On the uniqueness of uniform norms and $C^{\ast }$-norms
Volume 191 / 2009
Studia Mathematica 191 (2009), 263-270
MSC: 46J05, 46K05, 22D15.
DOI: 10.4064/sm191-3-7
Abstract
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.
