Calkin algebra

Remarks

Since $K(H)$ is a closed two-sided ideal of $B(H)$ that is closed under adjoints, $Q(H)$ is a unital C*-algebra. Calkin showed that for separable $H$, $K(H)$ is the only closed two-sided ideal of $B(H)$ other than $\set{0}$ and $B(H)$, so $Q(H)$ is a simple C*-algebra. An operator $T \in B(H)$ is Fredholm exactly when $\pi(T)$ is invertible in $Q(H)$. The spectrum of $\pi(T)$ in $Q(H)$ is the essential spectrum of $T$. The algebra $Q(H)$ is infinite-dimensional and not separable as a normed space. The construction makes sense for any Hilbert space, but for non-separable $H$ the ideal $K(H)$ is no longer the only nontrivial closed ideal.