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The Schatten space $S_{4}$ is a $Q$-algebra

Author: Christian Le Merdy
Journal: Proc. Amer. Math. Soc. 126 (1998), 715-719
MSC (1991): Primary 47D25; Secondary 47A80, 46B70
MathSciNet review: 1468194
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Abstract: For any $1 \leq p \leq \infty $, let $S_{p}$ denote the classical $p$-Schatten space of operators on the Hilbert space $\ell _{2}$. It was shown by Varopoulos (for $p \geq 2$) and by Blecher and the author (full result) that for any $1 \leq p \leq \infty , S_{p}$ equipped with the Schur product is an operator algebra. Here we prove that $S_{4}$ (and thus $S_{p}$ for any $2 \leq p \leq 4$) is actually a $Q$-algebra, which means that it is isomorphic to some quotient of a uniform algebra in the Banach algebra sense.

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  • 1. J. Bergh, J. Löfström, Interpolation spaces, Spinger Verlag, New York, 1976. MR 58:2349
  • 2. D.P. Blecher, Tensor products of operator spaces II, Canadian J. Math 44 (1992), 75-90. MR 93e:46084
  • 3. D.P. Blecher, The standard dual of an operator space, Pacific J. Math 153 (1992), 15-30. MR 93d:47083
  • 4. D.P. Blecher, C. Le Merdy, On quotients of function algebras and operator algebra structures on $\ell _{p}$, J. Operator Theory 34 (1995), 315-346. MR 96k:46028
  • 5. D.P. Blecher, V.I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), 262-292. MR 93d:46095
  • 6. A.M. Davie, Quotient algebras of uniform algebras, London Math. Soc. 7 (1973), 31-40. MR 48:2779
  • 7. E.G. Effros, Z.-J. Ruan, A new approach to operator spaces, Canad. Math. Bull 34 (1991), 329-337. MR 93a:47045
  • 8. G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conf. Series in Math., Vol 60, 1986. MR 88a:47020
  • 9. G. Pisier, The operator Hilbert space $OH$, complex interpolation and tensor norms. Memoirs A.M.S., Vol. 122 (1996). MR 97a:46024
  • 10. N.T. Varopoulos, Some remarks on $Q$-algebras, Ann. Inst. Fourier 22 (1972), 1-11. MR 49:3544
  • 11. N.T. Varopoulos, Sur les quotients des algèbres uniformes, C. R. Acad. Sci. Paris Série I Math. 274 (1972), 1344-1346. MR 45:2480
  • 12. N.T. Varopoulos, A theorem on operator algebras, Math. Scand. 37 (1975), 173-182. MR 53:1295
  • 13. J. Wermer, Quotient algebras of uniform algebras, in Symposium on function algebras and rational approximation, University of Michigan, 1969.

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Additional Information

Christian Le Merdy
Affiliation: Equipe de Mathématiques, Université de Franche-Comté, CNRS UMR 6623, F-25030 Besancon Cedex, France

Received by editor(s): June 26, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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