nLab
C-star algebra

A $C^{*}$-algebra is a complex Banach algebra with an involution compatible with complex conjugation (=$*$-algebra) and such that the $C^{*}$-identity holds:

$$\parallel A^{*}A\parallel =\parallel A{\parallel }^2.$$\| A^* A \| = \|A\|^2.

Every $C^{*}$-algebra can be realized (via GNS construction) as a norm closed $*$-subalgebra of the Banach $*$-algebra $B(H)=L(H,H)$ of bounded linear operators on a Hilbert space $H$ with the induced norm.

See operator algebras.