# Maximal abelian subalgebra in generated von Neumann algebra

Let $$D \subseteq A$$ be an abelian C*-subalgebra of the C*-algebra $$A$$ where $$A \subseteq B(H)$$ for some separable Hilbert space $$H$$. Assume that the von Neumann Algebra generated by $$D$$ is a maximal abelian subalgebra (masa) of the von Neumann algebra generated by $$A$$.

Does this already imply that $$D \subseteq A$$ is maximal abelian?

For instance, take $$A=\ell^\infty(\mathbb N)$$, and $$D=c_0(\mathbb N)$$.
For a different example, take $$A=C[0,1]$$ seen as multiplication operators on $$L^2[0,1]$$, and $$D=\{f\in A:\ f(0)=0\}.$$ Then $$D''=A''=L^\infty[0,1]$$.