Multiplicative and almost multiplicative maps in probabilistic normed algebras

Document Type : Research Paper


Department of Pure Mathematics, Faculty of Mathematical Sciences, Shahrekord University, P. O. Box 88186-34141, Shahrekord, Iran


Our main purpose of this paper is to study the relationship between multiplicative maps and almost multiplicative maps between probabilistic normed algebras. We first derive some properties of invertible elements and their relation with multiplicative maps. Then we show that every complex homomorphism on elements whose probabilistic norm is equal to 1, is bounded. In the following, we give an open problem about the functionally continuous of unital commutative probabilistic Banach algebra. Finally, we prove that every almost multiplicative map that is not a multiplicative map is continuous.


[1] E. Ansari-piri and N. Eghbali, Almost n-multiplicative maps, Afr. J. Math. Comput. Sci. Res. 5 (2012), 200–203.
[2] E. Ansari-piri, H. Shayanpour and Z. Heidarpour, Approximately n-multiplicative and approximately additive functions in normed algebras, Bull. Math. Anal. Appl. 7 (2015), no. 1, 12–19.
[3] J. Braˇciˇc and M.S. Moslehian, On automatic continuity of 3-homomorphisms on Banach algebras, Bull. Malays. Math. Sci. Soc. 30 (2007),195–200.
[4] S.S. Chang, Y.J. Cho and S.M. Kang, Nonlinear operator theory in probabilistic metric spaces, New York, Nova Science Publishers, Inc, 2001.
[5] S.S. Chang, B.S. Lee, Y.J. Cho, Y.Q. Chen, S.M. Kang and J.M. Jung, Generalized contraction mapping principle and differential equations in probabilistic metric spaces, Proc. Am. Math. Soc. 124 (1996), 2367–2376.
[6] H.G. Dales, Banach algebras and automatic continuity, London Mathematical Society, Monograph 24, Clarendon Press, Oxford, 2000.
[7] M.S. El Naschie, Fuzzy dodecahedron topology and E-infinity spacetimes as a model for quantum physics, Chaos Solitons Fractals 30 (2006), no. 5, 1025–1033.
[8] M.S. El Naschie, On gauge invariance, dissipative quantum mechanics and self-adjoint sets, Chaos Solitons Fractals 32 (2007), no. 2, 271–273.
[9] M.S. El Naschie, P-Adic analysis and the transfinite E8 exceptional Lie symmetry group unification, Chaos Solitons Fractals 38 (2008), no. 3, 612–614.
[10] M. Eshaghi Gordji, A. Jabbari and E. KarapinarAutomatic continuity of n-homomorphisms on Banach algebras, Bull. Iran. Math. Soc. 41 (2015), no. 5, 1207–1271.
[11] M. Fragoulopoulou, Topological algebras with involution, Elsevier, 2005.
[12] W. Gahler and S. Gahler, Contributions to fuzzy analysis, Fuzzy Sets Syst. 105 (1999), 201–224.
[13] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 125 (1989), 385–389.
[14] O. Hadˇzi´c and E. Pap, Fixed point theory in probabilistic metric spaces, Kluwer Academic Publishers, Dordrecht, 2001.
[15] Z. Heidarpour, E. Ansari-piri, H. Shayanpour and A. Zohri, A class of certain properties of approximately n-multiplicative maps between locally multiplicatively convex algebras, Int. J. Nonlinear Anal. Appl. 9 (2018), no. 2, 111–116.
[16] S. Hejazian, M. Mirzavaziri and M. S. Moslehian, n-Homomorphisms, Bull. Iran. Math. Soc. 31 (2005), 13–23.
[17] T.G. Honary and H. Shayanpour, Automatic continuity of n-homomorphisms between Banach algebras, Quaest. Math. 33 (2010), 189–196.
[18] T.G. Honary and H. Shayanpour, Automatic continuity of n-homomorphisms between topological algebras, Bull. Aust. Math. Soc. 83 (2011), 389–400.
[19] T.G. Honary, M.N. Tavani and H. Shayanpour, Automatic continuity of n-homomorphisms between Frechet algebras, Quaest. Math. 34 (2011), 265–274.
[20] R.A.J. Howey, Approximately multiplicative functionals on algebras of smooth functions, J. London Math. Soc. 68 (2003), 739–752.
[21] T. Husain, Multiplicative functionals on topological algebras, Pitman Books Limited, Boston, London, Melbourne, 1983.
[22] J. Im Kang and R. Saadati, Approximation of homomorphisms and derivations on non-Archimedean random Lie C*-algebras via fixed point method, J. Ineq. Appl. 2012 (2012), 251, 1–10. [23] K. Jarosz, Almost multiplicative functionals, Studia Math. 124 (1997), no. 1, 37–58.
[24] K. Jarosz, Perturbations of Banach algebras, Lecture Notes in Mathematics, Vol. 1120 Springer-Verlag, Berlin, 1985.
[25] B.E. Johnson, Approximately multiplicative functionals, J. London Math. Soc. 2 (1986), 489–510.
[26] B.E. Johnson, Approximately multiplicative maps between Banach algebras, J. London Math. Soc. 37 (1988), no. 2, 294–316.
[27] A. Mallios, Topological Algebras, North Holland, 1986.
[28] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942) 535–537.
[29] E.A. Michael, Locally multiplicatively convex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 1–82.
[30] A.K. Mirmostafaee, Perturbation of generalized derivations in fuzzy Menger normed algebras, Fuzzy Sets Syst. 195 (2012), 109—117.
[31] Ch. Park, M.E. Gordji, and R. Saadati, Random homomorphisms and random derivations in random normed algebras via fixed point method, J. Ineq. Appl. 2012 (2012), 194, 1–10.
[32] Ch. Park, S.Y. Jang and R. Saadati, Fuzzy approximate of homomorphisms, J. Comp. Anal. Appl. 14 (2012), no.1, 833–841.
[33] E. Park and J. Trout, On the nonexistence of nontrivial involutive n-homomorphisms of C*-algebras, Trans. Am. Math. Soc. 361 (2009), 1949–1961.
[34] R. Saadati and Ch. Park, Approximation of derivations and the superstability in random Banach ∗-algebras, Adv. Diff. Equ. 418 (2018), 1–12.
[35] K.P.R. Sastry, G.A. Naidu, V.M. Latha, S.S.A. Sastri and I.L. Gayatri, Products of Menger probabilistic normed spaces, Gen. Math. Notes 2 (2011), no. 7, 15–23.
[36] B. Schweizer and A. Sklar, Probabilistic metric spaces, P. N. 275 North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co. New York 1983.
[37] P. ˇSemrl, Almost multiplicative functions and almost linear multiplicative functionals, Aeq. Math. 63 (2002), 180–192.
[38] P. ˇSemrl, Non-linear perturbations of homomorphisms on C(X), Quart. J. Math. Oxford. 50 (1999), 87–109.
[39] H. Shayanpour, E. Ansari-piri, Z. Heidarpor and A. Zohri, Approximately n-multiplicative functionals on Banach algebras, Mediterr. J. Math. 13 (2016), 1907–1920.
[40] H. Shayanpour, T.G. Honary and M.S. Hashemi, Certain properties of n-characters and n-homomorphisms on topological algebras, Bull. Malays. Math. Sci. Soc. 38 (2015), 985–999.
[41] R. Thakur and S.K. Samanta, Fuzzy Banach algebra, J. Fuzzy Math. 18 (2010), no. 3, 687–696.
[42] R. Thakur and S.K. Samanta, Fuzzy Banach algebra with Felbin’s type fuzzy norm, J. Fuzzy Math. 18 (2011), no. 4, 943–954.
Volume 14, Issue 6
June 2023
Pages 99-108
  • Receive Date: 18 February 2022
  • Revise Date: 08 May 2022
  • Accept Date: 21 May 2022
  • First Publish Date: 03 October 2022