N}}(G) \otimes_{kG} M\bigr) = \lim_{n \to \infty} \frac{\dim_{k}\bigl(k \otimes_{kG_n} M\bigr)}{[G:G_n]}$$ provided $k$ is an algebraic number field. Here ${{\mathcal N}}(G)$ is the group von Neumann algebra, and $\dim_{{{\mathcal N}}(G)}$ is the von Neumann dimension. See [@Lueck(1994c)] for $k = \mathbb{Q}$ and [@Dodziuk-Linnell-Mathai-Schick_Yates(2003)] for the general case.

Let $k$ be a field of characteristic zero and let $u = \sum_{g \in G} x_g
\cdot g\in kG$ be an element. Let $F$ be the finitely generated field extension of ${{\mathbb Q}}$ given by $F = {{\mathbb Q}}(x_g \mid g \in G) \subset k$. Then $u$ is already an element in $FG$. The field $F$ embeds into ${{\mathbb C}}$: since $F$ is finitely generated, it is a finite algebraic extension of a transcendental extension $F'$ of ${{\mathbb Q}}$ [@Lang(2002_Algebra)]\*[Theorem 1.1 on p. 356]{}, and $F'$ has finite transcendence degree over ${{\mathbb Q}}$. Since the transcendence degree of ${{\mathbb C}}$ over ${{\mathbb Q}}$ is infinite, there exists an embedding $F'\hookrightarrow{{\mathbb C}}$ induced by an injection of a transcendence basis of $F'/{{\mathbb Q}}$ into a transcendence basis ${{\mathbb C}}/{{\mathbb Q}}$, which extends to $F\hookrightarrow{{\mathbb C}}$ because ${{\mathbb C}}$ is algebraically closed. This reduces the case of fields of characteristic zero to the case $k={{\mathbb C}}$. 