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Completely bounded isomorphisms of operator algebras

Author(s): Alvaro Arias
Journal: Proc. Amer. Math. Soc. 124 (1996), 1091-1101.
MSC (1991): Primary 47D25; Secondary 46L89
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Abstract: In this paper the author proves that any two elements from one of the following classes of operators are completely isomorphic to each other.

  1. $\{VN(F_{n}):n\geq 2\}$. The $II_{1}$ factors generated by the left regular representation of the free group on $n$-generators.
  2. $\{C_{\lambda }^{*}(F_{n}):n\geq 2\}$. The reduced $C^{*}$-algebras of the free group on $n$-generators.
  3. Some ``non-commutative'' analytic spaces introduced by G. Popescu in 1991.
The paper ends with some applications to Popescu's version of von Neumann's inequality.

References:

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A. Arias and G. Popescu, Factorization and reflexivity on Fock spaces, preprint.

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D. Blecher and V. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), 262--292. MR 93d:46095

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E. Christensen and A. M. Sinclair, Completely bounded isomorphisms of injective von Neumann algebras, Proc. Edinburgh Math. Soc. (2) 32 (1989), 317--327. MR 90k:46135

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E. Effros, Advances in quantized functional analysis, Proceedings of the International Congress of Math., Berkeley, 1986, pp. 906--916. MR 89e:46064

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E. Effros and Z. J. Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), 329--337. MR 93a:47045

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A. Figa-Talamanca and M. Picardello, Harmonic analysis of free groups, Lecture notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker, New York, 1983. MR 85j:43001

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U. Haagerup and G. Pisier, Bounded linear operators between $C^{*}$-algebras, Duke Math. J. 71 (1993), 889--925. MR 94k:46112

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M. Pimsner and D. Voiculescu, $K$-groups of reduced crossed products by free groups, J. Operator Theory 8 (1982), 131--156. MR 84d:46092

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G. Pisier, The operator Hilbert space $OH$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. (to appear). CMP 95:15

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G. Popescu, von Neumann inequality for $(B(H)^{n})_{1}$, Math. Scand. 68 (1991), 292--304. MR 92k:47073

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A. G. Robertson and S. Wassermann, Completely bounded isomorphisms of injective operator systems, Bull. London Math. Soc. 21 (1989), 285--290. MR 90c:47077

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S. Sakai, $C^{*}$-algebras and $W^{*}$-algebras, Springer-Verlag, New York, 1971. MR 56:1082

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Additional Information:

Alvaro Arias
Affiliation: Division of Mathematics and Statistics, The University of Texas at San Antonio, San Antonio, Texas 78249
Email: arias@ringer.cs.utsa.edu

DOI: 10.1090/S0002-9939-96-03060-2
PII: S 0002-9939(96)03060-2
Received by editor(s): February 21, 1994
Received by editor(s) in revised form: August 19, 1994
Additional Notes: Supported in part by NSF DMS 93-21369.
Communicated by: Dale Alspach
Copyright of article: Copyright 1996, American Mathematical Society

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