The first published proof of the Mazur-Gelfand theorem, due to Gelfand himself (though previously announced without proof by Mazur), is based on the vector-valued version of the Liouville theorem, which was further extended by Arens to cover a more general situation (see  and references therein).
 R. Arens (1947), Linear topological division algebras, Bull. AMS, Vol. 53, pp. 623-630.