Unitization of an algebra

The unitization of a complex associative algebra $A$ is $A^{+}=A\oplus \mathbb{C}$ with productIts identity is $(0,1)$, and $a\mapsto (a,0)$ identifies $A$ with a two-sided ideal. If $A$ is a Banach algebra, the norm $\Vert (a,\lambda )\Vert =\Vert a\Vert +|\lambda |$ makes $A^{+}$ a unital Banach algebra with the original norm on $A$. The product formula directly proves submultiplicativity.^[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.