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 $c_0{otimes}_hc_0{cap}mathbb{B}({ell}^2)$(resp. $c_0{otimes}_{eh}c_0{cap}mathbb{B}({ell}^2)$). For $C^*2$-algebras A, B, it is shown that $A{otimes}_hB=A{otimes}_{eh}B$ if and only if A or B is finite-dimensional.