Spectral synthesis for operator space projective tensor product of $C^*$-algebras

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.

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