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)

References

[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