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SOME REDUCED FREE PRODUCTS OF ABELIAN C*

  • Received : 2009.03.25
  • Published : 2010.09.30

Abstract

We prove that the reduced free product of $k\;{\times}\;k$ matrix algebras over abelian $C^*$-algebras is not the minimal tensor product of reduced free products of $k\;{\times}\;k$ matrix algebras over abelian $C^*$-algebras. It is shown that the reduced group $C^*$-algebra associated with a group having the property T of Kazhdan is not isomorphic to a reduced free product of abelian $C^*$-algebras or the minimal tensor product of such reduced free products. The infinite tensor product of reduced free products of abelian $C^*$-algebras is not isomorphic to the tensor product of a nuclear $C^*$-algebra and a reduced free product of abelian $C^*$-algebra. We discuss the freeness of free product $II_1$-factors and solidity of free product $II_1$-factors weaker than that of Ozawa. We show that the freeness in a free product is related to the existence of Cartan subalgebras in free product $II_1$-factors. Finally, we give a free product factor which is not solid in the weak sense.

Keywords

free product of $C^*$-algebras;Powers' group;minimal tensor product;stable rank 1;prime factor;property T;Cartan subalgebra

References

  1. D. Avitzour, Free products of $C^*$-algebras, Trans. Amer. Math. Soc. 271 (1982), no. 2, 423-435.
  2. A. Connes, A factor of type $II_1$ with countable fundamental group, J. Operator Theory 4 (1980), no. 1, 151-153.
  3. A. Connes and V. Jones, Property T for von Neumann algebras, Bull. London Math. Soc. 17 (1985), no. 1, 57-62. https://doi.org/10.1112/blms/17.1.57
  4. K. Dykema, Simplicity and the stable rank of some free product $C^*$-algebras, Trans. Amer. Math. Soc. 351 (1999), no. 1, 1-40. https://doi.org/10.1090/S0002-9947-99-02180-7
  5. L. Ge, On maximal injective subalgebras of factors, Adv. Math. 118 (1996), no. 1, 34-70. https://doi.org/10.1006/aima.1996.0017
  6. L. Ge, Applications of free entropy to finite von Neumann algebras. II, Ann. of Math. (2) 147 (1998), no. 1, 143-157. https://doi.org/10.2307/120985
  7. J. Heo, On outer automorphism groups of free product factors, Internat. J. Math. 13 (2002), no. 1, 31-41. https://doi.org/10.1142/S0129167X02001198
  8. A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, Cambridge University Press, Cambridge, 2006.
  9. N. Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111-117. https://doi.org/10.1007/BF02441087
  10. N. Ozawa and S. Popa, Some prime factorization results for type $II_1$ factors, Invent. Math. 156 (2004), no. 2, 223-234. https://doi.org/10.1007/s00222-003-0338-z
  11. S. Popa, Orthogonal pairs of *-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), no. 2, 253-268.
  12. R. Powers, Simplicity of the $C^*$-algebra associated with the free group on two generators, Duke Math. J. 42 (1975), 151-156. https://doi.org/10.1215/S0012-7094-75-04213-1
  13. S. Sakai, Asymptotically abelian $II_1$-factors, Publ. Res. Inst. Math. Sci. Ser. A 4 (1968/1969), 299-307.
  14. A. Valette, Old and new about Kazhdan’s property (T), Representations of Lie groups and quantum groups (Trento, 1993), 271-333, Pitman Res. Notes Math. Ser., 311, Longman Sci. Tech., Harlow, 1994.
  15. D. Voiculescu, Symmetries of some reduced free product $C^*$-algebras, Operator algebras and their connections with topology and ergodic theory, Lect. Notes in Math. 1132 (1985), 556-588. https://doi.org/10.1007/BFb0074909
  16. D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992.

Acknowledgement

Supported by : Korea Science and Engineering Foundation (KOSEF)