Uniqueness of the Uniform Norm with an Application to Topological Algebras
S. J. Bhatt, D. J. Karia · Proceedings of the American Mathematical Society · 1992
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.