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.
