\begin{definition}[Definition:B*-Algebra]
Let $A$ be a Banach algebra.
Let $A$ have a conjugate-linear anti-automorphic involution $*$ satisfying the following axioms:
{{begin-axiom}}
{{axiom | n = \text B^* 1
        | q = \forall x \in A
        | ml= x^{**}
        | mo= =
        | mr= x
}}
{{axiom | n = \text B^* 2
        | q = \forall x \in A
        | ml= x^* + y^*
        | mo= =
        | mr= \paren {x + y}^*
}}
{{axiom | n = \text B^* 3
        | q = \forall x, y \in A
        | ml= x^* y^* 
        | mo= =
        | mr= \paren {y x}^*
}}
{{axiom | n = \text B^* 4
        | q = \forall x, c \in A
        | ml= \paren {c x}^*
        | mo= =
        | mr= \overline c \paren x^*
}}
{{axiom | n = \text B^* 5
        | q = \forall x \in A
        | ml= \norm {x x^*}
        | mo= =
        | mr= \norm x^2
}}
{{end-axiom}}
Then $A$ is referred to as a '''$\text B^*$-algebra'''.
\end{definition}