Suppose that $\ell_\phi$ is a reflexive Orlicz sequence space such that its dual space $\ell_\phi^*$ is isomorphic to $\ell_\phi$.

Is $\ell_\phi$ isomorphic to $\ell_2$?

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Suppose that $\ell_\phi$ is a reflexive Orlicz sequence space such that its dual space $\ell_\phi^*$ is isomorphic to $\ell_\phi$.

Is $\ell_\phi$ isomorphic to $\ell_2$?

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For a given $1<p$ and $\frac{1}{p}+\frac{1}{q}=1$ you can construct a *universal* Orlicz sequence space $\ell_M$ so that every Orlicz function $N$ *in between* $p$ and $q$ is equivalent to a function in $E_M$ which corresponds to the Orlicz subspaces spanned by constant block bases of $\ell_M$ thus complemented. Such a space is unique up to isomorphism (depending only to $p$), and $E_{M^*}$ has the same property so $\ell_{M^*}$ is isomorphic to $\ell_{M}$ by the uniqueness. This construction is given in [LT, Theorem 4.b.12]. See also the remark after.

This answer was first given by Bill Johnson in comments, I just added the reference.

*Lindenstrauss, Joram; Tzafriri, Lior*, Classical Banach spaces. 1: Sequence spaces. 2. Function spaces., Classics in Mathematics. Berlin: Springer-Verlag. xx, 432 p. (1996). ZBL0852.46015.

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