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

Author:
Yoshimichi Ueda

Journal:
Trans. Amer. Math. Soc. **355** (2003), 349-371

MSC (2000):
Primary 46L54; Secondary 37A20

Published electronically:
September 5, 2002

MathSciNet review:
1928091

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Ultra-product 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.**Lance Barnett,*Free product von Neumann algebras of type 𝐼𝐼𝐼*, Proc. Amer. Math. Soc.**123**(1995), no. 2, 543–553. MR**1224611**, 10.1090/S0002-9939-1995-1224611-7**2.**Etienne F. Blanchard and Kenneth J. Dykema,*Embeddings of reduced free products of operator algebras*, Pacific J. Math.**199**(2001), no. 1, 1–19. MR**1847144**, 10.2140/pjm.2001.199.1**3.**A. Connes,*Almost periodic states and factors of type 𝐼𝐼𝐼₁*, J. Functional Analysis**16**(1974), 415–445. MR**0358374****4.**Alain Connes,*Outer conjugacy classes of automorphisms of factors*, Ann. Sci. École Norm. Sup. (4)**8**(1975), no. 3, 383–419. MR**0394228****5.**Alain Connes,*Sur la classification des facteurs de type 𝐼𝐼*, C. R. Acad. Sci. Paris Sér. A-B**281**(1975), no. 1, Aii, A13–A15 (French, with English summary). MR**0377534****6.**A. Connes,*A factor not anti-isomorphic to itself*, Ann. Math. (2)**101**(1975), 536–554. MR**0370209****7.**A. Connes, J. Feldman, and B. Weiss,*An amenable equivalence relation is generated by a single transformation*, Ergodic Theory Dynamical Systems**1**(1981), no. 4, 431–450 (1982). MR**662736****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.**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****10.**Olli Lehto,*Quasiconformal mappings and singular integrals*, Symposia Mathematica, Vol. XVIII (Convegno sulle Transformazioni Quasiconformi e Questioni Connesse, INDAM, Rome, 1974) Academic Press, London, 1976, pp. 429–453. MR**0492241****11.**Uffe Haagerup,*An example of a nonnuclear 𝐶*-algebra, which has the metric approximation property*, Invent. Math.**50**(1978/79), no. 3, 279–293. MR**520930**, 10.1007/BF01410082**12.**H. Kosaki, Free products of measured equivalence relations, Preprint (2001).**13.**George W. Mackey,*Borel structure in groups and their duals*, Trans. Amer. Math. Soc.**85**(1957), 134–165. MR**0089999**, 10.1090/S0002-9947-1957-0089999-2**14.**Dusa McDuff,*Central sequences and the hyperfinite factor*, Proc. London Math. Soc. (3)**21**(1970), 443–461. MR**0281018****15.**Adrian Ocneanu,*Actions of discrete amenable groups on von Neumann algebras*, Lecture Notes in Mathematics, vol. 1138, Springer-Verlag, Berlin, 1985. MR**807949****16.**Sorin Popa,*Maximal injective subalgebras in factors associated with free groups*, Adv. in Math.**50**(1983), no. 1, 27–48. MR**720738**, 10.1016/0001-8708(83)90033-6**17.**Sorin Popa,*Markov traces on universal Jones algebras and subfactors of finite index*, Invent. Math.**111**(1993), no. 2, 375–405. MR**1198815**, 10.1007/BF01231293**18.**Sorin Popa,*Classification of subfactors and their endomorphisms*, CBMS Regional Conference Series in Mathematics, vol. 86, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1995. MR**1339767****19.**K. R. Parthasarathy and K. Schmidt,*On the cohomology of a hyperfinite action*, Monatsh. Math.**84**(1977), no. 1, 37–48. MR**0457680****20.**Florin Rădulescu,*An invariant for subfactors in the von Neumann algebra of a free group*, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 213–239. MR**1426841****21.**Klaus Schmidt,*Asymptotically invariant sequences and an action of 𝑆𝐿(2,𝑍) on the 2-sphere*, Israel J. Math.**37**(1980), no. 3, 193–208. MR**599454**, 10.1007/BF02760961**22.**Klaus Schmidt,*Amenability, Kazhdan’s property 𝑇, strong ergodicity and invariant means for ergodic group-actions*, Ergodic Theory Dynamical Systems**1**(1981), no. 2, 223–236. MR**661821****23.**Klaus Schmidt,*Algebraic ideas in ergodic theory*, CBMS Regional Conference Series in Mathematics, vol. 76, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR**1074576****24.**S. M. Agajan,*Characters of the representations of discrete series of the group 𝑆𝐿(2,𝐾), where 𝐾 is a discrete disconnected locally compact field, in the case of a quadratic character*, Vestnik Moskov. Univ. Ser. I Mat. Mekh.**3**(1979), 3–9, 84 (Russian, with English summary). MR**545762****25.**Yoshimichi Ueda,*Amalgamated free product over Cartan subalgebra*, Pacific J. Math.**191**(1999), no. 2, 359–392. MR**1738186**, 10.2140/pjm.1999.191.359**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, 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):
46L54,
37A20

Retrieve articles in all journals with MSC (2000): 46L54, 37A20

Additional Information

**Yoshimichi Ueda**

Affiliation:
Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

Address at time of publication:
Graduate School of Mathematics, Kyushu University, Fukuoka 810-8560, Japan

Email:
ueda@math.sci.hiroshima-u.ac.jp, ueda@math.kyushu-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-02-03100-8

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