Algebras containing a divisible unimodular sequence with dense self adjoint span

We characterize the uniform algebras A on a compact Hausdorff space X which contain a sequence {u_j}_{j = 0}^∞ of unimodular elements with $\text{u}{}_{\mathrm{j}}\text{u}{}_{\mathrm{j\; ?\; 1}}\text{? A}$ and closed span $\text{{u}{}_{\mathrm{j}}\text{u}\text{}}{}_{\mathrm{j\; =\; 0}}{}^{\mathrm{\infty }}\text{= C(X)}$ in terms of the maximal ideal space of A. Roughly, the essential set of A looks like (at most) countably many copies of the boundary of the unit disk, and A looks like the disk algebra on each.