Summary: Recently, the first author gave a characterization of operator algebras up to complete isomorphism. We give here some characterizations of quotients of function algebras (\(Q\)-algebras), again up to complete isomorphism. Using these, we examine which operator space structures on \(\ell_p\) (with pointwise product) correspond to operator algebras, and which to \(Q\)-algebras. We also give a new approach to the long outstanding similarity problem of Halmos, studying operator space structures on the disc algebra. Finally, we show that the Banach algebra of von Neumann-Schatten \(p\)-class operators on a Hilbert space is an operator algebra for all \(1\leq p\leq \infty\) (with either the usual or the Schur product). That is, these algebras are bicontinuously and algebraically isomorphic to a norm closed algebra of operators on some Hilbert space. However, with the usual operator space structures they are not completely bicontinuously isomorphic to any closed algebra of operators.