C*-algebra

Remarks

Example: the algebra $B(H)$ of bounded operators on a complex Hilbert space $H$, with the operator norm and the adjoint, is a C*-algebra. For clause 4: $\norm{T^* T} \le \norm{T}^2$, and $\norm{Tx}^2 = \ip{T^* T x}{x} \le \norm{T^* T} \norm{x}^2$ gives $\norm{T}^2 \le \norm{T^* T}$.

Example: for a compact Hausdorff space $X$, the algebra $C(X)$ of continuous functions $X \to \C$ with $f^* = \overline{f}$ and the supremum norm is a commutative C*-algebra, since $\norm{\overline{f} f} = \sup \abs{f}^2 = \norm{f}^2$.

Non-example: $M_2(\C)$ with the conjugate transpose and the Frobenius norm $\norm{A}_F = (\sum \abs{a_{ij}}^2)^{1/2}$ is a Banach algebra with involution, but it fails clause 4. For the identity matrix $I$, $\norm{I^* I}_F = \sqrt{2}$, while $\norm{I}_F^2 = 2$.

The commutative Gelfand–Naimark theorem says that every commutative C*-algebra with unit is isometrically $*$-isomorphic to $C(X)$ for a compact Hausdorff space $X$. The Gelfand–Naimark theorem says that every C*-algebra is isometrically $*$-isomorphic to a norm-closed $*$-subalgebra of some $B(H)$.