Definition:Banach *-Algebra

Definition

Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.

Let $\norm {\, \cdot \,}$ be an algebra norm on $A$ such that $\struct {A, \norm {\, \cdot \,} }$ is a Banach algebra and:

$\norm {a^\ast} = \norm a$

for each $a \in A$.

We say that $\struct {A, \ast, \norm {\, \cdot \,} }$ a Banach $\ast$-algebra.

Also see

• Results about Banach $\ast$-algebras can be found here.

Source of Name

This entry was named for Stefan Banach.

Sources

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