free orbit-dimension* $\mathfrak{K}_{0}(\mathcal{M})$ of $\mathcal{M}$ is defined by $$\mathfrak{K}_{0}(\mathcal{M})=\sup\{\mathfrak{K}_{0}(x_{0},\ldots ,x_{n})\ |\ \text{$x_{0},\ldots,x_{n}$ generate }\mathcal{M}\text{ as a von Neumann algebra}\},$$ 