Character on Unital Banach Algebra is Unital Algebra Homomorphism

Theorem

Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra over $\C$.

Let $\phi : A \to \C$ be a character on $A$.

Then $\phi$ is a unital algebra homomorphism.

Proof

By the definition of a character, $\phi$ is a non-zero algebra homomorphism.

We only need to verify that:

$\map \phi { {\mathbf 1}_A} = 1$

We have:

$\map \phi { {\mathbf 1}_A} = \map \phi { {\mathbf 1}_A^2} = \paren {\map \phi { {\mathbf 1}_A} }^2$

So, we have:

$\map \phi { {\mathbf 1}_A} \in \set {0, 1}$

Note that for all $x \in A$, we have:

$\map \phi x = \map \phi { {\mathbf 1}_A} \map \phi x$

Hence if we had $\map \phi { {\mathbf 1}_A} = 0$, we would have $\phi = 0$.

Since $\phi \ne 0$, we must therefore have $\map \phi { {\mathbf 1}_A} = 1$.

So $\phi$ is a unital algebra homomorphism.

$\blacksquare$

Sources

• 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $4.10$: The continuity of characters