ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES

Title & Authors
ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES
Itoh, Takashi; Nagisa, Masaru;

Abstract
We describe the Haagerup tensor product $\small{{\ell}^{\infty}{\otimes}_h{\ell}^{\infty}}$ and the extended Haagerup tensor product $\small{{\ell}^{\infty}{\otimes}_{eh}{\ell}^{\infty}}$ in terms of Schur product maps, and show that $\small{{\ell}^{\infty}{\otimes}_h{\ell}^{\infty}{\cap}\mathbb{B}({\ell}^2)}$(resp. $\small{{\ell}^{\infty}{\otimes}_{eh}{\ell}^{\infty}{\cap}\mathbb{B}({\ell}^2)}$) coincides with $\small{c_0{\otimes}_hc_0{\cap}\mathbb{B}({\ell}^2)}$(resp. $\small{c_0{\otimes}_{eh}c_0{\cap}\mathbb{B}({\ell}^2)}$). For $\small{C^*2}$-algebras A, B, it is shown that $A{\otimes}_hB Keywords operator space;Haagerup tensor product;extended Haagerup tensor product;Schur product; Language English Cited by References 1. D. P. Blecher and C. Le Merdy, Operator Algebras and Their Modules, London Math. Soc. Monogr. New Ser. 30, Oxford Univ. Press, Oxford, 2004. 2. D. P. Blecher and V. I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), no. 2, 262-292. 3. D. P. Blecher and R. R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. 45 (1992), no. 1, 126-144. 4. E. G. Effros and A. Kishimoto, Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J. 36 (1987), no. 2, 257-276. 5. E. G. Effros and Z.-J. Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), no. 3, 329-337. 6. E. G. Effros and Z.-J. Ruan, Self-duality for the Haagerup tensor product and Hilbert space factorizations, J. Funct. Anal. 100 (1991), no. 2, 257-284. 7. E. G. Effros and Z.-J. Ruan, Operator spaces, J. London Math. Soc. Monogr. New Ser. 23, Oxford Univ. Press, New York, 2000. 8. E. G. Effros and Z.-J. Ruan, Operator space tensor products and Hopf convolution algebras, J. Operator Theory 50 (2003), no. 1, 131-156. 9. T. Itoh and M. Nagisa, Schur products and module maps on B(${\mathcal{H}}\$), Publ. Res. Inst. Math. Sci. 36 (2000), no. 2, 253-268.

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