Definition 33.1.5 (Unitisation).label Let $A$ be a Banach algebra over $\complex$, and $\tilde A = \complex \oplus A$ with
\[\iota: A \to \complex \oplus A \quad x \mapsto 0 + x\]
For each $\lambda + x, \mu + y \in \tilde A$, define
\[(\lambda + x)(\mu + y) = \lambda \mu + (\lambda x + \mu x + xy)\]
and
\[\norm{\lambda + x}_{\tilde A}= |\lambda| \norm{x}_{A}\]
then
- (1)
$\tilde A$ is a unital associative algebra over $\complex$.
- (2)
$\iota: A \to \tilde A$ is a homomorphism.
- (U)
For any pair $(B, \phi)$ satisfying (1) and (2), there exists a unique continuous unital homomorphism $\tilde \phi: \tilde A \to B$ such that $\phi(1) = 1$ and the following diagram commutes: A @->[r]^ & B
A @->[u]^ @->[ru]_ & - (4)
$\iota(A)$ is a closed two-sided ideal of $\tilde A$.
The algebra $\tilde A$ is the unitisation of $A$.
Proof. (U): For each $\lambda + x \in \tilde A$, let $\tilde \phi(\lambda + x) = \lamdba + \phi(x)$.$\square$
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