A characteristic property of the algebra $C(\Omega)_\beta$

We study some properties of the algebras of continuous functions on a locally compact space whose topology is defined by the family of all multiplication operators ($\beta$-uniform algebras). We introduce the notion of a $\beta$-amenable algebra and show that a $\beta$-uniform algebra is $\beta$-amenable if and only if it coincides with the algebra of bounded functions on a locally compact space (an analog of M. V. Sheinberg's theorem for uniform algebras).