Given a Banach algebra $A$.
Consider a onesided approximate unit: $$e:\Lambda\to A:\quad ae_\lambda\stackrel{\lambda}\to a\quad(\forall a\in A)$$
Does it follow that it is eventually bounded: $$\exists\lambda_0\in\Lambda:\quad\sup_{\lambda\geq\lambda_0}\|e_\lambda\|<\infty$$ What about it in the C*-algebra case?
Clearly the restriction to eventual boundedness is inevitable as
otherwise there are trivial counterexamples merely due to the order on the index set.
(I couldn't come up with construction that works so far..)