Unital C*-Algebra is Unital Banach Algebra

Theorem

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra that is unital as an algebra, with identity element ${\mathbf 1}_A \ne {\mathbf 0}_A$.

Then $A$ is unital as a Banach algebra.

Proof

We have:

\(\ds \norm { {\mathbf 1}_A}^2\)

\(=\)

\(\ds \norm { {\mathbf 1}_A {\mathbf 1}_A^\ast}\)

Definition of C*-Algebra

\(\ds \)

\(=\)

\(\ds \norm { {\mathbf 1}_A^2}\)

Identity Element in Unital *-Algebra is Hermitian

\(\ds \)

\(=\)

\(\ds \norm { {\mathbf 1}_A}\)

From Norm Axiom $\text N 1$: Positive Definiteness, we have $\norm { {\mathbf 1}_A} \ne 0$.

Hence $\norm { {\mathbf 1}_A} = 1$.

$\blacksquare$

Sources

• 1990: Gerard J. Murphy: C*-Algebras and Operator Theory ... (previous) ... (next): $2.1$: $\text C^\ast$-Algebras