Definition (Involution) An involution on an algebra $A$ over an involutory field $F$ is a map $\cdot^* : A \to A : a \mapsto a^*$ such that for every $\{a, b\} \subset A$ and $\lambda \in F$ we have

1. $a^{**} = a$
2. $(ab)^* = b^* a^*$ and
3. $(\lambda a+b)^* = \lambda^*a^* + b^*$ where $\lambda^*$ denotes the involution of $\lambda$ in $F$

Definition ($*$ Algebra) A $*$ algebra is an algebra with an involution.

Definition (Topological $*$ algebra) A topological $*$ algebra is a $*$ algebra which is also a topological vector space such that its algebra multiplication and involution are continuous.

Remarks:

• $*$ algebras are a particular type of involutory rings.
• The involutory field $F$ is often taken as $\mathbb{C}$ where the involution is given by complex conjugation. In this case, condition 3 could be rewritten as:

  3.$\;(\lambda a +b)^*= \overline{\lambda}a^*+b^*$

• Banach *-algebras are topological $*$ algebras.