• Uniqueness of the uniform norm and adjoining identity in Banach algebras

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      https://www.ias.ac.in/article/fulltext/pmsc/105/04/0405-0409

    • Keywords

       

      Adjoining identity to a Banach algebra; unique uniform norm property; spectral extension property; regular norm; weakly regular Banach algebra

    • Abstract

       

      LetAe be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, ∥ · ∥). Unlike the case for aC*-norm on a Banach *-algebra,Ae admits exactly one uniform norm (not necessarily complete) if so doesA. This is used to show that the spectral extension property carries over fromA to Ae. Norms onAe that extend the given complete norm ∥ · ∥ onA are investigated. The operator seminorm ∥ · ∥op onAe defined by ∥ · ∥ is a norm (resp. a complete norm) iffA has trivial left annihilator (resp. ∥ · ∥op restricted toA is equivalent to ∥ · ∥).

    • Author Affiliations

       

      S J Bhatt1 H V Dedania1 2

      1. Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar - 388120, India
      2. Department of Mathematics, University of Leeds, Leeds - LS2 9JT, UK
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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