  We study the spectral synthesis for the Banach *-algebra $A\oop B$, the
operator space projective tensor product of $C^*$-algebras $A$ and $B$. It is
shown that if $A$ or $B$ has finitely many closed ideals, then $A\oop B$ obeys
spectral synthesis. The Banach algebra $A \oop A$ with the reverse involution
is also studied.
