Spectral gap characterization of full type III factors
Spectral gap characterization of full type III factors
复制标题
全 III 类因子的光谱间隙表征
DOI：
10.1515/crelle-2016-0071
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
A. Marrakchi
中科院分区：
文献类型：
--
作者：
A. Marrakchi
我们给出了{\mathm{III}}型因子的满度的谱间隙刻画，它类似于示踪情形下的Connes定理。利用这个判据，我们推广了Jones的一个定理，证明了如果M是一个全因子，且{\sigma：G\right tarrow\mathm{aut}(M)}是一个离散群G的外作用，而它在{\mathm{out}(M)}中的像是离散的，则交叉积von Neumann代数{M\rTimes_{\sigma}G}也是一个全因子.我们利用这一结果证明了Tomatsu-Ueda的如下猜想：{\mathm{III}_{1}}型因子M的连续核是满的当且仅当M是满的，且它的τ不变量是{\mathbb{R}}上的通常拓扑。
We give a spectral gap characterization of fullness for type {\mathrm{III}} factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if M is a full factor and {\sigma:G\rightarrow\mathrm{Aut}(M)} is an outer action of a discrete group G whose image in {\mathrm{Out}(M)} is discrete, then the crossed product von Neumann algebra {M\rtimes_{\sigma}G} is also a full factor. We apply this result to prove the following conjecture of Tomatsu–Ueda: the continuous core of a type {\mathrm{III}_{1}} factor M is full if and only if M is full and its τ invariant is the usual topology on {\mathbb{R}}.