Definition 33.1.2 (Unital Banach Algebra).label Let $(A, \norm{\cdot}_{A})$ be a Banach algebra, then $A$ is unital if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, there exists an equivalent norm $\norm{\cdot}_{1}: A \to [0, \inftu)$ such that $\norm{1}_{1} = 1$, and $A$ is always assumed to be equipped with this norm.
Proof, [Proposition I.1.3, Tak01]. For each $x \in A$, let $L_{x} \in L(A; A)$ be defined by $y \mapsto xy$, and let $\norm{x}_{1} = \norm{L_x}_{L(A; A)}$, then $\norm{x}_{1} \le \norm{x}_{A}$ and $\norm{1}_{1} = 1$. On the other hand, $\frac{\norm{x}_{A}}{\norm{1}_{A}}\le \norm{x}_{1}$, so $\norm{\cdot}_{1}$ is equivalent to $\norm{\cdot}_{A}$.$\square$
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