Definition:Unitization of Normed Algebra

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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


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