where M_{-1} = N and M_{0}=M , and each M_{n+1} = \langle M_n, e_{n+1} \rangle is generated by the previous algebra and a projection. The union of all these algebras has a tracial state tr whose restriction to each M_n is the tracial state, and so the closure of the union is another type {\rm II}_1 von Neumann algebra M_{\infty} .
