Let $A$ be a complex $C^*$-algebra. If $A$ is commutative, then $A$ is nuclear. If $A$ is finite-dimensional as a complex [vector space](/page/Vector%20Space), then $A$ is nuclear.

Here nuclearity is understood in the tensor-norm sense: for every complex $C^*$-algebra $B$, the canonical quotient map $A\otimes_{\max}B\to A\otimes_{\min}B$ is isometric. Equivalently, the minimal and maximal $C^*$-tensor norms on the algebraic $*$-[tensor product](/page/Tensor%20Product) $A\odot B$ coincide, so every $C^*$-cross norm on $A\odot B$ extending the given norms on the two factors is the same norm.