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.
@article{10_4064_sm191_3_7,
author = {P. A. Dabhi and H. V. Dedania},
title = {On the uniqueness of uniform norms and $C^{\ast }$-norms},
journal = {Studia Mathematica},
pages = {263--270},
year = {2009},
volume = {191},
number = {3},
doi = {10.4064/sm191-3-7},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm191-3-7/}
}
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AU - H. V. Dedania
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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
