Geodesic
On the uniqueness of uniform norms and $C^{\ast }$-norms
P. A. Dabhi 1   ; H. V. Dedania 1
1 Department of Mathematics Sardar Patel University Vallabh Vidyanagar 388 120 Gujarat, India
Studia Mathematica, Tome 191 (2009) no. 3, pp. 263-270 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We prove that a semisimple, commutative Banach algebra has either exactly one uniform norm or infinitely many uniform norms; this answers a question asked by S. J. Bhatt and H. V. Dedania [Studia Math. 160 (2004)]. A similar result is proved for $C^\ast $-norms on $\ast $-semisimple, commutative Banach $\ast $-algebras. These properties are preserved if the identity is adjoined. We also show that a commutative Beurling $\ast $-algebra $L^1(G,\omega )$ has exactly one uniform norm if and only if it has exactly one $C^\ast $-norm; this is not true in arbitrary $\ast $-semisimple, commutative Banach $\ast $-algebras.
DOI : 10.4064/sm191-3-7
Keywords: prove semisimple commutative banach algebra has either exactly uniform norm infinitely many uniform norms answers question asked bhatt dedania studia math similar result proved ast norms ast semisimple commutative banach ast algebras these properties preserved identity adjoined commutative beurling ast algebra omega has exactly uniform norm only has exactly ast norm arbitrary ast semisimple commutative banach ast algebras

P. A. Dabhi  1   ; H. V. Dedania  1

1 Department of Mathematics Sardar Patel University Vallabh Vidyanagar 388 120 Gujarat, India 
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P. A. Dabhi; H. V. Dedania. On the uniqueness of uniform norms and $C^{\ast }$-norms. Studia Mathematica, Tome 191 (2009) no. 3, pp. 263-270. doi: 10.4064/sm191-3-7

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