It is shown that the Bourgain algebra $A{\left(\right)}_b$ of the disk algebra A() with respect to $H^{\infty }\left(\right)$ is the algebra generated by the Blaschke products having only a finite number of singularities. It is also proved that, with respect to $H^{\infty }\left(\right)$, the algebra QA of bounded analytic functions of vanishing mean oscillation is invariant under the Bourgain map as is $A{\left(\right)}_b$.

An example is given of a semisimple commutative Banach algebra that has the strong spectral extension property but fails the multiplicative Hahn-Banach property. This answers a question posed by M. J. Meyer in [4].