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On function and operator modules

Authors: David Blecher and Christian Le Merdy
Journal: Proc. Amer. Math. Soc. 129 (2001), 833-844
MSC (2000): Primary 47L30, 47L25; Secondary 46H25, 46J10, 46L07
Published electronically: August 30, 2000
MathSciNet review: 1802002
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Abstract: Let $A$ be a unital Banach algebra. We give a characterization of the left Banach $A$-modules $X$ for which there exists a commutative unital $C^{*}$-algebra $C(K)$, a linear isometry $i\colon X\to C(K)$, and a contractive unital homomorphism $\theta \colon A\to C(K)$ such that $i(a\cdotp x) =\theta (a)i(x)$ for any $a\in A, x\in X$. We then deduce a ``commutative" version of the Christensen-Effros-Sinclair characterization of operator bimodules. In the last section of the paper, we prove a $w^{*}$-version of the latter characterization, which generalizes some previous work of Effros and Ruan.

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  • [AE] E.M. Alfsen and E.G. Effros, Structure in real Banach spaces I, II, Annals of Math. 96 (1972), 98-173. MR 50:5432
  • [B1] E. Behrends, M-structure and the Banach-Stone theorem, Springer Lecture Notes 736, Berlin-Heidelberg-New-York, 1979. MR 81b:46002
  • [B2] E. Behrends, Multiplier representations and an application to the problem whether $A\otimes _{\varepsilon } X$ determines $A$ and/or $X$, Math. Scand. 52 (1983), 117-144. MR 84m:46050
  • [Bl1] D.P. Blecher, Commutativity in operator algebras, Proc. Amer. Math. Soc 109 (1990), 709-715. MR 90k:46128
  • [Bl2] D.P. Blecher, Tensor products of operator spaces II, Canad. J. Math. 44 (1992), 75-90. MR 93e:46084
  • [Bl3] D.P. Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), 15-30. MR 93d:47083
  • [Bl4] D.P. Blecher, A generalization of Hilbert modules, J. Funct. Anal. 136 (1996), 365-421. MR 97g:46071
  • [Bl5] D.P. Blecher, The Shilov boundary of an operator space - and the characterization theorems, Preprint (1999).
  • [BP] D.P. Blecher and V.I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), 262-292. MR 93d:46095
  • [BRS] D.P. Blecher, Z.-J. Ruan and A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990), 188-201. MR 91b:47098
  • [CES] E. Christensen, E.G. Effros and A.M. Sinclair, Completely bounded multilinear maps and $C^{*}$-algebraic cohomology, Invent. Math. 90 (1987), 279-296. MR 89k:46084
  • [CS] E. Christensen and A.M. Sinclair, Representation of completely bounded multilinear operators, J. Funct. Anal. 72 (1987), 151-181. MR 89f:46113
  • [Cu] F. Cunningham, M-structure in Banach spaces, Math. Proc. Camb. Philos. Soc. 63 (1967), 613-629. MR 35:3415
  • [DF] A. Defant and K. Floret, Tensor norms and operator ideals, North-Holland, Amsterdam, 1993. MR 94e:46130
  • [ER1] E.G. Effros and Z.-J. Ruan, Representations of operator bimodules and their applications, J. Operator Theory 19 (1988), 137-157. MR 91e:46077
  • [ER2] E.G. Effros and Z.-J. Ruan, A new approach to operator spaces, Canadian Math. Bull. 34 (1991), 329-337. MR 93a:47045
  • [ER3] E.G. Effros and Z.-J. Ruan, Operator convolution algebras: an approach to quantum groups, Unpublished (1991).
  • [J] K. Jarosz, Multipliers in complex Banach spaces and structure of the unit balls, Studia Math. T. LXXXVII (1987), 197-213. MR 89j:46017
  • [L1] C. Le Merdy, An operator space characterization of dual operator algebras, Amer. J. Math. 121 (1999), 55-63. CMP 99:16
  • [L2] C. Le Merdy, Finite rank approximation and semidiscreteness for linear operators, Annales Inst. Fourier 49 (1999), 1869-1901. CMP 2000:08
  • [PS] V.I. Paulsen and R.R. Smith, Multilinear maps and tensor norms on operator systems, J. Funct. Anal. 73 (1987), 258-276. MR 89m:46099
  • [Pi1] G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Series 60 (Amer. Math. Soc., Providence, R.I.), 1986. MR 88a:47013
  • [Pi2] G. Pisier, An introduction to the theory of operator spaces, Preprint (1997).
  • [T] A.M. Tonge, Banach algebras and absolutely summing operators, Math. Proc. Cambridge Philos. Soc. 80 (1976), 465-473. MR 55:11071

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Additional Information

David Blecher
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3476

Christian Le Merdy
Affiliation: Département de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France

Received by editor(s): May 24, 1999
Published electronically: August 30, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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