A concrete example of an orthonormal basis of a Hilbert module over $K(H)$, the algebra of compact operators

Question

Suppose $H$ be a Hilbert space, $K(H)$ be the set of compact operators on $H$, $E$ be Hilbert module over $K(H)$ and $(e_{\lambda})_{\lambda \in I}$ be a orthonormal bass for $E$,

Can you mention example for $H, K(H), E$ $\hspace{0.1cm}$ and $(e_{\lambda})_{\lambda \in I}$?

Answer 1

Why don't you take any finite-dimensional Hilbert space $H$ and $E=K(H)$. Then $E$ is just the algebra of matrices $M_n$ with $n=\dim H$. The standard matrix units in $M_n$ certainly form an orthonormal basis.