Let \(\Omega \) be a compact space and let \(A \subset C(\Omega )\) be a uniform algebra. Motivated by a recent paper of Feinstein and Izzo, we give a different proof of the result that weakly sequentially complete uniform algebras are finite dimensional. Our general formulation uses a classical result of Rudin and is different from the peak-set analysis of Feinstein and Izzo. Our approach also covers several conditions involving the interplay of weak*-weak topologies on the unit sphere of \(A^*\) and shows that under these conditions, A is finite dimensional. Some of our results also work in the case of point-separating closed subalgebras of \(C(\Omega )\) . This work also complements some of the work from (Nygaard, O., Werner, D.: Arch. Math. (Basel) 76, 441–444 (2001)), where a classification of uniform algebras, based on the interplay of weak and norm topologies of A was carried out.
