BANACH ALGEBRAS WEAK^* GENERATED BY THEIR IDEMPOTENTS

Abstract

For a closed set $E$ contained in the closed unit interval, we show that the big Lipschitz algebra $\varLambda_{\gamma}(E)$ $(0\lt\gamma\lt1)$ is sequentially weak$^{\ast}$ generated by its idempotents if and only if it is weak$^{\ast}$ generated by its idempotents if and only if the little Lipschitz algebra $\lambda_{\gamma}(E)$ is generated by its idempotents, and we describe a class of perfect symmetric sets for which this holds. Moreover, we prove that $\varLambda_1(E)$ is sequentially weak$^{\ast}$ generated by its idempotents if and only if $E$ is of measure zero. Finally, we show that the quotient algebras

$$ \mathcal{A}_{\beta}/\overline{J_{\beta}(E)}^{\text{weak}^{\ast}} $$

of the Beurling algebras need not be weak$^{\ast}$ generated by their idempotents, when $E$ is of measure zero and $\beta\ge\tfrac{1}{2}$.

AMS 2000 Mathematics subject classification: Primary 46J10; 46J30; 26A16; 42A16