## Modified Trif’s functional equations in Banach modules over a $$C$$-algebra and approximate algebra homomorphisms.(English)Zbl 1046.39022

Let $$d$$ and $$r$$ be positive integers, $$A$$ be a unital $$C^*$$-algebra and $$a \in A$$ be fixed. The operator $$D_a$$ acting on the set of mappings $$f: {_A\mathcal{B}}\to {_A\mathcal{C}}$$, where both sets are left Banach $$A$$-modules which may have different norms, is defined by the formula: $\begin{split} (D_af)(x_1, \dots , x_n) =\\ =\frac{d}{r} \cdot {_{n-2}C_{k-2}}af \left(\frac{1}{d}\sum_{i=1}^n {rx_i}\right) +{_{n-2}C_{k-1}}\sum_{i=1}^n {af(x_i)} - k \cdot \sum_{1 \leq i_1 < \dots <i_k \leq n} f\left( \frac{1}{k} \sum_{j=1}^k {ax_{i_j}} \right). \end{split} \tag{1}$ Several theorems are proved saying that if the norm of $$(D_af)(x_1,\dots ,x_n)$$ is bounded by a nonnegative $$\varphi (x_1,\dots ,x_n), x_j \in{_A{\mathcal B}}$$, then there is the unique $$A$$-linear mapping $$T:{_A\mathcal{B}}\to {_A\mathcal{C}}$$ whose closedness to $$f$$ is controlled by a function $$\psi$$ defined with the aid of $$\varphi$$. Similar theorems concerning the stability of homomorphisms between Banach algebras contain the counterpart: if the algebra is unital, then the solution of the relevant inequality is itself an algebra homomorphism (the superstability phenomenon).
Trif’s equation referred to in the title of the paper reads: $$(D_1f)(x_1, \dots , x_n) = 0$$, where $$r = 1, d = n$$, and $$f$$ maps one vector space into another. T. Trif [J. Math. Anal. Appl. 272, 604–616 (2002; Zbl 1036.39021)] proved that if $$f$$ is a solution of the equation and $$f(0) = 0$$ then it is necessarily an additive function.

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 39B52 Functional equations for functions with more general domains and/or ranges

Zbl 1036.39021
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