Stability of the Apollonius Type Additive Functional Equation in Modular Spaces and Fuzzy Banach Spaces

In this work, we investigate the generalized Hyers-Ulam stability of the Apollonius type additive functional equation in modular spaces with or without Δ 2 -conditions. We study the same problem in fuzzy Banach spaces and β -homogeneous Banach spaces. We show the hyperstability of the functional equation associated with the Jordan triple product in fuzzy Banach algebras. The obtained results can be applied to differential and integral equations with kernels of non-power types.

Jordan Triple Product Homomorphisms on Triangular Matrices to and from Dimension One

A map $\Phi$ is a Jordan triple product (JTP for short) homomorphism whenever $\Phi(A B A)= \Phi(A) \Phi(B) \Phi(A)$ for all $A,B$. We study JTP homomorphisms on the set of upper triangular matrices $\mathcal{T}_n(\mathbb{F})$, where $\Ff$ is the field of real or complex numbers. We characterize JTP homomorphisms $\Phi: \mathcal{T}_n(\mathbb{C}) \to \mathbb{C}$ and JTP homomorphisms $\Phi: \mathbb{F} \to \mathcal{T}_n(\mathbb{F})$. In the latter case we consider continuous maps and the implications of omitting the assumption of continuity.

Maps preserving 2-idempotency of certain products of operators

Let A,B be standard operator algebras on complex Banach spaces X and Y of dimensions at least 3, respectively. In this paper we give the general form of a surjective (not assumed to be linear or unital) map ? : A ? B such that ?2 : M2(C)?A ? M2(C)?B defined by ?2((sij)2x2) = (?(sij))2x2 preserves nonzero idempotency of Jordan product of two operators in both directions. We also consider another specific kinds of products of operators, including usual product, Jordan semi-triple product and Jordan triple product. In either of these cases it turns out that ? is a scalar multiple of either an isomorphism or a conjugate isomorphism.

Maps preserving Jordan triple product on the self-adjoint elements of C∗-algebras

Let [Formula: see text] and [Formula: see text] be two unital [Formula: see text]-algebras with unit [Formula: see text]. It is shown that the mapping [Formula: see text] which preserves arithmetic mean and Jordan triple product is a difference of two Jordan homomorphisms provided that [Formula: see text]. The structure of [Formula: see text] is more refined when [Formula: see text] or [Formula: see text]. Furthermore, if [Formula: see text] is a [Formula: see text]-algebra of real rank zero and [Formula: see text] is additive and preserves absolute value of product, then [Formula: see text] such that [Formula: see text] (respectively, [Formula: see text]) is a complex linear (respectively, antilinear) ∗-homomorphism.