> [!definition] > > Let $(\alg, \times, ||\cdot||)$ be a [[Banach Algebra|Banach algebra]], and $*$ be an [[Involution|involution]] such that $(\alg, \times, *)$ is a $*$[[Star Algebra|-algebra]]. Then $(\alg, \times, ||\cdot||, *)$ is a $C^*$-algebra if $||xx^*|| = ||x||^2$. > [!theorem] > > Let $\alg$ be a $C^*$-algebra, then $||x^*|| = ||x||$. > > *Proof*. Since $\alg$ is also a Banach algebra > $ > ||x||^2 = ||x^*x|| \le ||x^*|| \cdot ||x|| \Rightarrow ||x^*|| \ge ||x|| > $ > we apply the involution again to get $||x|| \ge ||x^*||$. > [!theorem] > > Let $\alg = \text{Lend}(\ch)$ where $\ch$ is a [[Hilbert Space|Hilbert space]], and $x \in \alg$ such that $x = x^*$. > 1. The pure point spectrum $\sigma_p(x) = \bracs{\lambda \in \complex: \lambda e - x \text{ not injective}}$ is real. > 2. For any $v, w \in \ch$ such that $xv = \lambda v$, $xw = \mu w$, and $\lambda \ne \mu$, then $v \perp w$. > > *Proof*. > $ > \lambda\angles{v, w} = \angles{xv, w} = \angles{v, xw} = \mu \angles{v, w} > $