The classical identification of the predual of B(H) (the algebra of all bounded operators on a Hilbert space H) with the projective operator space tensor product is extended to the context of Hilbert modules over commutative von Neumann algebras. Each bounded module homomorphism b between Hilbert modules over a general C*-algebra is shown to be completely bounded with . The so called projective operator tensor product of two operator modules X and Y over an abelian von Neumann algebra C is introduced and if Y is a Hilbert module, this product is shown to coincide with the Haagerup tensor product of X and Y over C.
@article{bwmeta1.element.bwnjournal-article-bcpv38i1p227bwm,
author = {Magajna, Bojan},
title = {Hilbert modules and tensor products of operator spaces},
journal = {Banach Center Publications},
volume = {38},
year = {1997},
pages = {227-246},
zbl = {0879.46024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p227bwm}
}
Magajna, Bojan. Hilbert modules and tensor products of operator spaces. Banach Center Publications, Tome 38 (1997) pp. 227-246. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p227bwm/
[000] [1] P. Ara and M. Mathieu, On the central Haagerup tensor product, Proc. Edinburgh Math. Soc. (2) 37 (1993), 161-174. | Zbl 0793.46036
[001] [2] D. P. Blecher, Tensor products of operator spaces II, Canad. J. Math. 44 (1992), 75-90. | Zbl 0787.46059
[002] [3] D. P. Blecher, A generalization of Hilbert modules, J. Funct. Anal. 136 (1996), 365-421. | Zbl 0951.46033
[003] [4] D. P. Blecher, On selfdual Hilbert modules, to appear in Fields Inst. Commun.
[004] [5] D. P. Blecher, P. S. Muhly and V. I. Paulsen, Categories of operator modules (Morita equivalence and projective modules), to appear in Mem. Amer. Math. Soc. | Zbl 0966.46033
[005] [6] D. P. Blecher and V. I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), 262-292. | Zbl 0786.46056
[006] [7] D. P. Blecher and R. R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. (2) 45 (1992), 126-144. | Zbl 0712.46029
[007] [8] L. G. Brown, P. Green and M. A. Rieffel, Stable isomorphisms and strong Morita equivalence of C*-algebras, Pacific J. Math. 71 (1977), 349-363. | Zbl 0362.46043
[008] [9] A. Chatterjee and A. M. Sinclair, An isometry from the Haagerup tensor product into completely bounded operators, J. Operator Theory 28 (1992), 65-78. | Zbl 0817.46053
[009] [10] A. Chatterjee and R. R. Smith, The central Haagerup tensor product and maps between von Neumann algebras, J. Funct. Anal. 112 (1993), 97-120. | Zbl 0778.46041
[010] [11] E. Christensen and A. M. Sinclair, A survey of completely bounded operators, Bull. London Math. Soc. 21 (1989), 417-448. | Zbl 0698.46044
[011] [12] E. G. Effros and R. Exel, On multilinear double commutant theorems, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser. 135 (1988), 81-94. | Zbl 0698.46049
[012] [13] E. G. Effros and A. Kishimoto, Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J. (1987), 257-276. | Zbl 0635.46062
[013] [14] E. G. Effros and Z.-J. Ruan, Representation of operator bimodules and their applications, J. Operator Theory 19 (1988), 137-157. | Zbl 0705.46026
[014] [15] E. G. Effros and Z.-J. Ruan, Self-duality for the Haagerup tensor product and Hilbert space factorizations, J. Funct. Anal. 100 (1991), 257-284. | Zbl 0761.47022
[015] [16] E. G. Effros and Z.-J. Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), 329-337. | Zbl 0769.46037
[016] [17] E. G. Effros and Z.-J. Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1993), 579-584. | Zbl 0808.46026
[017] [18] E. G. Effros and Z.-J. Ruan, Mapping spaces and liftings for operator spaces, Proc. London Math. Soc. (3) 69 (1994), 171-197. | Zbl 0814.47053
[018] [19] H. Halpern, Module homomorphisms of a von Neumann algebra into its center, Trans. Amer. Math. Soc. 140 (1969), 183-193. | Zbl 0183.42202
[019] [20] K. Jensen and K. Thomsen, Elements of KK-theory, Birkhäuser, Basel, 1991. | Zbl 1155.19300
[020] [21] R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, Vols. 1, 2, Academic Press, London, 1983, 1986. | Zbl 0518.46046
[021] [22] I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839-858. | Zbl 0051.09101
[022] [23] E. C. Lance, Hilbert C*-modules, London Math. Soc. Lecture Note Ser. 210 (1995).
[023] [24] B. Magajna, The Haagerup norm on the tensor product of operator modules, J. Funct. Anal. 129 (1995), 325-348. | Zbl 0828.46048
[024] [25] B. Magajna, Strong operator modules and the Haagerup tensor product, Proc. London Math. Soc. (3) 74 (1997), 201-240. | Zbl 0899.46036
[025] [26] G. J. Murphy, C*-algebras and operator theory, Academic Press, London, 1990.
[026] [27] W. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468. | Zbl 0239.46062
[027] [28] V. I. Paulsen, Completely bounded maps and dilations, Research Notes in Math. 146, Pitman, London, 1986. | Zbl 0614.47006
[028] [29] C. Pearcy, On unitary equivalence of matrices over the ring of continuous complex-valued functions on a Stonian space, Canad. J. Math. 15 (1963), 323-331. | Zbl 0144.37704
[029] [30] M. A. Rieffel, Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51-96. | Zbl 0295.46099
[030] [31] Z. J. Ruan, Subspaces of C*-algebras, J. Funct. Anal. 76 (1988), 217-230. | Zbl 0646.46055
[031] [32] A. M. Sinclair and R. R. Smith, Hochschild cohomology of von Neumann algebras, London Math. Soc. Lecture Note Ser. 203 (1995). | Zbl 0826.46050
[032] [33] R. R. Smith, Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal. 102 (1991), 156-175. | Zbl 0745.46060