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