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Let $A$ be a Banach algebra. Is there a Banach algebra $B$ and a non-trivial closed ideal $I$ of $B$ such that $\frac{B}{I}\cong A$?

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    $\begingroup$ How about $B=A\oplus A$, $I=A\oplus 0$. $\endgroup$
    – Christian Remling
    Commented Oct 1, 2015 at 16:17

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Sure. Let $K$ be any compact Hausdorff space that contains at least two points and for $B$ take the space of $A$ valued continuous functions on $K$. Take any $p$ in $K$. Let $I$ be the ideal of all functions in $B$ that vanish at $p$.

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    $\begingroup$ Bill, I really don't think one should encourage questions like this on MO. Cf. mathoverflow.net/questions/217749/… $\endgroup$
    – Yemon Choi
    Commented Oct 1, 2015 at 18:30
  • $\begingroup$ You are right, Yemon. Sorry. $\endgroup$
    – Bill Johnson
    Commented Oct 2, 2015 at 20:03

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