Definition:Unitization of Normed Algebra

Definition
Let $\GF \in \set {\R, \C}$.

Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$ that is not unital as an algebra.

Let $A_+$ be the unitization of $A$.

Define $\norm {\, \cdot \,}_{A_+} : A_+ \to \hointr 0 \infty$ by:


 * $\norm {\tuple {x, \lambda} }_{A_+} = \norm x + \cmod \lambda$

for each $\tuple {x, \lambda} \in A_+$.

We call $\struct {A_+, \norm {\, \cdot \,}_{A_+} }$ the unitization of $\struct {A, \norm {\, \cdot \,} }$.

Also see

 * Unitization of Normed Algebra is Unital Normed Algebra
 * Unitization of Normed Algebra is Banach Algebra iff Original Algebra is Banach Algebra
 * Normed Algebra Embeds into Unitization as Closed Ideal