Let $H$ be an infinite dimensional separable Hilbert space. I'm looking for an easy way to show that the dimension of the quotientalgebra $B(H)/K(H)$ is infinite.
Attempt:
If the dimension is finite, the quotient is separable. Also $K(H)$ is separable, so we get that $B(H)$ is separable, which is not the case.
(1) Is my argument correct?
(2) Are there other easy ways to see this?
Your argument seems correct, but I believe there are details to be filled out to conclude that $B(H)$ is separable.
A fun way to see this (but an overkill): if $B(H)/K(H)$ is finite dimensional, then it is a nuclear $C^*$-algebra. Since $K(H)$ is also nuclear and we have a short-exact sequence of $C^*$-algebras $$0\to K(H)\to B(H)\to B(H)/K(H)\to0 $$ we have that $B(H)$ is nuclear, which is not the case, since $H$ is infinite dimensional.