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*-norms on *-semisimple, commutative Banach *-algebras. These properties are preserved if the identity is adjoined. We also show that a commutative Beurling *-algebra L¹(G,ω) has exactly one uniform norm if and only if it has exactly one C*-norm; this is not true in arbitrary *-semisimple, commutative Banach *-algebras.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-3-7,
author = {P. A. Dabhi and H. V. Dedania},
title = {On the uniqueness of uniform norms and C*-norms},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {263-270},
zbl = {1173.46030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-3-7}
}
P. A. Dabhi; H. V. Dedania. On the uniqueness of uniform norms and C*-norms. Studia Mathematica, Tome 192 (2009) pp. 263-270. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm191-3-7/