# Nuclear C*-algebra

In mathematics, a nuclear C*-algebra is a C*-algebra A such that the injective and projective C*-cross norms on AB are the same for every C*-algebra B. These were first studied by Takesaki (1964) under the name "Property T" (this is unconnected with Kazhdan's property T).

## Characterizations

Nuclearity admits the following equivalent characterizations:

• A C*-algebra is nuclear if the identity map, as a completely positive map, approximately factors through matrix algebras. Abelian C*-algebras are nuclear. One might say that nuclearity means that the C*-algebra admits noncommutative "partitions of unity."
• A C*-algebra is nuclear if and only if its enveloping von Neumann algebra is injective.
• A separable C*-algebra is nuclear if and only if it is isomorphic to a C*-subalgebra B of the Cuntz algebra $\mathcal{O}_2$ with the property that there exists a conditional expectation from $\mathcal{O}_2$ to B.
• A C*-algebra is nuclear if and only if it is amenable as a Banach algebra.