The relationship between a commutative Banach algebra and its maximal ideal space

Our main result is an extension of a theorem due to Novodvorskii and Taylor; we give some special cases. Let A be a commutative Banach algebra with identity, and let Δ be its maximal ideal space. Let B be a Banach algebra with identity; let B^?1 denote the invertible group in B and id B denote the set of idempotents in B. Let $\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$] denote the set of path components of $\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$, and Δ, B^?1] denote the set of homotopy classes of continuous maps of Δ into B^?1. We prove that the Gelfand transform on A induces a bijection of $\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$] onto Δ, B^?1], and extend this result to prove a theorem of Davie. We show that the Gelfand transform induces a bijection of $\text{id}\text{(A}\text{}\text{?}\text{bo B)}$] onto Δ, id B], and investigate consequences of this result for specific examples of the Banach algebra B.