The Nilpotency of the Nil Metric F-Algebras

Let F be a normed field. In this work, we prove that every nil complete metric F-algebra is nilpotent when F has characteristic zero. This result generalizes Grabiner's Theorem for Banach algebras, first proved in 1969. Furthermore, we show that a metric F-algebra A and its completion C(A) satisfy the same polynomial identities, and consequently, if char(F)=0 and C(A) is nil, then A is nilpotent. Our results allow us to resolve K\"othe's Problem affirmatively for complete metric algebras over normed fields of characteristic zero.