Fullness, Connes' groups, and ultraproducts of amalgamated free products over Cartan subalgebras
Author:
Yoshimichi Ueda
Journal:
Trans. Amer. Math. Soc. 355 (2003), 349371
MSC (2000):
Primary 46L54; Secondary 37A20
Published electronically:
September 5, 2002
MathSciNet review:
1928091
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Abstract: Ultraproduct algebras associated with amalgamated free products over Cartan subalgebras are investigated. As applications, their Connes' groups are computed in terms of ergodic theory, and also we clarify what condition makes them full factors (i.e., their inner automorphism groups become closed).
 1.
L. Barnett, Free product von Neumann algebras of type III, Proc. Amer. Math. Soc., 123 (1995), 543553. MR 95c:46096
 2.
E. F. Blanchard and K. J. Dykema, Embeddings of reduced free products of operator algebras, Pacific J. Math., 199 (2001), 119. MR 2002f:46115
 3.
A. Connes, Almost periodic states and factors of type III, J. Funct. Anal., 16 (1974), 415445. MR 50:10840
 4.
A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. (4) 8 (1975), 383419. MR 52:15031
 5.
A. Connes, Sur la classification des facteurs de type II, C. R. Acad. Sci. Paris, Sér. A Math., 281 (1975), 1315. MR 51:13706
 6.
A. Connes, A factor not antiisomorphic to itself, Ann. Math. 101 (1975), 536554. MR 51:6438
 7.
A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory and Dynam. Systems, 1 (1981), 431450. MR 84h:46090
 8.
K.J. Dykema, Free products of finitedimensional and other von Neumann algebras with respect to nontracial states, in Free probability theory (Waterloo, ON, 1995), 4188, Fields Inst. Commun., 12 (1997), Amer. Math. Soc., Providence, RI.
 9.
J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, II, Trans. Amer. Math. Soc., 234 (1977), 289324, 325359. MR 98c:46131
 10.
U. Haagerup, The standard form of von Neumann algebras, Math. Scand., 37 (1975), 271283. MR 58:11387
 11.
U. Haagerup, An example of a nonnuclear algebra, which has the metric approximation property, Invent. Math., 50 (19781979), 279293. MR 80j:46094
 12.
H. Kosaki, Free products of measured equivalence relations, Preprint (2001).
 13.
G. W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc., 85 (1957), 134165. MR 19:752b
 14.
D. McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc., (3) 21 (1970), 443461. MR 43:6737
 15.
A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Math., 1138 (1985), SpringerVerlag. MR 87e:46091
 16.
S. Popa, Maximal injective subalgebras in factors associated with free groups, Adv. Math., 50 (1983), 2748. MR 85h:46084
 17.
S. Popa, Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math., 111 (1993), 375405. MR 94c:46128
 18.
S. Popa, Classification of Subfactors and Their Endomorphisms, CBMS, Regional Conference Series in Math., vol. 86, Amer. Math. Soc., (1995). MR 96d:46085
 19.
K. R. Parthasarathy and K. Schmidt, On the cohomology of a hyperfinite action, Monatsh. Math., 84 (1977), 3748. MR 56:15884
 20.
F. Radulescu, An invariant for subfactors in the von Neumann algebra of a free group, in Free probability theory (Waterloo, ON, 1995), 213239, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR 98h:46068
 21.
K. Schmidt, Asymptotically invariant sequences and an action of on the 2sphere, Israel J. of Math., 37 (1980), 193208. MR 82e:28023a
 22.
K. Schmidt, Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic groupactions, Ergodic Theory and Dynam. Systems, 1 (1981), 223236. MR 83m:43001
 23.
K. Schmidt, Algebraic Ideas in Ergodic Theory, CBMS Regional Conference Series in Mathematics, 76. Published for the Conference Board of the Mathematical Sciences, Washington, DC, Amer. Math. Soc., Providence, RI, 1990, vi+94 pp. MR 92k:28029
 24.
C. Series, An application of groupoid cohomology, Pacific J. Math., 92 (1981), No. 2, 415432. MR 82f:22014
 25.
Y. Ueda, Amalgamated free product over Cartan subalgebra, Pacific J. Math., 191, No. 2 (1999), 359392. MR 2001a:46063
 26.
Y. Ueda, Amalgamated free product over Cartan subalgebra, II, Supplementary Results and Examples, Preprint (2000).
 27.
D.V. Voiculescu, K.J. Dykema and A. Nica, Free Random Variables, CRM Monograph Series I, Amer. Math. Soc., Providence, RI, 1992. MR 94c:46133
 1.
 L. Barnett, Free product von Neumann algebras of type III, Proc. Amer. Math. Soc., 123 (1995), 543553. MR 95c:46096
 2.
 E. F. Blanchard and K. J. Dykema, Embeddings of reduced free products of operator algebras, Pacific J. Math., 199 (2001), 119. MR 2002f:46115
 3.
 A. Connes, Almost periodic states and factors of type III, J. Funct. Anal., 16 (1974), 415445. MR 50:10840
 4.
 A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. (4) 8 (1975), 383419. MR 52:15031
 5.
 A. Connes, Sur la classification des facteurs de type II, C. R. Acad. Sci. Paris, Sér. A Math., 281 (1975), 1315. MR 51:13706
 6.
 A. Connes, A factor not antiisomorphic to itself, Ann. Math. 101 (1975), 536554. MR 51:6438
 7.
 A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory and Dynam. Systems, 1 (1981), 431450. MR 84h:46090
 8.
 K.J. Dykema, Free products of finitedimensional and other von Neumann algebras with respect to nontracial states, in Free probability theory (Waterloo, ON, 1995), 4188, Fields Inst. Commun., 12 (1997), Amer. Math. Soc., Providence, RI.
 9.
 J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I, II, Trans. Amer. Math. Soc., 234 (1977), 289324, 325359. MR 98c:46131
 10.
 U. Haagerup, The standard form of von Neumann algebras, Math. Scand., 37 (1975), 271283. MR 58:11387
 11.
 U. Haagerup, An example of a nonnuclear algebra, which has the metric approximation property, Invent. Math., 50 (19781979), 279293. MR 80j:46094
 12.
 H. Kosaki, Free products of measured equivalence relations, Preprint (2001).
 13.
 G. W. Mackey, Borel structure in groups and their duals, Trans. Amer. Math. Soc., 85 (1957), 134165. MR 19:752b
 14.
 D. McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc., (3) 21 (1970), 443461. MR 43:6737
 15.
 A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Math., 1138 (1985), SpringerVerlag. MR 87e:46091
 16.
 S. Popa, Maximal injective subalgebras in factors associated with free groups, Adv. Math., 50 (1983), 2748. MR 85h:46084
 17.
 S. Popa, Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math., 111 (1993), 375405. MR 94c:46128
 18.
 S. Popa, Classification of Subfactors and Their Endomorphisms, CBMS, Regional Conference Series in Math., vol. 86, Amer. Math. Soc., (1995). MR 96d:46085
 19.
 K. R. Parthasarathy and K. Schmidt, On the cohomology of a hyperfinite action, Monatsh. Math., 84 (1977), 3748. MR 56:15884
 20.
 F. Radulescu, An invariant for subfactors in the von Neumann algebra of a free group, in Free probability theory (Waterloo, ON, 1995), 213239, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR 98h:46068
 21.
 K. Schmidt, Asymptotically invariant sequences and an action of on the 2sphere, Israel J. of Math., 37 (1980), 193208. MR 82e:28023a
 22.
 K. Schmidt, Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic groupactions, Ergodic Theory and Dynam. Systems, 1 (1981), 223236. MR 83m:43001
 23.
 K. Schmidt, Algebraic Ideas in Ergodic Theory, CBMS Regional Conference Series in Mathematics, 76. Published for the Conference Board of the Mathematical Sciences, Washington, DC, Amer. Math. Soc., Providence, RI, 1990, vi+94 pp. MR 92k:28029
 24.
 C. Series, An application of groupoid cohomology, Pacific J. Math., 92 (1981), No. 2, 415432. MR 82f:22014
 25.
 Y. Ueda, Amalgamated free product over Cartan subalgebra, Pacific J. Math., 191, No. 2 (1999), 359392. MR 2001a:46063
 26.
 Y. Ueda, Amalgamated free product over Cartan subalgebra, II, Supplementary Results and Examples, Preprint (2000).
 27.
 D.V. Voiculescu, K.J. Dykema and A. Nica, Free Random Variables, CRM Monograph Series I, Amer. Math. Soc., Providence, RI, 1992. MR 94c:46133
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Additional Information
Yoshimichi Ueda
Affiliation:
Department of Mathematics, Graduate School of Science, Hiroshima University, HigashiHiroshima, 7398526, Japan
Address at time of publication:
Graduate School of Mathematics, Kyushu University, Fukuoka 8108560, Japan
Email:
ueda@math.sci.hiroshimau.ac.jp, ueda@math.kyushuu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994702031008
PII:
S 00029947(02)031008
Received by editor(s):
October 30, 2000
Received by editor(s) in revised form:
February 7, 2002
Published electronically:
September 5, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
