We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its Banach space structure.
Clearly, finite-dimensionality and other cardinality properties can be detected. Thus, to avoid these cases, let us restrict to separable, infinite-dimensional $C^*$-algebras.
For commutative $C^*$-algebras the problem has been solved by Milutin. In particular, $C(X)$ and $C(Y)$ are isomorphic as Banach spaces for any uncountable, compact, metric spaces $X$ and $Y$.
In the paper
Hamana. On linear topological properties of some $C^*$-algebras, Tohoku Math. J., II. Ser. 29, 157-163 (1977).
it is shown that the Banach space structure also reflects if all irreducible representations of the $C^*$-algebra are finite-dimensional.
Let us ask what happens at the 'opposite' end of the scale:
Question: Are all simple, separable, infinite-dimensional $C^*$-algebras isomorphic as Banach spaces?