×

Homomorphisms and derivations in \(C^{*}\)-algebras. (English) Zbl 1157.39017

The paper deals with the following question due to S. M. Ulam [A collection of mathematical problems. New York, London: Interscience Publishers (1960; Zbl 0086.24101)]: Given a group \(G\), a metric group \(G'\) with metric \(\rho(\cdot,\cdot)\), and \(\varepsilon>0\), does there exist a \(\delta>0\) such that if \(f:G\to G'\) satisfies \(\rho(f(xy),f(x)f(x))<\delta\) for all \(x,y\in G\), then a homomorphism \(h:G\to G'\) exists with \(\rho(f(x),h(x))<\varepsilon\) for all \(x\in G\)?
The authors provide an affirmative answer to this question in some certain easy cases. Let \(f:A\to B\) be a mapping such that
\[ \big\| f(z-\mu x)+\mu f(z-y)+\tfrac12f(x+y)\big\| \leq\big\| 2f \big(z-\tfrac{x+y}4\big)\big\|, \]
for all \(\mu\in\mathbb{T}:=\{\lambda\in\mathbb{C}:| \lambda| =1\}\) and all \(x,y,z\in A\). It is proved that if \(A\) and \(B\) are both \(C^*\)-algebras and \(f\) satisfies \(\| f(xy)-f(x)f(y)\| \leq\theta\| x\| ^r \| y\| ^r\) and \(\| f(x^*)-f(x)^*\| \leq2\theta\| x\| ^r\) for some \(r>1\) and \(\theta\geq0\), then \(f\) is a \(*\)-homomorphism. They also show that if \(A\) and \(B\) are both \(C^*\)-algebras and \(f\) satisfies \(\| f(xy)-f(x)y-xf(y)\| \leq\theta\| x\| ^r\| y\| ^r\) for some \(r>1\) and \(\theta\geq0\), then \(f\) is a linear derivation. The same results concerning homomorphisms and Lie (respectively, Jordan) derivations when \(A\) and \(B\) are both Lie \(C^*\)-algebras (respectively, \(JC^*\)-algebras) are given by exactly the same reasoning.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46L05 General theory of \(C^*\)-algebras
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras

Citations:

Zbl 0086.24101
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, NY, USA, 1960. · Zbl 0086.24101
[2] D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[4] Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 · doi:10.2307/2042795
[5] J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268-273, 1989. · Zbl 0672.41027 · doi:10.1016/0021-9045(89)90041-5
[6] Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,” Aequationes Mathematicae, vol. 39, no. 2-3, pp. 292-293, 309, 1990.
[7] Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics & Mathematical Sciences, vol. 14, no. 3, pp. 431-434, 1991. · Zbl 0739.39013 · doi:10.1155/S016117129100056X
[8] Th. M. Rassias and P. \vSemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989-993, 1992. · Zbl 0761.47004 · doi:10.2307/2159617
[9] P. G\uavruta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431-436, 1994. · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211
[10] S.-M. Jung, “On the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 221-226, 1996. · Zbl 0888.46018 · doi:10.1006/jmaa.1996.0433
[11] P. G\uavruta, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” in Advances in Equations and Inequalities, Hadronic Math. Ser., pp. 67-71, Hadronic Press, Palm Harbor, Fla, USA, 1999.
[12] M. A. Sibaha, B. Bouikhalene, and E. Elqorachi, “Ulam-G\uavruta-Rassias stability for a linear functional equation,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 157-168, 2007, Euler’s Tri-centennial Birthday Anniversary Issue in FIDA. · Zbl 1137.39018
[13] K. Ravi and M. Arunkumar, “On the Ulam-G\uavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 143-156, 2007, Euler’s Tri-centennial Birthday Anniversary Issue in FIDA.
[14] C. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,” 2007, to appear in Bulletin des Sciences Mathématiques. · Zbl 1141.17302
[15] J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185-190, 1992. · Zbl 0753.39003
[16] J. M. Rassias, “On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces,” Journal of Mathematical and Physical Sciences, vol. 28, no. 5, pp. 231-235, 1994. · Zbl 0840.46024
[17] J. M. Rassias, “On the stability of the general Euler-Lagrange functional equation,” Demonstratio Mathematica, vol. 29, no. 4, pp. 755-766, 1996. · Zbl 0884.47040
[18] J. M. Rassias, “Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings,” Journal of Mathematical Analysis and Applications, vol. 220, no. 2, pp. 613-639, 1998. · Zbl 0928.39014 · doi:10.1006/jmaa.1997.5856
[19] J. M. Rassias, “On the stability of the multi-dimensional Euler-Lagrange functional equation,” The Journal of the Indian Mathematical Society, vol. 66, no. 1-4, pp. 1-9, 1999. · Zbl 1141.39310
[20] J. M. Rassias, “Asymptotic behavior of mixed type functional equations,” The Australian Journal of Mathematical Analysis and Applications, vol. 1, no. 1, 21 pages, 2004, article no. 10. · Zbl 1060.39033
[21] K.-W. Jun and H.-M. Kim, “Ulam stability problem for Euler-Lagrange-Rassias quadratic mappings,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 82-95, 2007, Euler’s Tri-centennial Birthday Anniversary Issue in FIDA.
[22] C. Park, “Stability of an Euler-Lagrange-Rassias type additive mapping,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 101-111, 2007, Euler’s Tri-centennial Birthday Anniversary Issue in FIDA.
[23] M. J. Rassias and J. M. Rassias, “On the Ulam stability for Euler-Lagrange type quadratic functional equations,” The Australian Journal of Mathematical Analysis and Applications, vol. 2, no. 1, 10 pages, 2005, article no. 11. · Zbl 1094.39027
[24] C. Park, “Lie - *homomorphisms between Lie C*-algebras and Lie - *derivations on Lie C*-algebras,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 419-434, 2004. · Zbl 1051.46052 · doi:10.1016/j.jmaa.2003.10.051
[25] C. Park, “Homomorphisms between Lie JC*-algebras and Cauchy-Rassias stability of Lie JC*-algebra derivations,” Journal of Lie Theory, vol. 15, no. 2, pp. 393-414, 2005. · Zbl 1091.39006
[26] C. Park, “Homomorphisms between Poisson JC*-algebras,” Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79-97, 2005. · Zbl 1091.39007 · doi:10.1007/s00574-005-0029-z
[27] C. Park, Y. Cho, and M. Han, “Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations,” Journal of Inequalities and Applications, vol. 2007, Article ID 41820, 13 pages, 2007. · Zbl 1133.39024 · doi:10.1155/2007/41820
[28] C. Park and J. Cui, “Generalized stability of C*-ternary quadratic mappings,” Abstract and Applied Analysis, vol. 2007, Article ID 23282, 6 pages, 2007. · Zbl 1158.39020 · doi:10.1155/2007/23282
[29] C. Park, J. C. Hou, and S. Q. Oh, “Homomorphisms between JC*-algebras and Lie C*-algebras,” Acta Mathematica Sinica, vol. 21, no. 6, pp. 1391-1398, 2005. · Zbl 1121.39030 · doi:10.1007/s10114-005-0629-y
[30] J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126-130, 1982. · Zbl 0482.47033 · doi:10.1016/0022-1236(82)90048-9
[31] J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques, vol. 108, no. 4, pp. 445-446, 1984. · Zbl 0599.47106
[32] J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95-103, 1992. · Zbl 0779.47005
[33] J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,” Discussiones Mathematicae, vol. 14, pp. 101-107, 1994. · Zbl 0819.39012
[34] J. M. Rassias, “Alternative contraction principle and Ulam stability problem,” Mathematical Sciences Research Journal, vol. 9, no. 7, pp. 190-199, 2005.
[35] J. M. Rassias, “Solution of the Hyers-Ulam stability problem for quadratic type functional equations in several variables,” The Australian Journal of Mathematical Analysis and Applications, vol. 2, no. 2, 9 pages, 2005, article no. 11. · Zbl 1094.39026
[36] J. M. Rassias, “On the general quadratic functional equation,” Boletín de la Sociedad Matemática Mexicana, vol. 11, no. 2, pp. 259-268, 2005. · Zbl 1094.39024
[37] J. M. Rassias, “On the Ulam problem for Euler quadratic mappings,” Novi Sad Journal of Mathematics, vol. 35, no. 2, pp. 57-66, 2005. · Zbl 1164.39334
[38] J. M. Rassias, “On the Cauchy-Ulam stability of the Jensen equation in C*-algebras,” International Journal of Pure Applied Mathematics & Statistics, vol. 2, pp. 92-101, 2005.
[39] J. M. Rassias, “Alternative contraction principle and alternative Jensen and Jensen type mappings,” International Journal of Applied Mathematics & Statistics, vol. 4, no. 5, pp. 1-10, 2006.
[40] J. M. Rassias, “Refined Hyers-Ulam approximation of approximately Jensen type mappings,” Bulletin des Sciences Mathématiques, vol. 131, no. 1, pp. 89-98, 2007. · Zbl 1112.39025 · doi:10.1016/j.bulsci.2006.03.011
[41] J. M. Rassias and M. J. Rassias, “On some approximately quadratic mappings being exactly quadratic,” The Journal of the Indian Mathematical Society, vol. 69, no. 1-4, pp. 155-160, 2002. · Zbl 1104.39300
[42] J. M. Rassias and M. J. Rassias, “Asymptotic behavior of Jensen and Jensen type functional equations,” PanAmerican Mathematical Journal, vol. 15, no. 4, pp. 21-35, 2005. · Zbl 1082.39026
[43] J. M. Rassias and M. J. Rassias, “Asymptotic behavior of alternative Jensen and Jensen type functional equations,” Bulletin des Sciences Mathématiques, vol. 129, no. 7, pp. 545-558, 2005. · Zbl 1081.39028 · doi:10.1016/j.bulsci.2005.02.001
[44] Th. M. Rassias, “The problem of S. M. Ulam for approximately multiplicative mappings,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 352-378, 2000. · Zbl 0958.46022 · doi:10.1006/jmaa.2000.6788
[45] Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264-284, 2000. · Zbl 0964.39026 · doi:10.1006/jmaa.2000.7046
[46] Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23-130, 2000. · Zbl 0981.39014 · doi:10.1023/A:1006499223572
[47] Th. M. Rassias, Ed., Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. · Zbl 1047.39001
[48] F. Skof, “Proprietà locali e approssimazione di operatori,” Rendiconti del Seminario Matemàtico e Fisico di Milano, vol. 53, pp. 113-129, 1983. · Zbl 0599.39007 · doi:10.1007/BF02924890
[49] K.-W. Jun and H.-M. Kim, “On the stability of Appolonius’ equation,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 11, no. 4, pp. 615-624, 2004. · Zbl 1076.39026
[50] A. Gilányi, “Eine zur Parallelogrammgleichung äquivalente Ungleichung,” Aequationes Mathematicae, vol. 62, no. 3, pp. 303-309, 2001. · Zbl 0992.39026 · doi:10.1007/PL00000156
[51] J. Rätz, “On inequalities associated with the Jordan-von Neumann functional equation,” Aequationes Mathematicae, vol. 66, no. 1-2, pp. 191-200, 2003. · Zbl 1078.39026 · doi:10.1007/s00010-003-2684-8
[52] W. Fechner, “Stability of a functional inequality associated with the Jordan-von Neumann functional equation,” Aequationes Mathematicae, vol. 71, no. 1-2, pp. 149-161, 2006. · Zbl 1098.39019 · doi:10.1007/s00010-005-2775-9
[53] A. Gilányi, “On a problem by K. Nikodem,” Mathematical Inequalities & Applications, vol. 5, no. 4, pp. 707-710, 2002. · Zbl 1036.39020
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.