We show that the uniform compactification of a uniform space (X, U) can be considered as a space of filters on X. We apply these filters to study the LUC-compactification of a topological group.2010 Mathematics Subject Classification: Primary 54D35; Secondary 54E15.

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Let F be a lattice of real-valued functions on a non-empty set X such that F contains the constant functions. Using certain filters on X determined by F, we construct a compact Hausdorff topological space δX with the property that every bounded member of F extends to δX and these extensions form a dense subspace of C(δX). If A is any C *-subalgebra of ℓ ∞ (X) containing the constant functions, then our construction gives a representation of the spectrum of A as a space of filters on X.

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We present a study of C * -algebras SO(ϕ) of slowly oscillating functions in the direction of filters ϕ on a locally compact topological group G. We show that SO(ϕ) is an m-admissible subalgebra of C(G) if and only if the closure of the filter ϕ in the LU C-compactification G LU C of G is an ideal of G LU C and that the semigroup compactification of G determined by SO(ϕ) always contains right zero elements. Using this, we characterize a new interesting C * -algebra of bounded continuous functions on G. The spectrum of this C * -algebra determines the universal semigroup compactification of G with respect to the property that the semigroup compactification contains a right zero element. We show that the topological center of this universal compactification is G. As an application of the previous results, we show that every closed ideal of G LU C contained in the ideal U (G) of uniform points of G LU C can be decomposed into 2 2 κ(G) closed left ideals of G LU C .