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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...