  The analytic von Neumann regular closure $R(\Gamma)$ of a complex group
algebra $\C\Gamma$ was introduced by Linnell and Schick. This ring is the
smallest $*$-regular subring in the algebra of affiliated operators $U(\Gamma)$
containing $\C\Gamma$. We prove that all the algebraic von Neumann regular
closures corresponding to sofic representations of an amenable group are
isomorphic to $R(\Gamma)$. This result can be viewed as a structural
generalization of L\"uck's Approximation Theorem.
  \noindent The main tool of the proof which might be of independent interest
is that an amenable group algebra $K\Gamma$ over any field $K$ can be embedded
to the rank completion of an ultramatricial algebra.
