# Nuclear-C*-algebra

A -algebra with the following property: For any -algebra there is on the algebraic tensor product a unique norm such that the completion of with respect to this norm is a -algebra. Thus, relative to tensor products, nuclear -algebras behave similarly to nuclear spaces (cf. Nuclear space) (although infinite-dimensional nuclear -algebras are not nuclear spaces). The class of nuclear -algebras includes all type I -algebras. This class is closed with respect to the inductive limit. If is a closed two-sided ideal in a -algebra , then is nuclear if and only if and are. A subalgebra of a nuclear -algebra need not be a nuclear -algebra. The tensor product of two -algebras and is nuclear if and only if and (both) are nuclear. If is an amenable locally compact group, then the enveloping -algebra of the group algebra is nuclear (the converse is not true). Each factor representation of a nuclear -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 -subalgebra of a -algebra can be extended to a factor state on the whole algebra.

Let be the -algebra of all bounded linear operators on a Hilbert space , and let be a -algebra of operators on . If is nuclear, then its weak closure is an injective von Neumann algebra, that is, there is a projection with norm one; in this case the commutant of is also injective. An arbitrary -algebra is nuclear if and only if its enveloping von Neumann algebra is injective.

A -algebra is nuclear if and only if it has the completely positive approximation property, i.e. the identity operator in 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 -algebra has the approximation and bounded approximation properties (see Nuclear operator). There is, however, a non-nuclear -algebra with the bounded approximation property. The -algebra of all bounded operators on an infinite-dimensional Hilbert space does not have the completely positive approximation property, or even the approximation property, so that is not nuclear.

#### References

 [1] E.C. Lance, "Tensor products and nuclear -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)