Structure of Commutative Normed Rings

In a complex commutative normed ring the unit sphere at the origin has a vertex at the unit element. If the ring is finite dimensional, the radical translated to the unit element intersects this sphere only at the unit element. A finite dimensional ring containing an element of nilpotency degree equal to the dimension or the radical is a direct sum of a ring with a scalar product and a ring with a convolution product. Using this decomposition the conjugate space is made into normed ring, and a duality theory is obtained. General properties are given or completely continuous and weakly completely continuous elements or various types of rings. In a star ring, if uniform convergence with respect to the maximal ideals implies weak convergence, then the square or a weakly completely continuous operator is completely continuous. Some of the consequences of this result are: (a) no infinite dimensional ring of this type is reflexive as a Banach space, (b) all weakly completely continuous elements or infinite dimensional indecomposable rings of this type lie in the radical, (c) a new proof or Dunford's theorem that the square of a weakly completely continuous operator from L into L is completely continuous is given.