Uniqueness of the uniform norm with an application to Topological Algebras

Any square-preserving linear seminorm on a unital commutative algebra is submultiplicative; and the uniform norm on a uniform Banach algebra is the only uniform Q-algebra norm on it. This is proved and is used to show that (i) uniform norm on a regular uniform Banach algebra is unique among all uniform (not necessarily complete) norms and (ii) a complete uniform topological algebra that is a Q-algebra is a uniform Banach algebra. Relevant examples, showing that the respective assumptions regarding regularity, Q-algebra norm, and uniform property of topology cannot be omitted, have been discussed