= Unitization of an algebra
= Unitization
{synonym}
The <unitization> of a complex <associative algebra> $A$ is $A^+=A\oplus\mathbb C$ with product
$$
(a,\lambda)(b,\mu)=(ab+\lambda b+\mu a,\lambda\mu).
$$
Its identity is $(0,1)$, and $a\mapsto(a,0)$ identifies $A$ with a two-sided <ideal>. If $A$ is a <Banach algebra>, the <norm> $\|(a,\lambda)\|=\|a\|+|\lambda|$ makes $A^+$ a unital <Banach algebra> with the original <norm> on $A$. The product formula directly proves <submultiplicativity>.<Lectures on C-star algebras by Vahid Shirbisheh>[Definition 2.1.12, printed p.11]{ref}
For a noncompact <Hausdorff space> $X$ that is a <locally compact space>, adjoining constant <functions> identifies the unitization of $C_0(X)$ algebraically with $C(X^+)$, where $X^+$ is the <one-point compactification>. This does not identify the displayed sum <norm> with the <supremum norm>: those are equivalent <norms>, but the <C-star unitization> uses the latter. Unitization lets definitions such as the <spectrum of an element> apply to algebras that initially have no identity.
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