Banach algebras with unique uniform norm II

S. J. Bhatt¹ ; H. V. Dedania¹

¹ Department of Mathematics Sardar Patel University Vallabh Vidyanagar 388120, India

Studia Mathematica, Tome 147 (2001) no. 3, pp. 211-235

Résumé

Semisimple commutative Banach algebras ${\cal A}$ admitting exactly one uniform norm (not necessarily complete) are investigated. ${\cal A}$ has this Unique Uniform Norm Property iff the completion $U({\cal A})$ of ${\cal A}$ in the spectral radius $r(\cdot )$ has UUNP and, for any non-zero spectral synthesis ideal ${\cal I}$ of $U({\cal A})$, ${\cal I}\cap {\cal A}$ is non-zero. ${\cal A}$ is regular iff $U({\cal A})$ is regular and, for any spectral synthesis ideal ${\cal I}$ of ${\cal A}$, ${\cal A}/{\cal I}$ has UUNP iff $U({\cal A})$ is regular and for any spectral synthesis ideal ${\cal I}$ of $U({\cal A})$, ${\cal I} = k(h({\cal A} \cap {\cal I}))$ (hulls and kernels in $U({\cal A})$). ${\cal A}$ has UUNP and the Shilov boundary coincides with the Gelfand space iff ${\cal A}$ is weakly regular in the sense that, given a proper, closed subset $F$ of the Gelfand space, there exists a non-zero $x$ in ${\cal A}$ having its Gelfand transform vanishing on $F$. Several classes of Banach algebras that are weakly regular but not regular, as well as those that are not weakly regular but have UUNP are exhibited. The UUNP is investigated for quotients, tensor products, and multiplier algebras. The property UUNP compares with the unique $C^{*}$-norm property on (not necessarily commutative) Banach $^*$-algebras. The results are applied to multivariate holomorphic function algebras as well as to the measure algebra of a locally compact abelian group $G$. For a continuous weight $\omega $ on $G$, the Beurling algebra $L^1(G,\omega )$ (assumed semisimple) has UUNP iff it is regular.

Détail

DOI : 10.4064/sm147-3-2

Keywords: semisimple commutative banach algebras cal admitting exactly uniform norm necessarily complete investigated cal has unique uniform norm property completion cal cal spectral radius cdot has uunp non zero spectral synthesis ideal cal cal cal cap cal non zero cal regular cal regular spectral synthesis ideal cal cal cal cal has uunp cal regular spectral synthesis ideal cal cal cal cal cap cal hulls kernels cal cal has uunp shilov boundary coincides gelfand space cal weakly regular sense given proper closed subset gelfand space there exists non zero cal having its gelfand transform vanishing several classes banach algebras weakly regular regular those weakly regular have uunp exhibited uunp investigated quotients tensor products multiplier algebras property uunp compares unique * norm property necessarily commutative banach * algebras results applied multivariate holomorphic function algebras measure algebra locally compact abelian group continuous weight omega beurling algebra omega assumed semisimple has uunp regular