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In this paper, we give examples of maximal amenable subalgebras of the freegroup factor of two generators. More precisely, we consider two copies of thehyperfinite factor $R_i$ of type $\mathrm{II}_1$. From each $R_i$, we take aHaar unitary $u_i$ which generates a Cartan subalgebra of it. We show that thevon Neumann subalgebra generated by the self-adjoint operator$u_1+u_1^{-1}+u_2+u_2^{-1}$ is maximal amenable in the free product. Thisprovides infinitely many non-unitary conjugate maximal amenable MASAs.

Maximal amenable MASAs of the free group factor of two generators arising from the free products of hyperfinite factors