stability of $K_0$ ($C^*$-algebras), question about the tensor product $K(H)\otimes A$.

Question

I have a small question about stability of $K_0$:

If $A$ is a $C^*$-algebra and $H$ is a separable infinite dimensional Hilbert space then $$K_0(A)\cong K_0(K(H)\otimes A),$$ where $K(H)$ denotes the compact operators on $H$.

I'm not sure which tensor product $K(H)\otimes A$ is meant here. It must be the tensor product as a $C^*$-algebra, but it doesn't matter if it's the tensor product with respect to the minimal or the maximal tensor norm because $K(H)$ is nuclear, or am I wrong?

Answer 1

The notation in the C$^*$ literature for tensor products is far from uniform. But yes, as $K(H)$ is nuclear, it doesn't matter which C$^*$-norm you put on $K(H)\otimes A$, they are all the same.