Gleason-Kahane-Żelazko theorem for bilinear maps
Abstract
Let A and B be two unital Banach algebras and 𝖀=A×B. We prove that the bilinear mapping ϕ:𝖀→ℂ is a bi-Jordan homomorphism if and only if ϕ is unital, invertibility preserving and jointly continuous. Additionally, if 𝖀 is commutative, then ϕ is a bi-homomorphism.
Keywords
Gleason–Kahane–Żelazko theorem; preserves invertibility; bihomomorphism; bi-Jordan homomorphism
References
F.F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, 1973.
A.M. Gleason, A characterization of maximal ideals, J. Analyse Math. 19 (1967), 171–172.
N. Jacobson and C.E. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), no. 3, 479–502.
K. Jarosz, Generalizations of the Gleason–Kahane–Żelazko theorem, Rocky Mountain J. Math. 21 (1991), no. 3, 915–921.
J.-P. Kahane and W. Żelazko, A characterization of maximal ideals in commutative Banach algebras, Studia Math. 29 (1968), 339–343.
S.H. Kulkarni, Gleason–Kahane–Żelazko theorem for real Banach algebras, J. Math. Phys. Sci. 18 (1983/84), Special Issue, S19–S28.
T. Miura, S.-E. Takahasi, and G. Hirasawa, Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras, J. Inequal. Appl. 2005, no. 4, 435–441.
A.R. Sourour, The Gleason–Kahane–Żelazko theorem and its generalizations, in: J. Zemánek (ed.), Functional Analysis and Operator Theory, Banach Center Publ., 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994, pp. 327–331.
E.C. Titchmarsh, The Theory of Functions, Reprint of the second (1939) edition, Oxford University Press, Oxford, 1976.
W. Żelazko, A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math. 30 (1968), 83–85.
A. Zivari-Kazempour, A characterisation of 3-Jordan homomorphisms on Banach algebras, Bull. Aust. Math. Soc. 93 (2016), no. 2, 301–306.
A. Zivari-Kazempour, Automatic continuity of n-Jordan homomorphisms on Banach algebras, Commun. Korean Math. Soc. 33 (2018), no. 1, 165–170.
A. Zivari-Kazempour, When is a bi-Jordan homomorphism bi-homomorphism?, Kragujevac J. Math. 42 (2018), no. 4, 485–493.
A. Zivari-Kazempour, Characterization of n-Jordan homomorphisms and their automatic continuity on Banach algebras, Ann. Univ. Ferrara (2022). DOI: 10.1007/s11565-022-00425-6
Department of Mathematics, Ayatollah Borujerdi University, Iran Iran, Islamic Republic of
https://orcid.org/0000-0001-8362-8490
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