# Tensor product of traces of von Neumann algebras

I am trying to understand how to define tensor product of normal semifinite faithful (in short n.s.f) traces between two von Neumann algebras $(M_1,\tau_1)$ and $(M_2,\tau_2),$ where $\tau_i$ is the n.s.f. trace on $M_i$ for $i=1,2.$ I am following the lecture notes of Xu www.cim.nankai.edu.cn/activites/conferences/hy200707/summer.../Xu-long.pdf. Actually he defines this via GNS representations of the von Neumann algebras which I do not know. Since I am very new, can anyone explain to me what is he trying to tell there? I know GNS representation of $C^*$-algebras. But what is he telling in page 27 by "Using GNS construction we can view $M_k$ as a von Neumann algebra acting on $H_k = L_2(M_k)$ by left multiplication."? I have very basic knowledge on von Neumann algebras (as much as is there in there lecture note of Xu I mentioned). Also how do I show that that $\tau_1\otimes\tau_2$ is again an n.s.f trace? Also I want to know how to do tensor products for countable no. of n.s.f. traces.

• Do you know how it works for finite traces? If you do then note that from semifinitess you get two sequences of projections -- $(p_k)\subset M_1$ and $(q_k)\subset M_2$ -- such that the traces $p_k\tau_1 p_k$ and $q_k \tau_2 q_k$ are finite, so you can form their tensor product and then pass to the limit with $k\to \infty$. Mar 20, 2018 at 19:06