Let $H$ be a Hilbert space. I want to view $H$ as a Hilbert $C^*$ module over $\mathcal{B}(H)$. First let me take $H=\ell^2$. Identify $x \in \ell^2$ as $1 \times \infty$ matrix $(x_1, x_2, \ldots)$. Regard $H$ as $B(H)$ module by matrix multiplication. Define the $B(H)$ valued innerproduct on $H$ by $\langle x,y\rangle =(\overline{x_i}y_j)$. I want to prove that the Hilbert $C^*$ norm on $H$ is nothing but the usual $\ell^2$ norm. I am not able to do that.
By definition, the Hilbert $C^*$ norm is $\|x\|=\sqrt{\|\langle x,x\rangle\|}$. By the above definition, $\langle x,x\rangle $ is the matrix $(\overline{x_i}x_j)$. I want to find its operator norm. Or Am I making any mistake?