Questions about $B(H)$ and $B(H)/K(H)$ as Banach space

Question

I am trying to investigate the relation between Uniformly Convexity and existence of Schauder Basis for a Banach space.

I read in a Handbook article that $B(H)$ (the algebra of all bounded operators on a Hilbert space $H$) and $B(H)/K(H)$ (the Calkin algebra for the same Hilbert space) are both naturally occurring examples of space which fails the Approximation Property.

I am wondering if we know:

a) if the above algebras are separable Banach spaces;

b) the convexity of them as Banach spaces (e.g. strictly convex, uniformly convex, ...?)

I apologize if this is really trivial functional analysis problem. Thank you!

Answer 1

$B(H)$ is not separable since it contains isometric copy of $\ell_\infty$ which is non-separable. $B(H)$ is not strctly convex since it contains isometric copy of $\ell_\infty$ which is not strictly convex. A fortiori $B(H)$ is not uniformly convex because every uniformly convex space is stricly convex.

The same result holds for $C(H):=B(H)/K(H)$ since it contains isometric copy of $\ell_\infty/c_0$ which is not separable and not strictly convex.