Definition:Unitization of Algebra over Field

Definition

Let $K$ be a field.

Let $A$ be an algebra over $K$ that is not unital.

Let $A_+ = A \times K$ be the direct product of $A$ and $K$ as vector spaces over $K$ with vector addition $+_{A \times K}$ and scalar multiplication $\cdot_{A \times K}$.

Define multiplication $\circ_{A \times K}$ on $A \times K$ by:

$\tuple {a, \lambda} \circ_{A \times K} \tuple {b, \mu} = \tuple {a b + \lambda b + \mu a, \lambda \mu}$

for each $\tuple {a, \lambda}, \tuple {b, \mu} \in A \times K$.

We say that $\struct {A \times K, +_{A \times K}, \cdot_{A \times K}, \circ_{A \times K} }_K$ is the unitization of $A$ and write $A_+$ for $A \times K$ equipped with these operations.

If $A$ is already unital, we set $A_+ = A$.

Also see

• Unitization of Algebra over Field is Unital Algebra over Field
• Algebra over Field Embeds into Unitization as Ideal

Sources

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