Nuclear C*-algebra

In the mathematical field of functional analysis, a nuclear C*-algebra is a C*-algebra $A$ such that for every C*-algebra $B$ the injective and projective C*-cross norms coincides on the algebraic tensor product $A⊗B$ and the completion of $A⊗B$ with respect to this norm is a C*-algebra. This property was first studied by under the name "Property T", which is not related to Kazhdan's property T.

Characterizations
Nuclearity admits the following equivalent characterizations:
 * The identity map, as a completely positive map, approximately factors through matrix algebras. By this equivalence, nuclearity can be considered a noncommutative analogue of the existence of partitions of unity.
 * The enveloping von Neumann algebra is injective.
 * It is amenable as a Banach algebra.
 * (For separable algebras) It is isomorphic to a C*-subalgebra $B$ of the Cuntz algebra $𝒪_{2}$ with the property that there exists a conditional expectation from $𝒪_{2}$ to $B$.

Examples
The commutative unital C* algebra of (real or complex-valued) continuous functions on a compact Hausdorff space as well as the noncommutative unital algebra of $n×n$ real or complex matrices are nuclear.