I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties:

  1. There exists a bounded approximate identity in $I$ for $I$ i.e., a net $\{e_\alpha\}\subset I$ such that $$ae_\alpha\to a,\quad e_\alpha a\to a,\quad a\in I.$$
  2. For all of the $a\in A$ with $Ia=\{0\}$ or $aI=\{0\}$ we get to $a=0$.
  3. There exists an element $x\in A$ with $x\not\in I$ and $xA_1x=\overline{\{xax:a\in A, \|a\|\leq1\}}^{\|.\|}$ is compact.

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