Given a non-abelian C*-algebra $A$. I am wondering what are the possible abelian sub-C*-algebras of $A$. Let $K$ be the spectrum of $A$. Does $A$ contain an isomorphic copy (as a Banach space) of the space $C(K)$? (if $A$ is abelian and unital, then thay are of course isometrically indistinguishable).
If the answer for my question is negative, let $L$ be a compact metric space. Must $A$ contain $C(L)$?