If $A$ is a Banach algebra and $L$ a left ideal of $A$, consider the representation $T_{L}$ of $A$ into the algebra $B(A/L)$ of bounded linear operators on the quotient space $A/L$ defined by $T_{L}(a)(b+L)=ab+L$. When is $T_{L}(A)$ closed in the norm of $B(A/L)$? Certainly this happens if $A/L$ is finite-dimensional. Are there other cases? Thanks, Paul.
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4$\begingroup$ How did this question arise? $\endgroup$– Nik WeaverCommented Mar 22, 2020 at 18:34
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1$\begingroup$ I am currently studying the invariant subspaces of strictly cyclic Banach algebras and their reflexivity (as operator algebras). $\endgroup$– Paul VisoianuCommented Mar 22, 2020 at 18:54
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$\begingroup$ Well, it would be easy to give examples. In "reasonable" cases the kernel of $T_L$ is $L$ and the norm of $T_L(a)$ equals the norm of $a + L$ in $A/L$. So $T_L(A)$ is isometric to $A$ and therefore complete. There will be degenerate examples where that isn't true, though. A good question would be: is $T_L(A)$ always closed? $\endgroup$– Nik WeaverCommented Mar 22, 2020 at 19:58
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$\begingroup$ Thanks, but I do not know when this happens in general. It may happen even if the norms are not equal. $\endgroup$– Paul VisoianuCommented Mar 22, 2020 at 20:12
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$\begingroup$ The kernel is a two sided ideal, cannot equal L (a left ideal), $\endgroup$– Paul VisoianuCommented Mar 23, 2020 at 21:10
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