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Uniqueness of the uniform norm and adjoining identity in Banach algebras

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Abstract

LetA e be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, ∥ · ∥). Unlike the case for aC*-norm on a Banach *-algebra,A e 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 A e . Norms onA e that extend the given complete norm ∥ · ∥ onA are investigated. The operator seminorm ∥ · ∥op onA e defined by ∥ · ∥ is a norm (resp. a complete norm) iffA has trivial left annihilator (resp. ∥ · ∥op restricted toA is equivalent to ∥ · ∥).

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Authors and Affiliations

  1. Department of Mathematics, Sardar Patel University, 388120, Vallabh Vidyanagar, India

    S. J. Bhatt & H. V. Dedania

  2. Department of Mathematics, University of Leeds, LS2 9JT, Leeds, UK

    H. V. Dedania

Authors
  1. S. J. Bhatt
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  2. H. V. Dedania
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Bhatt, S.J., Dedania, H.V. Uniqueness of the uniform norm and adjoining identity in Banach algebras. Proc. Indian Acad. Sci. (Math. Sci.) 105, 405–409 (1995). https://doi.org/10.1007/BF02836876

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  • DOI: https://doi.org/10.1007/BF02836876

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