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Abstract

Let $\mathfrak M$ be the maximal ideal space of the algebra $H^\infty $ of bounded holomorphic functions on the unit disk $\mathbb {D}\subset \mathbb {C}$. The classical results of K. Hoffman describe complex-analytic maps from a connected complex-analytic space $X$ to $\mathfrak M$. In particular, the image of every such map belongs to a Gleason part, and the space of holomorphic maps from $X$ to $\mathbb {D}$ is dense in the topology of pointwise convergence in the space of holomorphic maps from $X$ to $\mathfrak M$. In this paper, we extend Hoffman’s results to other classes of continuous maps from certain topological spaces to $\mathfrak M$.