Uniqueness of the uniform norm and adjoining identity in Banach algebras

Bhatt, S. J. ; Dedania, H. V. (1995) Uniqueness of the uniform norm and adjoining identity in Banach algebras Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 105 (4). pp. 405-409. ISSN 0253-4142

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Official URL: http://www.ias.ac.in/j_archive/mathsci/105/4/405-4...

Related URL: http://dx.doi.org/10.1007/BF02836876

Abstract

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

Item Type:Article
Source:Copyright of this article belongs to Indian Academy of Sciences.
Keywords:Adjoining Identity to a Banach algebra; Unique Uniform Norm Property; Spectral Extension Property; Regular Norm; Weakly Regular Banach Algebra
ID Code:59676
Deposited On:07 Sep 2011 05:16
Last Modified:18 May 2016 10:09

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