Let $H$ a Hilbert Space, $B(H)=\{T:H\to H :T$ is linear and continuous$\}$, $K(H)=\{T\in B(H):T$ is compact$\}$.
I already know that $K(H)$ is a simple algebra, this is, the only closed ideals of $K(H)$ is itself and $\{0\}$ (even though $H$ don't need be separable). But how I check that if $H$ is separable then $K(H)$ is maximal in $B(H$)?
What I found here is that "is well-know statement", so where I can find a proof?