On function and operator modules

Blecher, David; Le Merdy, Christian

doi:10.1090/S0002-9939-00-05866-4

On function and operator modules
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by David Blecher and Christian Le Merdy PDF

Proc. Amer. Math. Soc. 129 (2001), 833-844 Request permission

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.

References

Erik M. Alfsen and Edward G. Effros, Structure in real Banach spaces. I, II, Ann. of Math. (2) 96 (1972), 98–128; ibid. (2) 96 (1972), 129–173. MR 352946, DOI 10.2307/1970895
Ehrhard Behrends, $M$-structure and the Banach-Stone theorem, Lecture Notes in Mathematics, vol. 736, Springer, Berlin, 1979. MR 547509
Ehrhard Behrends, Multiplier representations and an application to the problem whether $A\otimes _{\varepsilon }X$ determines $A$ and/or $X$, Math. Scand. 52 (1983), no. 1, 117–144. MR 697504, DOI 10.7146/math.scand.a-11999
David P. Blecher, Commutativity in operator algebras, Proc. Amer. Math. Soc. 109 (1990), no. 3, 709–715. MR 1009985, DOI 10.1090/S0002-9939-1990-1009985-X
David P. Blecher, Tensor products of operator spaces. II, Canad. J. Math. 44 (1992), no. 1, 75–90. MR 1152667, DOI 10.4153/CJM-1992-004-5
David P. Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), no. 1, 15–30. MR 1145913
David P. Blecher, A generalization of Hilbert modules, J. Funct. Anal. 136 (1996), no. 2, 365–421. MR 1380659, DOI 10.1006/jfan.1996.0034
D.P. Blecher, The Shilov boundary of an operator space - and the characterization theorems, Preprint (1999).
David P. Blecher and Vern I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), no. 2, 262–292. MR 1121615, DOI 10.1016/0022-1236(91)90042-4
David P. Blecher, Zhong-Jin Ruan, and Allan M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990), no. 1, 188–201. MR 1040962, DOI 10.1016/0022-1236(90)90010-I
Erik Christensen, Edward G. Effros, and Allan Sinclair, Completely bounded multilinear maps and $C^\ast$-algebraic cohomology, Invent. Math. 90 (1987), no. 2, 279–296. MR 910202, DOI 10.1007/BF01388706
Erik Christensen and Allan M. Sinclair, Representations of completely bounded multilinear operators, J. Funct. Anal. 72 (1987), no. 1, 151–181. MR 883506, DOI 10.1016/0022-1236(87)90084-X
F. Cunningham Jr., $M$-structure in Banach spaces, Proc. Cambridge Philos. Soc. 63 (1967), 613–629. MR 212544, DOI 10.1017/s0305004100041591
Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
Edward G. Effros and Zhong-Jin Ruan, Representations of operator bimodules and their applications, J. Operator Theory 19 (1988), no. 1, 137–158. MR 950830
Edward G. Effros and Zhong-Jin Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), no. 3, 329–337. MR 1127754, DOI 10.4153/CMB-1991-053-x
E.G. Effros and Z.-J. Ruan, Operator convolution algebras: an approach to quantum groups, Unpublished (1991).
Krzysztof Jarosz, Multipliers in complex Banach spaces and structure of the unit balls, Studia Math. 87 (1987), no. 3, 197–213. MR 927504, DOI 10.4064/sm-87-3-197-213
C. Le Merdy, An operator space characterization of dual operator algebras, Amer. J. Math. 121 (1999), 55-63.
C. Le Merdy, Finite rank approximation and semidiscreteness for linear operators, Annales Inst. Fourier 49 (1999), 1869–1901.
V. I. Paulsen and R. R. Smith, Multilinear maps and tensor norms on operator systems, J. Funct. Anal. 73 (1987), no. 2, 258–276. MR 899651, DOI 10.1016/0022-1236(87)90068-1
Joseph A. Ball and Thomas L. Kriete III, Operator-valued Nevanlinna-Pick kernels and the functional models for contraction operators, Integral Equations Operator Theory 10 (1987), no. 1, 17–61. MR 868573, DOI 10.1007/BF01199793
G. Pisier, An introduction to the theory of operator spaces, Preprint (1997).
Andrew M. Tonge, Banach algebras and absolutely summing operators, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 3, 465–473. MR 438152, DOI 10.1017/S030500410005310X