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arXiv 1812.04035math.OA
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C $^\ast$ -代数和von Neumann algebras的序拓扑

The order topology on duals of C$^\ast$-algebras and von Neumann algebras

Emmanuel Chetcuti, Jan Hamhalter

摘要:

对于冯·诺依曼代数$\mathcal M$,我们研究了与厄米部分$\mathcal M_*^s$和预对偶$\mathcal M_*$的区间有关的序拓扑.证明了$\mathcal M_*^s$上的序拓扑与范数诱导的拓扑是一致的。与此相反,证明了区间$[0,φ]$的序拓扑与每个$φ\in \mathcal M_*^+$的范数所诱导的序拓扑相等的条件是$\mathcal M$交换的充分必要条件.证明了若$φ$是C-$^\ast$ -代数$\mathcal A$上的有界线性泛函,则$[0,φ]$中的范数-空序列与$[0,φ]$上的关于序拓扑的空序列重合当且仅当von Neumann代数$π_φ(\mathcal A)'$是有限型的(其中$π_φ$表示相应的GNS表示).这一事实使我们能够给出有限vonNeumann代数的一个新的拓扑刻画。此外,我们还证明了对于所有不属于$I$型的C $^\ast$ -代数,对偶空间的序有界部分上的范数和序拓扑收敛到零是不等价的。

英文摘要:

For a von Neumann algebra $\mathcal M$ we study the order topology associated with the hermitian part $\mathcal M_*^s$ and to intervals of the predual $\mathcal M_*$. It is shown that the order topology on $\mathcal M_*^s$ coincides with the topology induced by the norm. In contrast to this, it is proved that the condition of having the order topology associated to the interval $[0,φ]$ equal to that induced by the norm for every $φ\in \mathcal M_*^+$, is necessary and sufficient for the commutativity of $\mathcal M$. It is also proved that if $φ$ is a positive bounded linear functional on a C$^\ast$-algebra $\mathcal A$, then the norm-null sequences in $[0,φ]$ coincide with the null sequences with respect to the order topology on $[0,φ]$ if and only if the von Neumann algebra $π_φ(\mathcal A)'$ is of finite type (where $π_φ$ denotes the corresponding GNS representation). This fact allows us to give a new topological characterization of finite von Neumann algebras. Moreover, we demonstrate that convergence to zero for norm and order topology on order-bounded parts of dual spaces are nonequivalent for all C$^\ast$-algebras that are not of Type $I$.

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