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Suppose $M_1, M_2$ are type II_1 factors, A is a masa in $M_1\otimes M_2$, can we find a *-automorphism $\phi$ on $M_1\otimes M_2$ such that there are two masas $A_1\subset M_1,A_2\subset M_2$ and $\phi(A)$ is isomorphism to $ A_1\otimes A_2$?

Here, 'masa' means a maximal abelian self-adjoint subalgebra.

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