Character on Unital Banach Algebra is Unital Algebra Homomorphism
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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