The author generalizes the Arzela-Ascoli theorem to the setting of matrix order unit spaces, extending the work of Antonescu-Christensen on unital C^＊-algebras. This gives an affirmative answer to a question of Antonescu and Christensen.
A C*-metric algebra consists of a unital C*-algebra and a Leibniz Lip-norm on the C*-algebra. We show that if the Lip-norms concerned are lower semicontinuous, then any unital *-homomorphism from a C*-metric algebra to another one is necessarily Lipschitz. We come to the result that the free product of two unital completely Lipschitz contractive *-homomorphisms from upper related C*-metric algebras coming from *-filtrations to those which are lower related is a unital Lipschitz *-homomorphism.
