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E-raamat: Stability of Functional Equations in Banach Algebras

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 26-Jun-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319187082
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Stability of Functional Equations in Banach AlgebrasSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 26-Jun-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319187082
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Some of the most recent and significant results on homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C*-algebras, C*-ternary algebras, non-Archimedean Banach algebras and multi-normed algebras are presented in this book. A brief introduction for functional equations and their stability is provided with historical remarks. Since the homomorphisms and derivations in Banach algebras are additive and R-linear or C-linear, the stability problems for additive functional equations and additive mappings are studied in detail. The latest results are discussed and examined in stability theory for new functional equations and functional inequalities in Banach algebras and C*-algebras, non-Archimedean Banach algebras, non-Archimedean C*-algebras, multi-Banach algebras and multi-C*-algebras.

Graduate students with an understanding of operator theory, functional analysis, functional equations and analytic inequalities will find this book useful for furthering their understanding and discovering the latest results in mathematical analysis. Moreover, research mathematicians, physicists and engineers will benefit from the variety of old and new results, as well as theories and methods presented in this book.

Arvustused

The book under review provides an account of some of the most recent and significant results on the stability of homomorphisms and derivations. ... This beautiful book is well written, in a clear self-contained and reader-friendly style. It will prove to be very useful for self-study and seminars. It belongs in every library collection. I recommend it to graduate students and specialists working in functional equations, in functional analysis as well as in physics and engineering. (Pac Gvru, zbMATH 1323.39025, 2015)

1 Introduction
1(18)
1.1 Fixed Point Theorems
7(2)
1.2 Quasi-Banach Algebras
9(1)
1.3 C*-Algebras
10(2)
1.4 C*-Ternary Algebras
12(1)
1.5 Non-Archimedean Normed Algebras
13(1)
1.6 Multi--normed Algebras
14(5)
2 Stability of Functional Equations in Banach Algebras
19(32)
2.1 Stability of 1/qƒ(qx + qy + qz) = ƒ(x) +ƒ(y) +ƒ(z)
20(4)
2.1.1 Isomorphisms in Banach Algebras
20(3)
2.1.2 Derivations in Banach Algebras
23(1)
2.2 Stability of m-Variable Functional Equations
24(7)
2.2.1 Stability of Homomorphisms in Banach Algebras
24(5)
2.2.2 Stability of Derivations in Banach Algebras
29(2)
2.3 Stability in Quasi-Banach Algebras
31(7)
2.3.1 Stability of Homomorphisms in Quasi-Banach Algebras
32(3)
2.3.2 Isomorphisms in Quasi-Banach Algebras
35(1)
2.3.3 Stability of Generalized Derivations in Quasi-Banach Algebras
36(2)
2.4 Stability of Cauchy--Jensen Functional Equations
38(13)
2.4.1 Stability of Homomorphisms in Real Banach Algebras
38(6)
2.4.2 Stability of Generalized Derivations in Real Banach Algebras
44(7)
3 Stability of Functional Equations in C* - Algebras
51(114)
3.1 Isomorphisms in Unital C*-Algebras
51(12)
3.1.1 C*-Algebra Isomorphisms in Unital C*-Algebras
54(7)
3.1.2 On the Mazur-Ulam Theorem in Modules over C* -Algebras
61(2)
3.2 Apollonius Type Additive Functional Equations
63(7)
3.2.1 Homomorphisms and Derivations on C*-Algebras
64(3)
3.2.2 Homomorphisms and Derivations in Lie C*-Algebras
67(2)
3.2.3 Homomorphisms and Derivations in JC*-Algebras
69(1)
3.3 Stability of Jensen Type Functional Equations in C*-Algebras
70(15)
3.3.1 Stability of Homomorphisms in C*-Algebras
70(9)
3.3.2 Stability of Derivations in C*-Algebras
79(2)
3.3.3 Stability of Homomorphisms in Lie C*-Algebras
81(2)
3.3.4 Stability of Lie Derivations in C*-Algebras
83(2)
3.4 Generalized Additive Mapping
85(13)
3.4.1 Hyers--Ulam Stability of Functional Equations in Banach Modules over a C*-Algebra
85(7)
3.4.2 Homomorphisms in Unital C*-Algebras
92(6)
3.5 Generalized Additive Mappings in Banach Modules
98(12)
3.5.1 Odd Functional Equations in Variables
99(1)
3.5.2 Stability of Odd Functional Equations in Banach Modules over a C*-Algebra
100(6)
3.5.3 Isomorphisms in Unital C*-Algebras
106(4)
3.6 Jordan *-Derivations and Quadratic Jordan *-Derivations
110(14)
3.6.1 Stability of Jordan *-Derivations
110(4)
3.6.2 Stability of Quadratic Jordan *-Derivations
114(5)
3.6.3 Stability of Jordan *-Derivations: The Fixed Point Method
119(5)
3.7 (α, β, γ)-Derivations on Lie C*-Algebras: The Direct Method
124(6)
3.8 Square Roots and 3rd Root Functional Equations: The Direct Method
130(7)
3.8.1 Stability of the Square Root Functional Equation
131(4)
3.8.2 Stability of the 3rd Root Functional Equation
135(2)
3.9 Square Root and 3rd Root Functional Equations: The Fixed Point Method
137(6)
3.9.1 Stability of the Square Root Functional Equation
137(4)
3.9.2 Stability of the 3rd Root Functional Equation
141(2)
3.10 Positive-Additive Functional Equation
143(8)
3.10.1 Stability of the Positive-Additive Functional Equations: The Fixed Point Method
144(4)
3.10.2 Stability of the Positive-Additive Functional Equations: The Direct Method
148(3)
3.11 Stability of *-Homomorphisms in JC*-Algebras
151(14)
3.11.1 *-Homomorphisms in JC*-Algebras
153(7)
3.11.2 Stability of *-Homomorphisms in JC*-Algebras
160(5)
4 Stability of Functional Inequalities in Banach Algebras
165(36)
4.1 Stability of Additive Functional Inequalities in Banach Algebras
166(18)
4.1.1 Stability of C-Linear Mappings in Banach Spaces
166(5)
4.1.2 Stability of Homomorphisms in Proper CQ*-Algebras
171(7)
4.1.3 Stability of Derivations in Proper CQ*-Algebras
178(6)
4.2 Stability of Functional Inequalities over C*-Algebras
184(17)
4.2.1 Functional Inequalities in Normed Modules over C*-Algebras
185(7)
4.2.2 On Additive Functional Inequalities in Normed Modules over C* -Algebras
192(4)
4.2.3 Generalization of Cauchy-Rassias Inequalities via the Fixed Point Method
196(5)
5 Stability of Functional Equations in C* -Ternary Algebras
201(28)
5.1 C*-Ternary 3-Homomorphism and C*-Ternary 3-Derivations
202(7)
5.1.1 C*-Ternary 3-Homomorphisms in C*-Ternary Algebras
203(4)
5.1.2 C*-Ternary 3-Derivations in C*-Ternary Algebras
207(2)
5.2 Apollonius Type Additive Functional Equations
209(10)
5.2.1 Homomorphisms in C*-Ternary Algebras
210(2)
5.2.2 Derivations in C*-Ternary Algebras
212(1)
5.2.3 Homomorphisms in JB*-Triples
213(1)
5.2.4 Derivations in JB*-Triples
214(1)
5.2.5 C*-Ternary Homomorphisms: Fixed Point Method
215(1)
5.2.6 C*-Ternary Derivations: The Fixed Point Method
216(1)
5.2.7 JB*-Triple Homomorphisms: The Fixed Point Method
217(1)
5.2.8 JB*-Triple Derivations: Fixed Point Method
218(1)
5.3 Bi-θ-Derivations in JB*-Triples
219(10)
6 Stability of Functional Equations in Multi-Banach Algebras
229(74)
6.1 Stability of m-Variable Additive Mappings
230(11)
6.1.1 Stability of Homomorphisms in Multi-Banach Algebras
230(8)
6.1.2 Stability of Derivations in Multi-Banach Algebras
238(3)
6.2 Ternary Jordan Homomorphisms and Derivations in Multi-C*-Ternary Algebras
241(11)
6.2.1 Stability of Homomorphisms in Multi-C*-Ternary Algebras
242(7)
6.2.2 Stability of Derivations in Multi-C*-Ternary Algebras
249(3)
6.3 Generalized Additive Mappings and Isomorphisms in Multi-C*-Algebras
252(12)
6.3.1 Stability of Odd Functional Equations in Multi-Banach Modules over a Multi-C*-Algebra
252(10)
6.3.2 Isomorphisms in Unital Multi-C*-Algebras
262(2)
6.4 Additive Functional Inequalities in Proper Multi-CQ*-Algebras
264(18)
6.4.1 Stability of C-Linear Mappings in Multi-Banach Spaces
264(5)
6.4.2 Stability of Homomorphisms in Proper Multi-CQ*-Algebras
269(8)
6.4.3 Stability of Derivations in Proper CQ*-Algebras
277(5)
6.5 Stability of Homomorphisms and Derivations in Multi-C*-Ternary Algebras
282(21)
6.5.1 Stability of Homomorphisms
282(15)
6.5.2 Stability of Derivations in Multi-C*-Ternary Algebras
297(6)
7 Stability of Functional Equations in Non-Archimedean Banach Algebras
303(24)
7.1 Stability of Jensen Type Functional Equations: The Fixed Point Approach
304(16)
7.1.1 Stability of Homomorphisms in Non-Archimedean C* -Algebras
304(10)
7.1.2 Stability of Derivations in Non-Archimedean C* -Algebras
314(2)
7.1.3 Stability of Homomorphisms in Non-Archimedean Lie C* -Algebras
316(2)
7.1.4 Stability of Non-Archimedean Lie Derivations in C*-Algebras
318(2)
7.2 Stability for m-Variable Additive Functional Equations
320(7)
7.2.1 Stability of Homomorphisms and Derivations in Non-Archimedean C*-Algebras
320(4)
7.2.2 Stability of Homomorphisms and Derivations in Non-Archimedean Lie C*-Algebras
324(3)
References 327(14)
Index 341
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