Closed ideals and Lie ideals of minimal tensor product of certain C*-algebras

For a locally compact Hausdorff space $X$ and a $C^{\ast }$-algebra $A$ with only finitely many closed ideals, we discuss a characterization of closed ideals of $C_0(X,A)$ in terms of closed ideals of $A$ and certain (compatible) closed subspaces of $X$. We further use this result to prove that a closed ideal of $C_0(X){\otimes }^{\min }A$ is a finite sum of product ideals. We also establish that for a unital $C^{\ast }$-algebra $A$, $C_0(X,A)$ has centre-quotient property if and only if $A$ has centre-quotient property. As an application, we characterize the closed Lie ideals of $C_0(X,A)$ and identify all closed Lie ideals of $C_0(X){\otimes }^{\min }B(H)$, $H$ being a separable Hilbert space.