= Renorming a bounded multiplicative semigroup
{title2=$\|a\|_S=\|L_a\|_{(A,p)},\quad p(b)=\sup_{s\in S\cup\{1\}}\|sb\|$}
Let $S$ be a bounded multiplicatively closed subset of a unital <Banach algebra>. The displayed auxiliary <norm> $p$ is equivalent to the original norm, and each left multiplication $L_s$ is a contraction for $s\in S$. Taking the <operator norm> of the faithful left regular representation produces an equivalent <algebra norm> with $\|1\|_S=1$ and $\|s\|_S\leq1$. If $M=\max(\|1\|,\sup_S\|s\|)$, then $\|a\|/M\leq\|a\|_S\leq M\|a\|$. The auxiliary norm is itself submultiplicative, since $p(ab)\leq p(a)\|b\|\leq p(a)p(b)$, but $p(1)=M$ can exceed one. The operator norm enforces the required unital normalization.
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