Is there a classification theorem for Banach algebras, or even for Banach *algebras, similar to the GNS representation theorem for $C^*$-algebras?
If yes, please provide a reference where I can read about it.
If not, is there some Banach algebra which is NOT isomorphic to $B(X)$, where $X$ is a Banach space?
Any finite dimensional Banach algebra of dimension not a square is not isomorphic to $B(X)$ for any Banach space $X$.