Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T12:36:31.780Z Has data issue: false hasContentIssue false

Restricting a compact action to an injective subfactor

Published online by Cambridge University Press:  19 September 2008

Roberto Longo
Affiliation:
Dipartimento di Matematica, Seconda Università di Roma ‘Tor Vergata’ Via Orazio Raimondo, 00173 Rome, Italy
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose we are given an action α: G → Aut (M) of a group G on a factor M; α possible way to analyse α may be to look at the invariant components where the action becomes more tractable. This point of view naturally leads to the study of the injective invariant subalgebras (recall for instance the good properties shared by amenable discrete or compact actions in the hyperfinite case [14]).

Type
Research Article
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[1]Araki, H., Haag, R., Kastler, D. & Takesaki, M.. Extension of states and chemical potential. Commun. Math. Phys. 53 (1977), 97134.CrossRefGoogle Scholar
[2]Batty, C. J. K.. G-central subalgebras and extension of K.MS states. J. Functional Anal. 66 (1986), 1120.CrossRefGoogle Scholar
[3]Bratteli, O. & Goodman, F. M., Derivations tangential to compact group actions: spectral conditions in the weak closure. Can. J. Math. 37 (1985), 160192.CrossRefGoogle Scholar
[4]Buchholz, D., Doplicher, S. & Longo, R.. In preparation.Google Scholar
[5]Christensen, E.. Extension of derivations. J. Functional Anal. 27 (1978), 234247.CrossRefGoogle Scholar
[6]Christensen, E. & Evans, D. E.. Cohomology of operator algebras and quantum dynamical semigroups. J. London Math. Soc. 2 20 (1979), 358368.CrossRefGoogle Scholar
[7]Connes, A. & Takesaki, M.. The flow of weights on a factor of type III. Tohoku Math. J. 29 (1977), 473575.CrossRefGoogle Scholar
[8]Cuntz, J.. Simple C*-algebras generated by isometries. Commun. Math. Phys. 57 (1977), 173185.CrossRefGoogle Scholar
[9]Doplicher, S. & –Longo, R.. Standard and split inclusions of von Neumann algebras. Invent. Math. 75 (1984), 493536.CrossRefGoogle Scholar
[10]Kadison, R. V.. Derivations of operator algebras. Ann. of Math. 83 (1966), 280293.CrossRefGoogle Scholar
[11]Longo, R.. Solution of the factorial Stone-Weierstrass conjecture. Invent. Math. 76 (1984), 145155.CrossRefGoogle Scholar
[12]Longo, R.. Simple injective subfactors. Adv. in Math. 63 (1987), 152171.CrossRefGoogle Scholar
[13]Nakagami, Y. & Takesaki, M.. Duality for crossed products of von Neumann algebras. Lecture Notes in Mathematics N. 731, Springer-Verlag: Berlin-Heidelberg, 1979.CrossRefGoogle Scholar
[14]Ocneanu, A.. Actions of amenable groups on factors. Lecture Notes in Mathematics 1138, Springer-Verlag: Berlin-Heidelberg, 1980.Google Scholar
Jones, V. F. R. & Takesaki, M., Actions of compact abelian groups on semifinite factors. Ada Math. 153 (1984), 213258.Google Scholar
[15]Popa, S., Hyperfinite subalgebras normalized by a given automorphism and related problems, Lecture Notes in Mathematics, 1132, pp. 421433. Springer-Verlag. Berlin-Heidelberg, 1985.Google Scholar
[16]Roberts, J. E.. Cross products of von Neumann algebras by group duals, Simpos. Math. 20 (1976), 335363.Google Scholar
[17]Popa, S.. Maximal abelian injective subalgebras in factors associated with free groups. Adv. in Math. 50 (1983), 2748.CrossRefGoogle Scholar
[18]'Antoni, C. D. & Longo, R.. Interpolation by type I factors and the flip automorphism. J. Functional Anal. 51 (1983), 361371.CrossRefGoogle Scholar