33.1 Banach Algebras

Definition 33.1.1 (Banach Algebra).label Let $A$ be an associative algebra over $\complex$ and $\norm{\cdot}_{A}: A \to [0, \infty)$ be a norm, then $A$ is a Banach algebra if:

1. (1)

   $A$ is complete with respect to $\norm{\cdot}_{A}$.

2. (2)

   For any $x, y \in A$, $\norm{xy}_{A} \le \norm{x}_{A}\norm{y}_{A}$.

Definition 33.1.2 (Unital Banach Algebra).label Let $(A, \norm{\cdot}_{A})$ be a Banach algebra, then $A$ is unital if there exists $1 \in A$ such that for any $x \in A$, $x1 = 1x = x$. In which case, there exists an equivalent norm $\norm{\cdot}_{1}: A \to [0, \inftu)$ such that $\norm{1}_{1} = 1$, and $A$ is always assumed to be equipped with this norm.

Definition 33.1.3 (Homomorphism).label Let $A, B$ be Banach algebras and $\phi: A \to B$, then $\phi$ is a homomorphism if:

1. (1)

   $\phi \in L(A; B)$.

2. (2)

   For each $x, y \in A$, $\phi(xy) = \phi(x)\phi(y)$.

Definition 33.1.4 (Unital Homomorphism).label Let $A, B$ be unital Banach algebras and $\phi: A \to B$ be a homomorphism, then $\phi$ is a unital homomorphism if $\phi(1) = 1$.

Definition 33.1.5 (Unitisation).label Let $A$ be a Banach algebra over $\complex$, and $\tilde A = \complex \oplus A$ with

For each $\lambda + x, \mu + y \in \tilde A$, define

and

then

1. (1)

   $\tilde A$ is a unital associative algebra over $\complex$.

2. (2)

   $\iota: A \to \tilde A$ is a homomorphism.

3. (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. (4)

   $\iota(A)$ is a closed two-sided ideal of $\tilde A$.

The algebra $\tilde A$ is the unitisation of $A$.