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 $\small{A{\otimes}_hB=A{\otimes}_{eh}B}$ if and only if A or B is finite-dimensional.
Keywords
operator space;Haagerup tensor product;extended Haagerup tensor product;Schur product;
Language
English
Cited by
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