On the Bicomplex Gleason–Kahane–Żelazko Theorem

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• 作者：M. E. Luna-Elizarrarás ; C. O. Pérez-Regalado…
• 关键词：Bicomplex modules ; Hyperbolic ; valued norm ; Bicomplex functionals ; Bicomplex holomorphic functions ; Gleason–Kahane–Żelazko theorem ; 46A22 ; 46A19 ; 30G35
• 刊名：Complex Analysis and Operator Theory
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：10
• 期：2
• 页码：327-352
• 全文大小：526 KB
• 参考文献：1.Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C.: Basics of functional analysis with bicomplex scalars, and bicomplex Schur analysis, p. 95. XV, Springer Briefs in Mathematics (2014)
  2.Helemskii, A.Ya.: Lectures and Exercises on Functional Analysis. Translations of Mathematical Monographs, vol. 233. AMS, Providence (2006)
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  4.Luna-Elizarrarás, M.E., Perez-Regalado, C.O., Shapiro, M.: On linear functionals and Hahn–Banach theorems for hyperbolic and bicomplex modules. Adv. Appl. Clifford Algebras 23(4), 1105–1129 (2014)CrossRef
  5.Luna, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex numbers and their elementary functions. Cubo A Math. J. 14(2), 61–80 (2012)CrossRef MATH
  6.Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Complex Laplacian and derivatives of bicomplex functions. Complex Anal. Oper. Theory 7(5), 1675–1711 (2013)CrossRef MathSciNet MATH
  7.Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers. Birkhäuser, Basel (2015, to appear)
  8.Price, G.B.: An Introduction to Multicomplex Spaces and Functions. Monographs and Textbooks in Pure and Applied Mathematics, vol. 140. Marcel Dekker Inc, New York (1991)
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  10.Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)MATH
  
• 作者单位：M. E. Luna-Elizarrarás (1)
  C. O. Pérez-Regalado (1)
  M. Shapiro (1)
  
  1. Escuela Superior de Fisica y Matemáticas, Instituto Politécnico Nacional, 07338, Mexico City, Mexico
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Operator Theory
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1661-8262

文摘

We prove that a bicomplex linear functional acting on a bicomplex Banach algebra (with a hyperbolic-valued norm) in such a way that invertible elements are transformed into invertible bicomplex numbers is, in fact, a multiplicative functional and thus, an algebra homomorphism. We give two proofs of this. The first of them is based on the theory of bicomplex holomorphic functions and we present here a number of previously not published facts; the second uses its complex antecedent (classic Gleason–Kahane–Żelazko theorem). Keywords Bicomplex modules Hyperbolic-valued norm Bicomplex functionals Bicomplex holomorphic functions Gleason–Kahane–Żelazko theorem