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Let $A,B$ be Banach algebras and $A\hat{\otimes}B$ be projective tensor product of them. Let $S$ be an ideal of $A\hat{\otimes}B$. Are there ideals $I$ of $A$ and $J$ of $B$ such that $S=I\hat{\otimes}J$?

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  • $\begingroup$ I'm voting to close this question as off-topic because it should have been though through more carefully, by considering finite-dimensional cases $\endgroup$
    – Yemon Choi
    Commented Sep 17, 2015 at 16:31

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I suppose that you mean closed ideals. Howver the result is not even true in the simplest non-trivial case, that of two dimensional algebras. This is most easily seen by dualising it to the obviously false claim that closed subsets of products are products of closed subsets.

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