Achievement of continuity of $(\varphi,\psi)$-derivations without linearity

Abstract

Suppose that A is a C * -algebra acting on a Hilbert space H, and that ϕ, ψ are mappings from A into B(H) which are not assumed to be necessarily linear or continuous. A (ϕ, ψ)-derivation is a linear mapping d :We prove that if ϕ is a multiplicative (not necessarily linear) * -mapping, then every * -(ϕ, ϕ)-derivation is automatically continuous. Using this fact, we show that every * -(ϕ, ψ)-derivation d from A into B(H) is continuous if and only if the * -mappings ϕ and ψ are left and right d-continuous, respectively. IntroductionRecently, a number of analysts [2,4,12,13,14] have studied various generalized notions of derivations in the context of Banach algebras. There are some applications in the other fields of research [7]. Such mappings have been extensively studied in pure algebra; cf. [1,3,9]. A generalized concept of derivation is as follows.Definition 1.1. Suppose that B is an algebra, A is a subalgebra of B, X is a B-bimodule, and ϕ, ψ :By a ϕ-derivation we mean a (ϕ, ϕ)-derivation. Note that we do not have any extra assumptions such as linearity or continuity on the mappings ϕ and ψ.The automatic continuity theory is the study of (algebraic) conditions on a category, e.g. C * -algebras, which guarantee that every mapping belonging to a certain

Jewell Theorem for Higher Derivations on C*-Algebras

“…Regarding to a celebrated theorem of Sakai [11,12], all derivations defined on a C * -algebra are automatically continuous. Some results concerning to the theorem are discussed in [8] and [3]. Regarding to the Sakai's theorem we can deduce that the higher derivation d n = δ n n!…”

Ultraweak Continuity of σ-derivations on von Neumann Algebras

“…For other approaches to generalized derivations and their applications see [1,2,3,11] and references therein. In particular, the automatic continuity problem for (σ, τ )-derivations is considered in [8,9,10] and an achievement of continuity of (σ, τ )-derivations without linearity is given in [5].…”

A Kadison–sakai-Type Theorem

“…These maps have been investigated c 2009 Australian Mathematical Society 0004-9727/2009 $16.00 extensively in pure algebra. Recently, they have been treated in the Banach algebra theory (see [3,5,7,8,10,13] and references therein).…”

“…These maps have been extensively investigated in pure algebra. Recently, they have been treated in the Banach algebra theory (see [1,3,5,6,7,12] and references therein).…”