We study those Köthe coechelon sequence spaces $k_p\left(V\right)$, 1 ≤ p ≤ ∞ or p = 0, which are locally convex (Riesz) algebras for pointwise multiplication. We characterize in terms of the matrix V = (vₙ)ₙ when an algebra $k_p\left(V\right)$ is unital, locally m-convex, a -algebra, has a continuous (quasi)-inverse, all entire functions act on it or some transcendental entire functions act on it. It is proved that all multiplicative functionals are continuous and a precise description of all regular and all degenerate maximal ideals...

The isomorphic classification problem for the Köthe models of some $C^{\infty }$ function spaces is considered. By making use of some interpolative neighborhoods which are related to the linear topological invariant $D_{\phi }$ and other invariants related to the “quantity” characteristics of the space, a necessary condition for the isomorphism of two such spaces is proved. As applications, it is shown that some pairs of spaces which have the same interpolation property $D_{\phi }$ are not isomorphic.