 there exist von Neumann subalgebras $\mathcal{N}_{0},\mathcal{N}_{0}$ of $\mathcal{M}$ with $\mathfrak{K}_{0}(\mathcal{N}_{0})=\mathfrak{K}_{0}(\mathcal{N}_{0})=0$ and $n$ vectors $\xi_{0},\ldots,\xi_{n}$ in $L^{0}