Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have $\phi(ab)=\phi(a)\phi(b)$. The set of all character is shown by $\sigma(A)$.

Now suppose that $A,B$ are Banach algebra such that $A$ is ideal of $B$. Let $\phi\in\sigma(A)$. Is there a $\psi\in\sigma(B)$ such that for all $a\in A, \psi(a)=\phi(a)$? If it exists, is it unique? If it doesn't exists ever, under which conditions it exists?


Since $\phi$ is non-zero you can choose $a\in A$ with $\phi(a)=1$. Then define $\psi(x)=\phi(ax)$. This is well-defined because $A$ is an ideal. It is clearly linear, and it is multiplicative because $\psi(xy)=\phi(axy)=\phi(axy)\phi(a) = \phi(axya)=\phi(ax)\phi(ya)=\psi(x)\phi(a)\phi(ya)=\psi(x)\phi(aya)=\psi(x)\phi(ay)\phi(a)=\psi(x)\psi(y).$

For $x\in A$ you have $\psi(x)=\phi(ax)=\phi(a)\phi(x)=\phi(x)$. Finally, $\psi$ is continuous because characters on Banach algebras are always continuous (or because of the continuity of $B\to A$, $x\mapsto ax$ which follows from the closed graph theorem if the inclusion $A\hookrightarrow B$ is continuous).

The extension is unique: If $\tilde \psi$ is any extension then $\tilde \psi(x)=\phi(a) \tilde \psi(x)=\tilde \psi(a)\tilde \psi(x)=\tilde \psi(ax)=\phi(ax)=\psi(x)$.

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    $\begingroup$ Impressive! Thanks! $\endgroup$ – Hamid Shafie Asl Apr 24 '14 at 8:32
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    $\begingroup$ I know this trick from E. Michael's thesis about locally m-convex algebras, but I guess that it is in fact even older. $\endgroup$ – Jochen Apr 24 '14 at 14:32
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    $\begingroup$ I'll check E. Michael's thesis! Thanks again. $\endgroup$ – Hamid Shafie Asl Apr 24 '14 at 19:42

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