on $\mathcal{B}(\mathcal{H})$ maps each bounded linear operator $T$ on $\mathcal{H}$ to its adjoint, $T^*$.

Note that the second example contains as a special case the algebra $\mathcal{M}_n (\mathbb{C})$ ($= \mathbb{C}^{k\times k}$) of all $k\times k$ matrices with complex entries, with the norm being the usual spectral norm and the involution mapping each matrix $A=[a_{ij}]$ to its Hermitian conjugate $A^* = [\, \overline{a_{ji}}\, ]$. This algebra is noncommutative for $k\ge 2$.

Examples 1 and 2 above provide, in a precise sense, the only examples of $C^*$-algebras. Indeed, every (unital) commutative $C^*$-algebra admits a faithful representation onto an algebra of the form $C(\mathcal{X})$ for a suitable (and essentially unique) compact Hausdorff space $\mathcal{X}$; and, similarly, every unital (possibly noncommutative) $C^*$-algebra can be faithfully represented as a norm-closed subalgebra of $\mathcal{B}(\mathcal{H})$ for a suitable complex Hilbert space $\mathcal{H}$.

More precisely, a map $\phi$ between two $C^*$-algebras is a $*$-[*homomorphism*]{} if $\phi$ is linear, multiplicative, and such that $\phi(a^*) = \phi(a)^*$; a $*$-[*isomorphism*]{} is a bijective $*$-homomorphism. Two $C^*$-algebras are said to be [*isometrically $*$-isomorphic*]{} if there is a norm-preserving $*$-isomorphism between them, in which case they are indistinguishable as $C^*$-algebras. A [*$*$-subalgebra*]{} $\mathcal{B}_0$ of a $C^*$-algebra is a subalgebra that is [$*$-closed]{}, i.e., 