Transactions of the American Mathematical Society

Fullness, Connes' $\chi$ -groups, and ultra-products of amalgamated free products over Cartan subalgebras

Author(s): Yoshimichi Ueda
Journal: Trans. Amer. Math. Soc. 355 (2003), 349-371.
MSC (2000): Primary 46L54; Secondary 37A20
Posted: September 5, 2002
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Abstract: Ultra-product algebras associated with amalgamated free products over Cartan subalgebras are investigated. As applications, their Connes' $\chi$ -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), 543-553. MR 95c:46096
2.: E. F. Blanchard and K. J. Dykema, Embeddings of reduced free products of operator algebras, Pacific J. Math., 199 (2001), 1-19. MR 2002f:46115
3.: A. Connes, Almost periodic states and factors of type III $_{1}$ , J. Funct. Anal., 16 (1974), 415-445. MR 50:10840
4.: A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. (4) 8 (1975), 383-419. MR 52:15031
5.: A. Connes, Sur la classification des facteurs de type II, C. R. Acad. Sci. Paris, Sér. A Math., 281 (1975), 13-15. MR 51:13706
6.: A. Connes, A factor not anti-isomorphic to itself, Ann. Math. 101 (1975), 536-554. 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), 431-450. MR 84h:46090
8.: K.-J. Dykema, Free products of finite-dimensional and other von Neumann algebras with respect to non-tracial states, in Free probability theory (Waterloo, ON, 1995), 41-88, 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), 289-324, 325-359. MR 98c:46131
10.: U. Haagerup, The standard form of von Neumann algebras, Math. Scand., 37 (1975), 271-283. MR 58:11387
11.: U. Haagerup, An example of a non-nuclear $C^{*}$ -algebra, which has the metric approximation property, Invent. Math., 50 (1978-1979), 279-293. 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), 134-165. MR 19:752b
14.: D. McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc., (3) 21 (1970), 443-461. MR 43:6737
15.: A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Math., 1138 (1985), Springer-Verlag. MR 87e:46091
16.: S. Popa, Maximal injective subalgebras in factors associated with free groups, Adv. Math., 50 (1983), 27-48. MR 85h:46084
17.: S. Popa, Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math., 111 (1993), 375-405. 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), 37-48. 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), 213-239, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR 98h:46068
21.: K. Schmidt, Asymptotically invariant sequences and an action of $SL(2,{\mathbb{Z} })$ on the 2-sphere, Israel J. of Math., 37 (1980), 193-208. MR 82e:28023a
22.: K. Schmidt, Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions, Ergodic Theory and Dynam. Systems, 1 (1981), 223-236. 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, 415-432. MR 82f:22014
25.: Y. Ueda, Amalgamated free product over Cartan subalgebra, Pacific J. Math., 191, No. 2 (1999), 359-392. 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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