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!