Spectral gap characterization of full type III factors
2019 ◽
Vol 2019
(753)
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pp. 193-210
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Keyword(s):
Type Iii
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AbstractWe 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}}.
1969 ◽
Vol 21
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pp. 1293-1308
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2016 ◽
Vol 27
(11)
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pp. 1650091
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1987 ◽
Vol 101
(2)
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pp. 363-373
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Keyword(s):
2007 ◽
Vol 1
(3)
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pp. 367-398
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1981 ◽
Vol 1
(4)
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pp. 419-429
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