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

No abstract

In this note we introduce the notions of generalized probabilistic metric spaces and generalized Menger probabilistic metric spaces. After making our elementary observations and proving some basic properties of these spaces, we are going to prove some fixed point result in these spaces.

Let M be a Hilbert C-module. A linear mapping d W M ! M is called a derivation if d.< x; y >´/ D< dx; y >´C < x; dy >´C < x; y > d´for all x; y;´2 M. We give some results for derivations and automatic continuity of them on M. Also, we will characterize generalized derivations and strong higher derivations on the algebra of compact operators and adjointable operators of Hilbert C-modules, respectively.

No abstract