On the classification of full factors of type III
Author:
Dimitri Shlyakhtenko
Journal:
Trans. Amer. Math. Soc. 356 (2004), 4143-4159
MSC (2000):
Primary 46L10; Secondary 46L54
DOI:
https://doi.org/10.1090/S0002-9947-04-03457-9
Published electronically:
April 16, 2004
MathSciNet review:
2058841
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We introduce a new invariant $\mathscr {S}(M)$ for type III factors $M$ with no almost-periodic weights. We compute this invariant for certain free Araki-Woods factors. We show that Connes’ invariant $\tau$ cannot distinguish all isomorphism classes of free Araki-Woods factors. We show that there exists a continuum of mutually non-isomorphic free Araki-Woods factors, each without almost-periodic weights.
- Lance Barnett, Free product von Neumann algebras of type ${\rm III}$, Proc. Amer. Math. Soc. 123 (1995), no. 2, 543–553. MR 1224611, DOI https://doi.org/10.1090/S0002-9939-1995-1224611-7
- A. Connes, Correspondences, unpublished notes.
- Alain Connes, Une classification des facteurs de type ${\rm III}$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252 (French). MR 341115
- A. Connes, Almost periodic states and factors of type ${\rm III}_{1}$, J. Functional Analysis 16 (1974), 415–445. MR 0358374, DOI https://doi.org/10.1016/0022-1236%2874%2990059-7
- ---, Noncommutative geometry, Academic Press, 1994.
- Kenneth J. Dykema, Free products of finite-dimensional and other von Neumann algebras with respect to non-tracial states, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 41–88. MR 1426835
- Kenneth J. Dykema and Florin Rădulescu, Compressions of free products of von Neumann algebras, Math. Ann. 316 (2000), no. 1, 61–82. MR 1735079, DOI https://doi.org/10.1007/s002080050004
- Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 238, Springer-Verlag, New York-Berlin, 1979. MR 550606
- Uffe Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), no. 2, 271–283. MR 407615, DOI https://doi.org/10.7146/math.scand.a-11606
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
- S. Popa, Correspondences, INCREST preprint, 1986.
- S. Popa and D. Shlyakhtenko, Universal properties of $L(\mathbb {F}_\infty )$ in subfactor theory, MSRI preprint 2000-032, to appear in Acta Math.
- Florin Rădulescu, A one-parameter group of automorphisms of ${\scr L}(F_\infty )\otimes B(H)$ scaling the trace, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 13, 1027–1032 (English, with French summary). MR 1168529
- Florin Rădulescu, A type ${\rm III}_\lambda $ factor with core isomorphic to the von Neumann algebra of a free group, tensor $B(H)$, Astérisque 232 (1995), 203–209. Recent advances in operator algebras (Orléans, 1992). MR 1372534
- Dimitri Shlyakhtenko, Free quasi-free states, Pacific J. Math. 177 (1997), no. 2, 329–368. MR 1444786, DOI https://doi.org/10.2140/pjm.1997.177.329
- Dimitri Shlyakhtenko, Some applications of freeness with amalgamation, J. Reine Angew. Math. 500 (1998), 191–212. MR 1637501, DOI https://doi.org/10.1515/crll.1998.066
- Dimitri Shlyakhtenko, $A$-valued semicircular systems, J. Funct. Anal. 166 (1999), no. 1, 1–47. MR 1704661, DOI https://doi.org/10.1006/jfan.1999.3424
- ---, Microstates free entropy and cost of equivalence relations, Duke Math. J. 118 (2003), 375–425.
- Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. II, Invent. Math. 118 (1994), no. 3, 411–440. MR 1296352, DOI https://doi.org/10.1007/BF01231539
- D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. III. The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), no. 1, 172–199. MR 1371236, DOI https://doi.org/10.1007/BF02246772
- Dan Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy, Internat. Math. Res. Notices 1 (1998), 41–63. MR 1601878, DOI https://doi.org/10.1155/S107379289800004X
- D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L10, 46L54
Retrieve articles in all journals with MSC (2000): 46L10, 46L54
Additional Information
Dimitri Shlyakhtenko
Affiliation:
Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095
MR Author ID:
606307
ORCID:
0000-0002-0221-7508
Email:
shlyakht@math.ucla.edu
Received by editor(s):
July 21, 2002
Received by editor(s) in revised form:
July 17, 2003
Published electronically:
April 16, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
