View source for Nuclear-C*-algebra

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A [[C*-algebra| $  C  ^ {*} $-
algebra]]  $  A $
with the following property: For any  $  C  ^ {*} $-
algebra  $  B $
there is on the algebraic tensor product  $  A \otimes B $
a unique norm such that the completion of  $  A \otimes B $
with respect to this norm is a  $  C  ^ {*} $-
algebra. Thus, relative to tensor products, nuclear  $  C  ^ {*} $-
algebras behave similarly to nuclear spaces (cf. [[Nuclear space|Nuclear space]]) (although infinite-dimensional nuclear  $  C  ^ {*} $-
algebras are not nuclear spaces). The class of nuclear  $  C  ^ {*} $-
algebras includes all type I  $  C  ^ {*} $-
algebras. This class is closed with respect to the inductive limit. If  $  I $
is a closed two-sided ideal in a  $  C  ^ {*} $-
algebra  $  A $, 
then  $  A $
is nuclear if and only if  $  I $
and  $  A/I $
are. A subalgebra of a nuclear  $  C  ^ {*} $-
algebra need not be a nuclear  $  C  ^ {*} $-
algebra. The tensor product of two  $  C  ^ {*} $-
algebras  $  A $
and  $  B $
is nuclear if and only if  $  A $
and  $  B $(
both) are nuclear. If  $  G $
is an amenable locally compact group, then the enveloping  $  C  ^ {*} $-
algebra of the group algebra  $  L _ {1} ( G) $
is nuclear (the converse is not true). Each [[Factor representation|factor representation]] of a nuclear  $  C  ^ {*} $-
algebra is hyperfinite, that is, the [[Von Neumann algebra|von Neumann algebra]] generated by this representation can be obtained from an increasing sequence of finite-dimensional factors (matrix algebras). Any factor state on a nuclear  $  C  ^ {*} $-
subalgebra of a  $  C  ^ {*} $-
algebra can be extended to a factor state on the whole algebra.

Let  $  L ( H) $
be the  $  C  ^ {*} $-
algebra of all bounded linear operators on a Hilbert space  $  H $, 
and let  $  A $
be a  $  C  ^ {*} $-
algebra of operators on  $  H $. 
If  $  A $
is nuclear, then its weak closure  $  \overline{A}\; $
is an injective von Neumann algebra, that is, there is a projection  $  L ( H) \rightarrow \overline{A}\; $
with norm one; in this case the commutant  $  A  ^  \prime  $
of  $  A $
is also injective. An arbitrary  $  C  ^ {*} $-
algebra  $  A $
is nuclear if and only if its enveloping von Neumann algebra is injective.

A  $  C  ^ {*} $-
algebra  $  A $
is nuclear if and only if it has the completely positive approximation property, i.e. the identity operator in  $  A $
can be approximated in the strong operator topology by linear operators of finite rank with norm not exceeding 1, and with the additional property of  "complete positivity"  [[#References|[1]]].

Every nuclear  $  C  ^ {*} $-
algebra has the approximation and bounded approximation properties (see [[Nuclear operator|Nuclear operator]]). There is, however, a non-nuclear  $  C  ^ {*} $-
algebra with the bounded approximation property. The  $  C  ^ {*} $-
algebra  $  L ( H) $
of all bounded operators on an infinite-dimensional Hilbert space  $  H $
does not have the completely positive approximation property, or even the approximation property, so that  $  L ( H) $
is not nuclear.

====References====
<table><tr><td valign="top">[1]</td> <td valign="top">  E.C. Lance,   "Tensor products and nuclear $C ^ { * }$-algebras"  R.V. Kadison (ed.) , ''Operator algebras and applications'' , ''Proc. Symp. Pure Math.'' , '''38''' , Amer. Math. Soc.  (1982)  pp. 379–399</td></tr><tr><td valign="top">[2]</td> <td valign="top">  O. Bratteli,   D.W. Robinson,   "Operator algebras and quantum statistical mechanics" , '''1''' , Springer  (1979)</td></tr></table>

====Comments====

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  R.V. Kadison,   J.R. Ringrose,   "Fundamentals of the theory of operator algebras" , '''1–2''' , Acad. Press  (1983)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  G.K. Pedersen,   "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press  (1979)  pp. Sect. 8.15.15</td></tr></table>