Definition:C*-Algebra

Definition

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a Banach $\ast$-algebra over $\C$ such that:

\((\text C^* 5)\)

$:$

\(\ds \forall x \in A:\)

\(\ds \norm {x x^*} \)

\(\ds = \)

\(\ds \norm x^2 \)

Then $\struct {A, \ast, \norm {\, \cdot \,} }$ is referred to as a $\text C^*$-algebra.

That is, a $\text C^*$-algebra is a Banach algebra $\struct {A, \norm {\, \cdot \,} }$ equipped with an involution $\ast : A \to A$ satisfying:

\((\text C^* 1)\)	$:$			\(\ds \forall x \in A:\)		\(\ds x^{**} \)	\(\ds = \)	\(\ds x \)
\((\text C^* 2)\)	$:$			\(\ds \forall x \in A:\)		\(\ds x^* + y^* \)	\(\ds = \)	\(\ds \paren {x + y}^* \)
\((\text C^* 3)\)	$:$			\(\ds \forall x, y \in A:\)		\(\ds x^* y^* \)	\(\ds = \)	\(\ds \paren {y x}^* \)
\((\text C^* 4)\)	$:$			\(\ds \forall x \in A, c \in \C:\)		\(\ds \paren {c x}^* \)	\(\ds = \)	\(\ds \overline c \paren x^* \)
\((\text C^* 5)\)	$:$			\(\ds \forall x \in A:\)		\(\ds \norm {x x^*} \)	\(\ds = \)	\(\ds \norm x^2 \)

We call $(\text C^\ast 5)$ the $\text C^*$ identity.

Also known as

An older term for $\text C^*$-algebra is $\text B^*$-algebra.

Also see

• C* Identity implies Involution is Isometry, in which is proven the redundancy of the requirement that $\norm a = \norm {a^\ast}$ for all $a \in A$, given in the definition of a Banach $\ast$-Algebra, in the presence of $(\text C^\ast 5)$

• Results about $\text C^*$-algebras can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): $\text B^*$-algebra
• 1990: Gerard J. Murphy: C*-Algebras and Operator Theory ... (previous) ... (next): $2.1$: $\text C^\ast$-Algebras
• 2017: Manfred Einsiedler and Thomas Ward: Functional Analysis, Spectral Theory, and Applications Section $11.2$: $C^*$-algebras: Definition $11.16$