# Nuclear-C*-algebra

A $C ^ {*}$- 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) (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 of a nuclear $C ^ {*}$- algebra is hyperfinite, that is, the 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" [1].

Every nuclear $C ^ {*}$- algebra has the approximation and bounded approximation properties (see 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

 [1] 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 [2] O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , 1 , Springer (1979)

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