Sufficient conditions for the projective freeness of Banach algebras (Q1044531)

The ring \(R\) is projective free if every finitely generated projective \(R\)-module is free. In this paper, it is shown that, if \(R\) is a semi-simple commutative unital complex Banach algebra for which the Banach algebra \(C(M(R))\) of complex continuous functions on the maximal ideal space \(M(R)\) is a projective free ring, then \(R\) itself is also a projective free ring. Moreover, if \(M(R)\) is connected and each complex vector bundle of finite rank on \(M(R)\) is topologically trivial, then \(C(M(R))\) is a projective free ring. Let \(U\) be an open set in a normal space \(X\). As an application, it is shown that the Hardy algebra \(H^{\infty}(U)\) of bounded holomorphic functions on \(U\) is projective free. Also, the algebras of almost periodic functions and of measures are considered.