On the spectra of C⁎C⁎-algebras of slowly oscillating functions on locally compact groups

We present a study of C⁎C⁎-algebras SO(φ)SO(φ) of slowly oscillating functions in the direction of filters φ on a locally compact topological group G  . We show that SO(φ)SO(φ) is an m  -admissible subalgebra of C(G)C(G) if and only if the closure of the filter φ   in the LUCLUC-compactification GLUCGLUC of G   is an ideal of GLUCGLUC and that the semigroup compactification of G   determined by SO(φ)SO(φ) always contains right zero elements. Using this, we characterize a new interesting C⁎C⁎-algebra of bounded continuous functions on G  . The spectrum of this C⁎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 GLUCGLUC contained in the ideal U(G)U(G) of uniform points of GLUCGLUC can be decomposed into 22κ(G)22κ(G) closed left ideals of GLUCGLUC.