Rings of continuous functions vanishing at infinity on a frame

Abstract

Let C(X) denote the ring of all real-valued continuous functions on a topological space X; and C∞(X) be the subring of all functions C(X) which vanish at infinity. In [2], the paper “Rings of continuous functions vanishing at infinity,” (Comment. Math. Univ. Carolin. 45(3) (2004), 519–533), by A.R. Aliabad, F. Azarpanah, and M. Namdari, it is shown that for every completely regular Hausdorff space X, whenever C∞(X) ̸= (0), then there exists a locally compact space Y such that C∞(X) ∼= C∞(Y ). In fact, the space Y may be considered as a nonempty open locally compact subspace of X. In the present paper, analogous results are derived in a pointfree context in which topological spaces are replaced by frames.

Key words: Frame, locally compact space, continuous frame, vanish at infinity.