Bourgin (Duke Math J 16:385–397, 1949) proved that every approximate ring homomorphism from a Banach algebra onto a unital Banach algebra is automatically a ring homomorphism. Martindale (Proc Am Math Soc 21:695–698, 1969) proved that every multiplicative isomorphism from a prime ring containing a nontrivial idempotent onto an arbitrary ring is automatically additive. We show these results for approximate multiplicative skew derivations and approximate skew derivations. Some closely related results are also discussed. Furthermore, we give some important consequences and illustrative examples from our main results.

We prove that every n -Jordan derivation in the sense of Herstein (Bull Am Math Soc 67:517–531, 1961, p. 528) on n !-torsion free unital commutative rings is a derivation. Furthermore, we prove that every continuous n -Jordan derivation on semiprime normed algebras is a derivation. The results of this paper improve and generalize the main results of Bridges and Bergen (Proc Am Math Soc 90:25–29, 1984), but under weaker assumptions. Some applications and examples of our results are also provided.

At the present paper, we investigate bounded approximately local derivations of l(1)-Munn algebra MI(A), where I is an arbitrary nonempty set and A is an approximately locally unital Banach algebra. Indeed, we show that if B-A(A, A*) and( B)A(A, A*) are reflexive, then every bounded approximately local derivation from MI(A) into any Banach M-I(A)-bimodule X is a derivation. Finally, we apply this result to study bounded approximately local derivations of the semigroup algebra l(1)(S), where S is a uniformly locally finite inverse semigroup.

In this paper, we obtain a characterization for QM(L(X)), the quasi-multiplier algebra of L(X), where X is a foundation hypergroup . It is shown that QM(L(X)) can be identified by M(X) for this class of hypergroups. We also investigate quasi-multipliers on the second dual Banach algebra L-c(X)**. Indeed, we show that QM(L-c(X)**) is isomorphic with M(X). As an application, we prove that QM(L-c(X)**) = L(X) if and only if X is discrete.

In this paper, we give a characterization of the kernel of $$\tau $$ in a trivolutive Banach algebra $$({{\mathcal {A}}}, \tau )$$ . The main results (under a certain condition) are: (a) The closed ideal $${\mathcal {I}} = \ker \tau $$ of a unital Banach algebra $${{\mathcal {A}}}$$ is principal. (b) The closed ideal $${\mathcal {I}}=\ker \tau $$ has a bounded approximate identity if $${{\mathcal {A}}}$$ has a bounded approximate identity. Also we give some concrete example which indicates that the necessary condition must occur. Moreover, we relate trivolutions on the algebra $${\mathcal {A}}\times _{\theta }{\mathcal {B}}$$ with $$\theta $$ -Lau product to the corresponding ones on $${{\mathcal {A}}}$$ and $${\mathcal {B}}$$ .

In this paper, we introduce and study the notion of quasi-multipliers on a semi-topological semigroup [Formula: see text]. The set of all quasi-multipliers on [Formula: see text] is denoted by [Formula: see text]. First, we study the problem of extension of quasi-multipliers on topological semigroups to its Stone–Čech compactification. Indeed, we prove if [Formula: see text] is a topological semigroup such that [Formula: see text] is pseudocompact, then [Formula: see text] can be regarded as a subset of [Formula: see text] Moreover, with an extra condition we describe [Formula: see text] as a quotient subsemigroup of [Formula: see text] Finally, we investigate quasi-multipliers on topological semigroups, its relationship with multipliers and give some concrete examples.

Let $${\mathcal {S}}$$ be a compactly cancellative foundation semigroup with identity and $$M_a({\mathcal {S}})$$ be its semigroup algebra. In this paper, we give some characterizations for $$ {{\mathfrak {Q}}}{{\mathfrak {M}}}(M_a({\mathcal {S}}))$$, the quasi-multipliers of $$M_a({\mathcal {S}})$$. It is shown that $$ {{\mathfrak {Q}}}{{\mathfrak {M}}}(M_a({\mathcal {S}}))$$ may be identified by $$M({\mathcal {S}})$$. We deal with the quasi-multipliers on the dual Banach algebra $$L_{0}^{\infty }({\mathcal {S}};M_a({\mathcal {S}}))$$ and prove that its quasi-multipliers is again $$M({\mathcal {S}})$$. We also discuss the bilinear mappings $${\mathfrak {m}} :M_a({\mathcal {S}})^{*} \times M_a({\mathcal {S}})^{*} \longrightarrow M_a({\mathcal {S}})^{*}$$ which commutes with translations and convolutions.

We prove that every bounded n-derivation of a commutative factorizable Banach algebra maps into its radical. Also, the nilpotency of eigenvectors of any bounded n-derivation corresponding to its eigenvalues is derived. We introduce the notion of approximate n-derivations on a Banach algebra A and show that the separating space of an approximate n-derivation (n > 2 ) is not necessarily an ideal, unless the Banach algebra A is factorizable. From this and some results on bounded n-derivations. we prove that every approximate n-derivation of a semisimple factorizable Banach algebra is automatically continuous and every approximate n-derivation of a commutative semisimple factorizable Banach algebra is identically zero. Some applications of our results are also provided.

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation \(*\)-semigroup S when it is infinite non-discrete cancellative, \(M_a(S)^{**}\) does not admit an involution, and \(M_a(S)^{**}\) has a trivolution with range \(M_a(S)\) if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only if G is finite. However, \(A(G)^{**}\) always admits trivolution.

Let A be an arbitrary Banach algebra and phi a homomorphism from A onto C. Our first purpose in this paper is to give some equivalent conditions under which guarantees a phi-mean of norm one. Then we find some conditions under which there exists a phi-mean in the weak* cluster of {a is an element of A; parallel to a parallel to = phi(a) = 1} in A**.

Rassias(2001) introduced the pioneering cubic functional equation in the history of mathematical analysis: and solved the pertinent famous Ulam stability problem for this inspiring equation. This Rassias cubic functional equation was the historic transition from the following famous Euler-Lagrange-Rassias quadratic functional equation: to the cubic functional equations. In this paper, we prove the Ulam-Hyers stability of the cubic functional equation: in fuzzy normed linear spaces. We use the definition of fuzzy normed linear spaces to establish a fuzzy version of a generalized Hyers-Ulam-Rassias stability for above equation in the fuzzy normed linear space setting. The fuzzy sequentially continuity of the cubic mappings is discussed.

In this paper, the stability of the general cubic-quartic functional equation, f(x+ky)+f(x−ky)=k2(f(x+y)+f(x−y))+2(1−k2)f(x)+[(k4−k2)/4](f(2y)−8f(y))+f̃(2x)−16f̃(x), where f̃(x)≔f(x)+f(−x) in the setting of Menger probabilistic normed spaces, is proved.