On nuclear C∗-algebras

AbstractA C∗-algebra is called nuclear if there is a unique way of forming its tensor product with any other C∗-algebra. Takesaki [17] showed that all C∗-algebras of type I and all inductive limits of such algebras are nuclear, but that the C∗-algebra Cr∗(G) generated by the left regular representation of G on l2(G) is nonnuclear, where G is the free group on two generators. In this paper an extension property for tensor products of C∗-algebras is introduced, and is characterized in terms of the existence of a certain family of weak expectations on the algebra. Nuclearity implies the extension property, and this is used to show that for a discrete group G, Cr∗(G) is nuclear if and only if G is amenable.An approximation property in the dual of a C∗-algebra is introduced, and shown to imply nuclearity. It is not clear whether the converse implication holds, but it is proved that the known nuclear C∗-algebras satisfy the approximation property