Structure of biprojective Banach algebras with non-trivial radical

We study the structure of biprojective Banach algebras. In contrast to earlier results of Selivanov, we admit the presence of nilpotent ideals in the algebras under consideration, and the structure theorem covers almost all known examples. As a corollary, we obtain a complete classification of finite-dimensional biprojective Banach algebras. A major role in the proof is played by the approximation property for certain Banach spaces related to the algebras under consideration.