Park, Chun-Gil Modified Trif’s functional equations in Banach modules over a \(C\)-algebra and approximate algebra homomorphisms. (English) Zbl 1046.39022 J. Math. Anal. Appl. 278, No. 1, 93-108 (2003). 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. Reviewer: Bogdan Choczewski (Kraków) Cited in 20 Documents 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 Keywords:Banach module over \(C\)-algebra; Stability; Real rank 0; Unitary group; Trif’s; functional equation; Approximate algebra homomorphism; \(C^*\)-algebra Citations:Zbl 1036.39021 PDFBibTeX XMLCite \textit{C.-G. Park}, J. Math. Anal. Appl. 278, No. 1, 93--108 (2003; Zbl 1046.39022) Full Text: DOI References: [1] Brezis, H.; Ambrosetti, A.; Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122, 519-543 (1994) · Zbl 0805.35028 [2] Bahri, A.; Berestycki, H., A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267, 1-31 (1981) · Zbl 0476.35030 [3] Bartsch, T.; Willem, M., On an elliptic equations with convex and concave nonlinearities, Proc. Amer. Math. Soc., 123, 3555-3561 (1995) · Zbl 0848.35039 [4] Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems (1993), Birkhäuser [5] Courant, R.; Hilbert, D., Methods of Mathematical Physics, Vol. I (1953), Interscience: Interscience New York · Zbl 0729.00007 [6] Kajikiya, R., Non-radial solutions with group invariance for the sublinear Emden-Fowler equation, Nonlinear Anal., 28, 567-597 (2002) [7] Spanier, E., Algebraic Topology (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0145.43303 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.