For a discrete infinite amenable group $G$ the family of left-invariant means on $G$ is quite large having cardinality $2^{2^{|G|}}$. See Chapter 7 of

A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, American Math. Soc., Providence, Rhode Island, 1988.

Although I am not sure how natural it is, there is a recent result of Farah and Katsura concerning the number of non-isomorphic hyperfinie $II_1$-factors with preduals having density character $\kappa>\omega$. This number is precisely $2^\kappa$. Since $2^\kappa$ is much bigger than 1, it is a pretty good reason to stick with separably acting von Neumann algebras. (Of course it may still happen that $2^\kappa = \mathfrak{c}$.)

For a discrete infinite amenable group $G$ the family of left-invariant means on $G$ is quite large having cardinality $2^{2^{|G|}}$. See Chapter 7 of

A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, American Math. Soc., Providence, Rhode Island, 1988.

Although I am not sure how natural it is, there is a recent result of Farah and Katsura concerning the number of non-isomorphic hyperfinie $II_1$-factors with preduals having density character $\kappa>\omega$. This number is precisely $2^\kappa$. Since $2^\kappa$ is much bigger than 1, it is a pretty good reason to stick with separably acting von Neumann algebras. (Of course it may still happen that $2^\kappa = \mathfrak{c}$.)

For a discrete infinite amenable group $G$ the family of left-invariant means on $G$ is quite large having cardinality $2^{2^{|G|}}$. See Chapter 7 of

A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, American Math. Soc., Providence, Rhode Island, 1988.

Although I am not sure how natural it is, there is a recent result of Farah and Katsura concerning the number of non-isomorphic hyperfinie $II_1$-factors with preduals having density character $\kappa>\omega$. This number is precisely $2^\kappa$. Since $2^\kappa$ is much bigger than 1, it is a pretty good reason to stick with separably acting von Neumann algebras.

For a discrete infinite amenable group $G$ the family of left-invariant means on $G$ is quite large having cardinality $2^{2^{|G|}}$. See Chapter 7 of

A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, American Math. Soc., Providence, Rhode Island, 1988.

Although I am not sure how natural it is, there is a recent result of Farah and Katsura concerning the number of non-isomorphic hyperfinie $II_1$-factors with preduals having density character $\kappa>\omega$. This number is precisely $2^\kappa$. Since $2^\kappa$ is much bigger than 1, it is a pretty good reason to stick with separably acting von Neumann algebras. (Of course it may still happen that $2^\kappa = \mathfrak{c}$.)