Journal of the
Korean Mathematical Society JKMS

We describe the Haagerup tensor product $\ell^\infty \otimes_h \ell^\infty$ and the extended Haagerup tensor product $\ell^\infty \otimes_{eh} \ell^\infty$ in terms of Schur product maps, and show that $\ell^\infty \otimes_h \ell^\infty \cap {\mathbb B}(\ell^2)$ (resp. $\ell^\infty \otimes_{eh} \ell^\infty \cap {\mathbb B}(\ell^2)$) coincides with $c_0 \otimes_{h} c_0 \cap {\mathbb B}(\ell^2)$ (resp. $c_0 \otimes_{eh} c_0 \cap {\mathbb B}(\ell^2)$). For C*-algebras $A, B$, it is shown that $A \otimes_h B = A \otimes_{eh} B$ if and only if $A$ or $B$ is finite-dimensional.

Keywords: operator space, Haagerup tensor product, extended Haagerup tensor product, Schur product

MSC numbers: Primary 46L06, 46L05, 47L25