  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 $H^\infty$ 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.
