Banach algebras with unique uniform norm II
Volume 147 / 2001
Abstract
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.
