Let $\pi_1: A_1 \to B_1$ and $\pi_2: A_2 \to B_2$ be positive linear maps between complex $*$-algebras. Is the mapping $$\pi_1 \otimes \pi_2: A_1 \otimes A_2 \to B_1 \otimes B_2$$ again positive?

I.e., if $\sum_{i=1}^n x_i \otimes y_i \in A_1 \otimes A_2$, do we have $$\sum_{i,j=1}^n \pi_1(x_i^*x_j)\otimes \pi_2(y_i^*y_j) \ge 0$$

I know the proof for the case that $B_1 = B_2 = \mathbb{C}$. In that case, we can work with diagonalisation of matrices.

Math. Scand.111(2012), 5-11. $\endgroup$ – Nik Weaver May 14 at 15:53