Mean dimension is a numerical invariant in topological dynamics and related to entropy. The von Neumann-Luck rank is a L^2-invariant and related to L^2 Betti number.

I will establish an equality between the von Neumann-Luck rank of a module of the integral group ring of a amenable group and the mean dimension of the associated algebraic action. Also I will give some applications of this equality. This is a joint work with Hanfeng Li.