---
tag: 7SB03M850M
kind: concept
name: Unitization
msc: [46H05, 16S99]
sources:
  - url: https://en.wikipedia.org/wiki/Banach_algebra
---

## Summary

The unitization of an algebra $A$ is the algebra $\mathbb{K} \oplus A$ with a product that makes $(1, 0)$ a unit.
It contains $A$ as an ideal of codimension one.
It lets one apply tools that need a unit, such as invertibility and the spectrum, to algebras without one.
For normed algebras and Banach algebras it carries a natural norm, and for C*-algebras a natural C*-norm.

## Explanations

### L1 human

Some algebras have no element that acts like the number $1$, for example the continuous functions on $\R$ that tend to $0$ at infinity.
The unitization adds such an element by hand.
Its elements are formal sums $\lambda + a$ of a scalar and an element of the algebra, multiplied by expanding the brackets.
For the functions vanishing at infinity, adding the constants gives the continuous functions on $\R$ that have the same limit at $+\infty$ and at $-\infty$.