Automatic Continuity of Almost Multiplicative Linear Functionals on Fréchet Algebras

Abstract

Abstract. A linear functional T on a Fréchet algebra (A, (pn)) is called almost multiplicative with respect to the sequence (pn), if there exists ε ≥ 0 such that |T ab − T aT b| ≤ εpn(a)pn(b) for all n ∈ N and for every a, b ∈ A.We show that an almost multiplicative linear functional on a Fréchet algebra is either multiplicative or it is continuous, and hence every almost multiplicative linear functional on a functionally continuous Fréchet algebra is continuous.

Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras

“…Definition 2.1. [4] If T : Λ → Γ is a linear map from a Frechet algebra Λ to a Frechet algebra Γ, then the separating space of T is defined by S(T ) = {q ∈ Γ : there exists (q n ) ∞ n=1 in Λ such that q n → 0 and T q n → q}. Theorem 2.2.…”

Automatic continuity of Jordan almost multiplicative maps on special Jordan Banach algebras

“…Let us use the arguments used in proof of Theorem 3.4 in [6], to establish the next Theorem 3.2. Theorem 3.2.…”

Almost multiplicative maps and ε-spectrum of an element in Fréchet Q-algebra

“…See, for example, [10]. In 2014, Honary, Omidi and Sanatpour investigated the problem of automatic continuity for (weakly) almost multiplicative linear maps between Fréchet algebras [7] and [6].…”