Algebra of bounded operators on a Hilbert space

Description

Let $H$ be a complex Hilbert space with inner product $\ip{\cdot}{\cdot}$. Let $B(H)$ be the set of bounded linear operators $T \colon H \to H$, with pointwise addition and scalar multiplication, composition as product, and the operator norm $$\norm{T} = \sup \set{\norm{T x} : x \in H, \norm{x} \le 1}.$$ The involution is the adjoint $T \mapsto T^*$, defined by $\ip{T x}{y} = \ip{x}{T^* y}$ for all $x, y \in H$. With these operations $B(H)$ is a C*-algebra with unit the identity operator.

Check: $B(H)$ is complete in the operator norm because $H$ is complete, and $\norm{S T} \le \norm{S} \norm{T}$, so it is a Banach algebra. The adjoint is conjugate linear, satisfies $(S T)^* = T^* S^*$ and $T^{**} = T$, and $\norm{T^*} = \norm{T}$. For the C* identity, $\norm{T x}^2 = \ip{T^* T x}{x} \le \norm{T^* T} \norm{x}^2$ gives $\norm{T}^2 \le \norm{T^* T}$, and $\norm{T^* T} \le \norm{T^*} \norm{T} = \norm{T}^2$ gives the reverse inequality.