Quotient of a Banach algebra

Question

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

How about $B=A\oplus A$, $I=A\oplus 0$. – Christian Remling Oct 01 '15 at 16:17 — Christian Remling, Oct 01 '15 at 16:17

score 3 · Accepted Answer · answered Oct 01 '15 at 16:17

3

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 $p$.

answered Oct 01 '15 at 16:17

Bill Johnson

31,077

3

Bill, I really don't think one should encourage questions like this on MO. Cf. http://mathoverflow.net/questions/217749/ideal-in-projective-tensor-product-of-banach-algebras – Yemon Choi Oct 01 '15 at 18:30
You are right, Yemon. Sorry. – Bill Johnson Oct 02 '15 at 20:03

Quotient of a Banach algebra

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