Editing C*-algebra (section)

=== Finite-dimensional C*-algebras ===
The algebra M(''n'', '''C''') of ''n'' × ''n'' [[matrix (mathematics)|matrices]] over '''C''' becomes a C*-algebra if we consider matrices as operators on the Euclidean space, '''C'''<sup>''n''</sup>, and  use the [[operator norm]] ||·|| on matrices. The involution is given by the [[conjugate transpose]].  More generally, one can consider finite [[direct sum of modules|direct sum]]s of matrix algebras. In fact, all C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The self-adjoint requirement means finite-dimensional C*-algebras are [[Semisimple algebra|semisimple]], from which fact one can deduce the following theorem of [[Artin–Wedderburn theorem|Artin–Wedderburn]] type:

<blockquote>'''Theorem.'''  A finite-dimensional C*-algebra, ''A'', is [[Canonical form|canonically]] isomorphic to a finite direct sum
:<math> A = \bigoplus_{e \in \min A } A e</math>
where min ''A'' is the set of minimal nonzero self-adjoint central projections of ''A''.</blockquote>

Each C*-algebra, ''Ae'', is isomorphic (in a noncanonical way) to the full matrix algebra M(dim(''e''), '''C'''). The finite family indexed on min ''A'' given by {dim(''e'')}<sub>''e''</sub> is called the ''dimension vector'' of ''A''.  This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language of [[operator K-theory|K-theory]], this vector is the [[ordered group|positive cone]] of the ''K''<sub>0</sub> group of ''A''.

A '''†-algebra''' (or, more explicitly, a ''†-closed algebra'') is the name occasionally used in [[physics]]<ref>John A. Holbrook, David W. Kribs, and Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction." ''Quantum Information Processing''. Volume 2, Number 5, pp. 381&ndash;419.  Oct 2003.</ref> for a finite-dimensional C*-algebra.  The [[dagger (typography)|dagger]], †, is used in the name because physicists typically use the symbol to denote a [[Hermitian adjoint]], and are often not worried about the subtleties associated with an infinite number of dimensions.  (Mathematicians usually use the asterisk, *, to denote the Hermitian adjoint.) †-algebras feature prominently in [[quantum mechanics]], and especially [[quantum information science]].

An immediate generalization of finite dimensional C*-algebras are the [[approximately finite dimensional C*-algebra]]s.