Let $M=A*P$ to be a free product von Neumann algebra, and $A$ is a finite dimensional subalgebra, for simplicity, we may assume $A=M_2(\mathbb{C})$, and $P\neq \mathbb{C}$.
A standard fact is that $M\cong A\bar{\otimes}(A'\cap M)$. So $A'\cap M\subset A$ is not true.
But intuitively, elements in $A'\cap M$ should lie in $A$, so my question is the following
Question:
Is it possible to construct some specific element in $A'\cap M$ but not in $A$?
Say, when $M=M_2(\mathbb{C})*M_2(\mathbb{C})$ or $M=M_2(\mathbb{C})*\mathbb{C}^2$ or $M=L(\mathbb{Z}/2\mathbb{Z})*L(\mathbb{Z})$ etc.
Thanks in advance!
For $M=M_2(\mathbb{C})*P$, denote $E_{i,j}$ to be the usual matrix units, then take any $a\in M$, then easy computation shows that
$$x=(E_{1,1}+E_{1,2})a(E_{1,1}+E_{2,1})+(E_{2,1}+E_{2,2})a(E_{1,2}+E_{2,2})$$ lies in $A'\cap M$.