Editing C*-algebra (section)

=== C*-algebras of compact operators ===
Let ''H'' be a [[separable space|separable]] infinite-dimensional Hilbert space. The algebra ''K''(''H'') of [[Compact operator on Hilbert space|compact operator]]s on ''H'' is a [[norm closed]] subalgebra of ''B''(''H''). It is also closed under involution; hence it is a  C*-algebra.

Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*-algebras:

<blockquote>'''Theorem.''' If ''A'' is a C*-subalgebra of ''K''(''H''), then there exists Hilbert spaces {''H<sub>i</sub>''}<sub>''i''∈''I''</sub> such that
:<math> A \cong \bigoplus_{i \in I } K(H_i),</math>
where the (C*-)direct sum consists of elements (''T<sub>i</sub>'') of the Cartesian product Π ''K''(''H<sub>i</sub>'') with ||''T<sub>i</sub>''|| → 0.</blockquote>

Though ''K''(''H'')  does not have an identity element, a sequential [[approximate identity]] for ''K''(''H'') can be developed. To be specific, ''H'' is isomorphic to the space  of square summable sequences ''l''<sup>2</sup>; we may assume that ''H'' = ''l''<sup>2</sup>.  For each natural number ''n'' let ''H<sub>n</sub>'' be the subspace of sequences of ''l''<sup>2</sup> which vanish for indices ''k'' ≥ ''n'' and let ''e<sub>n</sub>'' be the orthogonal projection onto ''H<sub>n</sub>''. The sequence {''e<sub>n</sub>''}<sub>''n''</sub> is an approximate identity for ''K''(''H'').

''K''(''H'') is a two-sided closed ideal of ''B''(''H''). For separable Hilbert spaces, it is the unique ideal. The [[quotient]] of ''B''(''H'') by ''K''(''H'') is the [[Calkin algebra]].