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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

Citations:

Zbl 1036.39021
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References:

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