  The paper studies projective freeness and Hermiteness of algebras of
complex-valued continuous functions on topological spaces, Stein algebras, and
commutative unital Banach algebras. New sufficient cohomology conditions on the
maximal ideal spaces of the algebras are given that guarantee the fulfilment of
these properties. The results are illustrated by nontrivial examples. Based on
the Borsuk theory of shapes, a new class $\mathscr{C}$ of commutative unital
complex Banach algebras is introduced (an analog of the class of local rings in
commutative algebra) such that the projective tensor product with algebras in
$\mathscr C$ preserves projective freeness and Hermiteness. Some examples of
algebras of class $\mathscr{C}$ and of other projective free and Hermite
function algebras are assembled. These include, e.g., Douglas algebras,
finitely generated algebras of symmetric functions, Bohr-Wiener algebras,
algebras of holomorphic semi-almost periodic functions, and algebras of bounded
holomorphic functions on Riemann surfaces.
