Editing C*-algebra (section)

== Examples ==

=== 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.

=== C*-algebras of operators ===
The prototypical example of a C*-algebra is the algebra ''B(H)'' of bounded (equivalently continuous) [[linear operator]]s defined on a complex [[Hilbert space]] ''H''; here ''x*'' denotes the [[adjoint operator]] of the operator ''x'' : ''H'' → ''H''. In fact, every C*-algebra, ''A'', is *-isomorphic to a norm-closed adjoint closed subalgebra of ''B''(''H'') for a suitable Hilbert space, ''H''; this is the content of the [[Gelfand–Naimark theorem]].

=== 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]].

=== Commutative C*-algebras ===
Let ''X'' be a [[locally compact]] Hausdorff space.  The space <math>C_0(X)</math> of complex-valued continuous functions on ''X'' that ''vanish at infinity'' (defined in the article on [[locally compact|local compactness]]) forms a commutative C*-algebra <math>C_0(X)</math> under pointwise multiplication and addition. The involution is pointwise conjugation. <math>C_0(X)</math> has a multiplicative unit element if and only if <math>X</math> is  compact.  As does any C*-algebra, <math>C_0(X)</math> has an [[approximate identity]]. In the case of <math>C_0(X)</math> this is immediate: consider the directed set of compact subsets of <math>X</math>, and for each compact <math>K</math> let <math>f_K</math> be a function of compact support which is identically 1 on <math>K</math>.  Such functions exist by the [[Tietze extension theorem]], which applies to locally compact Hausdorff spaces. Any such sequence of functions <math>\{f_K\}</math> is an approximate identity.

The [[Gelfand representation]] states that every commutative C*-algebra is *-isomorphic to the algebra <math>C_0(X)</math>, where <math>X</math> is the space of [[Character (mathematics)|characters]] equipped with the [[Weak topology|weak* topology]]. Furthermore, if <math>C_0(X)</math> is [[isomorphism|isomorphic]] to <math>C_0(Y)</math> as C*-algebras, it follows that <math>X</math> and <math>Y</math> are [[homeomorphism|homeomorphic]]. This characterization is one of the motivations for the [[noncommutative topology]] and [[noncommutative geometry]] programs.

=== C*-enveloping algebra ===
Given a Banach *-algebra ''A'' with an [[approximate identity]], there is a unique (up to C*-isomorphism) C*-algebra '''E'''(''A'') and *-morphism π from ''A'' into '''E'''(''A'') that is [[universal morphism|universal]], that is, every other continuous *-morphism {{nowrap|π ' : ''A'' → ''B''}} factors uniquely through π.  The algebra '''E'''(''A'') is called the '''C*-enveloping algebra''' of the Banach *-algebra ''A''.

Of particular importance is the C*-algebra of a [[locally compact group]] ''G''.  This is defined as the enveloping C*-algebra of the [[group algebra of a locally compact group|group algebra]] of ''G''.  The  C*-algebra of ''G''  provides context for general [[harmonic analysis]] of ''G'' in the case ''G'' is non-abelian.  In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*-algebra.  See [[spectrum of a C*-algebra]].

=== Von Neumann algebras ===

[[Von Neumann algebra]]s, known as W* algebras before the 1960s, are a special kind of C*-algebra. They are required to be closed in the [[weak operator topology]], which is weaker than the norm topology.

The [[Sherman–Takeda theorem]] implies that any C*-algebra has a universal enveloping W*-algebra, such that any homomorphism to a W*-algebra factors through it.