关于End_φ(M)的一点注记

A Note on End_φ(M)

假设 M是一个具有可分预对偶的 von Neumann代数（特别是有限的 von Neumann代数）， End(M)是它的自同态半群。给 End(M)赋以 U—拓扑，我们证明了当是正规的忠实态时， End (M)是 End(M)的 U—闭子集（在序列收敛意义下）。我们还证明了 End (M)中—不变条件期望的指标是下半连续的。这推广了已有的结果。

Let M be a von Neumann algebra with separable predual (esp. finite von Neumann algebra), End(M) be its endomorphism semigroup. With End(M) in U－ topology,we proved that for a normal faithful state , End (M) is a U－ closed subset of End(M) under sequence convergence. We then show that the index of the－ invariant conditional expectation in End (M) is lower semi－ continuous.This generalized the previous results.