode[STIX]{x1D6FC}$ is an $E_{0}$-semigroup over $P$ on $M$.

中文翻译：

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可测量的半群是连续的

让$G$是第二个可数局部紧致 Hausdorff 拓扑群，并且$P$是一个封闭的子半群$G$包含标识元素$e\in G$. 假设内部$P$密集在$P$. 让$\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$是单位正态的半群$\ast$-冯诺依曼代数的自同态$M$具有满足自然可测性假设的可分离预对偶。我们表明$\unicode[STIX]{x1D6FC}$是一个$E_{0}$-半组结束$P$在$M$.