A. Connes and V. Jones :
“A \( \mathrm{II}_1 \) factor with two nonconjugate Cartan subalgebras Bull. Am. Math. Soc. (N.S.)
6 : 2
(1982 ),
pp. 211–212 .
MR
640947 Zbl
0501.46056 article
Abstract
People
BibTeX
A maximal abelian subalgebra \( A \) of the von Neumann algebra \( M \) is called a Cartan subalgebra if the normalizer
\[ N(A) = \{\text{unitaries } u\in M \text{ with } uAu^* = A\} \]
generates \( M \) as a von Neumann algebra (see [Feldman and Moore 1977]). It is a corollary of the paper by Connes, Feldman and Weiss [1982] that any two Cartan subalgebras of the hyperfinite type \( \mathrm{II}_1 \) factor \( R \) are conjugate by an automorphism of \( R \) . In this paper we construct the first example of a (separable) \( \mathrm{II}_1 \) factor with two nonconjugate Cartan subalgebras, showing that the correspondence between ergodic theory (up to orbit equivalence) and von Neumann algebras is many to one.
@article {key640947m,
AUTHOR = {Connes, A. and Jones, V.},
TITLE = {A \$\mathrm{II}_1\$ factor with two nonconjugate
{C}artan subalgebras},
JOURNAL = {Bull. Am. Math. Soc. (N.S.)},
FJOURNAL = {Bulletin of the American Mathematical
Society. New Series},
VOLUME = {6},
NUMBER = {2},
YEAR = {1982},
PAGES = {211--212},
DOI = {10.1090/S0273-0979-1982-14981-3},
NOTE = {MR:640947. Zbl:0501.46056.},
ISSN = {0273-0979},
}
V. Jones and A. Connes :
“Property \( T \) for von Neumann algebras Bull. Lond. Math. Soc.
17 : 1
(January 1985 ),
pp. 57–62 .
MR
766450 Zbl
1190.46047 article
People
BibTeX
@article {key766450m,
AUTHOR = {Jones, V. and Connes, A.},
TITLE = {Property \$T\$ for von {N}eumann algebras},
JOURNAL = {Bull. Lond. Math. Soc.},
FJOURNAL = {Bulletin of the London Mathematical
Society},
VOLUME = {17},
NUMBER = {1},
MONTH = {January},
YEAR = {1985},
PAGES = {57--62},
DOI = {10.1112/blms/17.1.57},
NOTE = {MR:766450. Zbl:1190.46047.},
ISSN = {0024-6093},
}
A. Connes :
“Seminar Bourbaki, vol. 1984/85 Index of subfactors, Hecke algebras and knot theory (after Vaughan Jones) ],
pp. 289–308 .
Astérisque 133–134 .
Société Mathématique de France (Paris ),
1986 .
Exposé no. 647.
MR
837226 Zbl
0597.57005 incollection
People
BibTeX
@incollection {key837226m,
AUTHOR = {Connes, A.},
TITLE = {Seminar {B}ourbaki, vol. 1984/85 [Index
of subfactors, {H}ecke algebras and
knot theory (after {V}aughan {J}ones)]},
SERIES = {Ast\'erisque},
NUMBER = {133--134},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {1986},
PAGES = {289--308},
URL = {http://www.numdam.org/article/SB_1984-1985__27__289_0.pdf},
NOTE = {Expos\'e no. 647. MR:837226. Zbl:0597.57005.},
ISSN = {0303-1179},
}
A. Connes, G. Faltings, V. Jones, S. Smale, R. Thom, and J. Wagensberg :
“Round-table discussion 87–108
in
Mathematical research today and tomorrow
(Barcelona, 1991 ),
vol. 1525 .
Lecture Notes in Math.
Springer, Berlin ,
1992 .
MR
1247058 incollection
People
BibTeX
@incollection {key1247058m,
AUTHOR = {Connes, Alain and Faltings, Gerd and
Jones, Vaughan and Smale, Stephen and
Thom, Ren\'{e} and Wagensberg, Jorge},
TITLE = {Round-table discussion},
BOOKTITLE = {Mathematical research today and tomorrow},
VOLUME = {1525},
SERIES = {Lecture Notes in Math.},
PUBLISHER = {Springer, Berlin},
YEAR = {1992},
PAGES = {87--108},
DOI = {10.1007/BFb0089208},
URL = {https://doi.org/10.1007/BFb0089208},
NOTE = {(Barcelona, 1991). MR:1247058.},
}
A. Connes, M. Flato, H. Hironaka, A. Jaffe, and V. Jones :
“Editorial (dedication of this issue to Huzihiro Araki) Comm. Math. Phys.
155 : 1
(1993 ),
pp. 1–2 .
MR
1228522 article
People
BibTeX
@article {key1228522m,
AUTHOR = {Connes, Alain and Flato, Mosh\'{e} and
Hironaka, Heisuke and Jaffe, Arthur
and Jones, Vaughan},
TITLE = {Editorial (dedication of this issue
to {H}uzihiro {A}raki)},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {155},
NUMBER = {1},
YEAR = {1993},
PAGES = {1--2},
URL = {http://projecteuclid.org/euclid.cmp/1104253196},
NOTE = {MR:1228522.},
ISSN = {0010-3616},
}
V. Jones and H. Moscovici :
“Book review: Alain Connes, ‘Noncommutative geometry’ ,”
Notices Am. Math. Soc.
44 : 7
(1997 ),
pp. 792–799 .
MR
1460207 Zbl
0908.46041 article
People
BibTeX
@article {key1460207m,
AUTHOR = {Jones, Vaughan and Moscovici, Henri},
TITLE = {Book review: {A}lain {C}onnes, ``{N}oncommutative
geometry''},
JOURNAL = {Notices Am. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical
Society},
VOLUME = {44},
NUMBER = {7},
YEAR = {1997},
PAGES = {792--799},
NOTE = {MR:1460207. Zbl:0908.46041.},
ISSN = {0002-9920},
}
A. Guionnet, V. F. R. Jones, and D. Shlyakhtenko :
“Random matrices, free probability, planar algebras and subfactors ,”
pp. 201–239
in
Quanta of maths: Conference on non commutative geometry in honor of Alain Connes
(Paris, 29 March–6 April 2007 ).
Edited by E. Blanchard, D. Ellwood, M. Khalkhali, M. Marcolli, H. Moscovici, and S. Popa .
Clay Mathematics Proceedings 11 .
American Mathematical Society (Providence, RI ),
2010 .
MR
2732052 Zbl
1219.46057 ArXiv
0712.2904 incollection
Abstract
People
BibTeX
Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated von Neumann algebras are \( \mathrm{II}_1 \) factors whose inclusions realize the given planar algebra as a system of higher relative commutants. We thus give an alternative proof to a result of Popa that every planar algebra can be realized by a subfactor.
@incollection {key2732052m,
AUTHOR = {Guionnet, A. and Jones, V. F. R. and
Shlyakhtenko, D.},
TITLE = {Random matrices, free probability, planar
algebras and subfactors},
BOOKTITLE = {Quanta of maths: {C}onference on non
commutative geometry in honor of {A}lain
{C}onnes},
EDITOR = {Blanchard, Etienne and Ellwood, David
and Khalkhali, Masoud and Marcolli,
Matilde and Moscovici, Henri and Popa,
Sorin},
SERIES = {Clay Mathematics Proceedings},
NUMBER = {11},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2010},
PAGES = {201--239},
NOTE = {(Paris, 29 March--6 April 2007). ArXiv:0712.2904.
MR:2732052. Zbl:1219.46057.},
ISSN = {1534-6455},
ISBN = {9780821852033},
}
A. Connes, V. Jones, M. Musat, and M. Rørdam :
“Uffe Haagerup — in memoriam Notices Am. Math. Soc.
63 : 1
(January 2016 ),
pp. 48–49 .
MR
3410853 Zbl
1338.46005 article
People
BibTeX
@article {key3410853m,
AUTHOR = {Connes, Alain and Jones, Vaughan and
Musat, Magdalena and R\o rdam, Mikael},
TITLE = {Uffe {H}aagerup---in memoriam},
JOURNAL = {Notices Am. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical
Society},
VOLUME = {63},
NUMBER = {1},
MONTH = {January},
YEAR = {2016},
PAGES = {48--49},
DOI = {10.1090/noti1303},
NOTE = {MR:3410853. Zbl:1338.46005.},
ISSN = {0002-9920},
}
