Definition 7.1.2 ($C^{*}$-algebra).label An involution on an algebra $A$ is an anti-automorphism of $A$ of order 2, i.e., a map $x\mto x^{*}$ from $A$ to $A$ satisfying

1. (1)

   $x^{**}=x$

2. (2)

   $(x+y)^{*}=x^{*}+y^{*}$

3. (3)

   $(\al x)^{*}=\cl{\al}x^{*}$

4. (4)

   $(xy)^{*}=y^{*}x^{*}$

5. (5)

   $\norm{x^*}{}=\norm{x}{}$ (Remark: Some authors do not require this for example Folland in [Fol15])

for every $x,y\in A,\al\in\com$ then $A$ is called an involutive Banach algebra (or $*$-algebra) and the map $x\mto x^{*}$ the involution (or $*$-operation). If the involution of $A$ satisfies the following additional condition

1. (1)

   $\norm{x^*x}{}=\norm{x^*}{}\norm{x}{},x\in A$

then $A$ is called a $C^{*}$-algebra. If $A$ and $B$ are Banach algebras, a (Banach algebra) homomorphism from $A$ to $B$ is a bounded linear map $\phi:A\to B$ such that $\phi(xy)=\phi(x)\phi(y)$ for all $x,y\in A$. If $A$ and $B$ are $*$-algebras, a $*$-homomorphism from $A$ to $B$ is a homomorphism $\phi$ such that $\phi(x^{*})=\phi(x)^{*}$ for all $x\in A$. If $S$ is a subset of the Banach algebra $A$, we say that $A$ is generated by $S$ if the linear combination of products of elements of $S$ are dense in $A$.