On Homomorphisms of Douglas Algebras

The paper describes homomorphisms between Douglas algebras and some semisimple Banach algebras. The main tool is a result on the structure of the space $C(Z,\mathfrak{M})$ of continuous mappings from a connected first-countable $T_1$ space $Z$ to the maximal ideal space $\mathfrak{M}$ of the algebra $H^{\mathrm{\infty }}$ of bounded holomorphic functions on the unit disk $\mathbb{D}\subset \mathbb{C}$. In particular, it is shown that the space of continuous mappings from $Z$ to $\mathbb{D}$ is dense in the topology of pointwise convergence in $C(Z,\mathfrak{M})$ and the homotopy groups of $\mathfrak{M}$ are trivial.