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
- 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
Sources
- 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $4.3$: Elementary constructions