= Renorming a separately continuous Banach algebra
{title2=$\|x\|_*=\|L_x\|$}
If an associative unital <algebra> has a complete <norm> and separately continuous multiplication, the <Uniform boundedness principle> applied to $L_x(y)=xy$, for $\|x\|\leq1$, yields $\|xy\|\leq C\|x\|\|y\|$. The <operator norm> of $L_x$ then defines an equivalent <algebra norm>: $\|x\|/\|e\|\leq\|L_x\|\leq C\|x\|$, and $L_{xy}=L_xL_y$ proves <submultiplicativity>. It normalizes the identity to norm one.
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