Definition 6.4.7 ($C^{*}$-Subalgebra).label Let $A$ be a $C^{*}$-algebra. A subalgebra $B$ of $A$ is called a $C^{*}$-subalgebra of $A$ if

• $B$ is closed

• $\fall x\in B,x^{*}\in B$

In this case $B$ is unital if

• $A,B$ are unital

• $1_{B}=1_{A}$

Since the intersection of $C^{*}$-subalgebras of $A$ is again a $C^{*}$-subalgebra of $A$, for any subset $E\suf A$, $\exists! B$ the smallest $C^{*}$-subalgebra of $A$ containing $E$. This algebra $B$ is called the $C^{*}$-subalgebra of $A$ generated by $E$.