The algebra of compact operators

Prove that if $V$ is a separable Hilbert space, then $\mathcal{C}(V)$, the Banach space of compact operators on $V$, is separable.

If $A \in \mathcal{B}(\mathcal{H})$ and $AT=TA$ for every compact operator $T$, show that $A$ is a multiple of the identity operator.

If $\mathcal{H}$ is a Hilbert space, show that $\mathcal{B}_0(\mathcal{H})$ has an approximate identity.

If $\mathcal{H}$ is a Hilbert space and $P$ is a projection, show that $\mathcal{B}_0(\mathcal{H})P$ is a closed modular left ideal of $\mathcal{B}_0(\mathcal{H})$. What is the associated right modular unit?

Find the minimal closed proper left ideals of $\mathcal{B}_0(\mathcal{H})$, $\mathcal{H}$ a Hilbert space. How about for $\mathcal{B}_0(\mathcal{X})$, $\mathcal{X}$ a Banach space?

What are the maximal modular left ideals of $\mathcal{B}_0(\mathcal{H})$, $\mathcal{H}$ a Hilbert space?

If $\mathcal{A} = \mathbb {C} + \mathcal{B}_0(\mathcal{H})$ and $\pi : \mathcal{A} \to \mathcal{B}(\mathcal{H})$ is the identity representation, show that $\pi^{(\alpha )}$ is a cyclic representation.

Show that the set of operators in $\mathrm{GL}(E)$ comprising operators that are linear compact perturbations of the identity is a subgroup of $\operatorname {GL}(E)$. It is denoted by $\operatorname {GLc}(E)$.