Royden compactification of integers (Q674574)

The Gelfand theory of representations of commutative Banach algebras implies the existence of the Royden compactification of an infinite connected graph. This paper provides a method for constructing the Royden compactification for the special case of the discrete space of integers \(\mathbb{Z}\). It begins by examining the Royden algebra \(BD\) of bounded, Dirichlet finite functions and then constructs the Royden compactification of \(\mathbb{Z}\) as the Gelfand space of \(BD.\) This compactification is shown to be a quotient space of the Čech-Stone compactification of \(\mathbb{Z}\). An explicit description of both the equivalence relation and the quotient topology are also given.