In this paper, we study the continuity and boundedness of linear operators defined between probabilistic modular spaces in the probabilistic sense.

Let 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.

In this paper, we generalize some main results of (Jachymski in Proc. Am. Math. Soc. 136:1359Soc. 136: -1373Soc. 136: , 2008) from metric to uniform spaces endowed with an E -distance and a graph using a new type of contractions by employing a class of nondecreasing functions. MSC: 47H10; 05C40

No abstract

The aim of this paper is to present fixed point results of contractive mappings in the framework of cone b-metric spaces endowed with a graph and associated with a generalized c-distance. Some corollaries and an example are presented to support the main result proved herein. Our results unify, extend and generalize various comparable results in the literature.2010 MSC: 46A19; 47H10; 05C20.