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Let $(C_b\left(X\right),)$ 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 $(C_b\left(X\right),)$ 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 $(C_b\left(X\right),)$ has no proper non-zero closed principal ideals. Moreover if $X$ satisfy any of these two...