A spectral synthesis property for C_b (X, β)

Let (Cb (X) , ) be the algebra of all continuous bounded real or complex valued functions defined on a completely regular Hausdorff space X with the usual algebraic operations and with the strict topology . It is proved that (Cb (X) , β) has a spectral synthesis, i.e. every of its closed ideals is an intersection of closed maximal ideals of codimension 1. We give one necessary and two sufficient conditions over X in order that (Cb (X) , β) has no proper non-zero closed principal ideals. Moreover if X satisfy any of these two conditions and is also a k-space, then any non zero element of Cb (X) is invertible or a topological divisor of zero.