We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_s\left(X\right)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if X is non-separable,...

Let $T$ be a bounded representation of a commutative Banach algebra $B$. The following spectral sets are studied. ${\Lambda }_1\left(T\right)$: the Gelfand space of the quotient algebra $B/\mathrm{Ker}T$. ${\Lambda }_2\left(T\right)$: the Gelfand space of the operator algebra $\mathrm{Im}T$. ${\Lambda }_3\left(T\right)$: those characters $\phi$ of $B$ for which the inequalities $\parallel T_{b}x-\widehat{b}\left(\phi \right)x\parallel \<ϵ\parallel x\parallel$, $b\in F$, have a common solution $x\ne 0$, for any $ϵ\>0$ and any finite subset $F$ of $B$. A theorem of Beurling on the spectrum of $L^{\infty }$-functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of our theory by...