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4 Punctuation.

In

Stegall, C. Banach spaces whose duals contain $\ell_1(\Gamma)$ with applications to the study of dual $L_1(\mu)$ spaces. Trans. Amer. Math. Soc. 176 (1973), 463–477

Stegall proved that $\ell_2$ is isometrically isomorphic to a norm one complemented subspace of $X^*$ when $X= (\sum_{n=1}^\infty \ell_2^n)_1$, yet $\ell_2$ does not embed into $X$.

It would be interesting to have an example of this phenomenon with $X^*$ separable.

Here is a more interesting example because the dual is separable, but the range of the projection is not $\ell_p$. Take any separable reflexive space $X$ that fails the approximation property and let $(E_n)$ be an increasing sequence of finite dimensional subspaces of $X$ whose union is dense in $X$. Let $c(E_n)$ be the space of sequences $(x_n)$ with $x_n$ in $E_n$ and $\lim_n x_n$ existing in $X$, normed by $\|(x_n)\|= \sup \|x_n\|$.
Define $c_0(E_n)$ to be $(\sum_{n=1}^\infty E_n)_0$.

Consider the short exact sequence (ses)

$0\to c_0(E_n) \to c(E_n) \to X\to 0$

where the second arrow is the inclusion mapping and the third is the quotient mapping $Q$ defined by $(x_n)\mapsto \lim x_n$. This sequence locally splits (with constant one), hence $Q^*$ maps $X^*$ onto a norm one complemented subspace of $c(E_n)^*$. The ses itself does not split because $c(E_n)$ has the approximation property (even a finite dimensional decomposition).

This example is Proposition 2.4 in

Johnson, William B.; Oikhberg, Timur: Separable lifting property and extensions of local reflexivity, Illinois J. Math. 45 (2001), no. 1, 123–137.

The construction itself is due to W. Lusky

A note on Banach spaces containing $c_0$ or $C_\infty$, J. Funct. Anal. 62 (1985), no. 1, 1–7.

Lusky was interested in the case that $X$ has the bounded approximation property. Then the resulting ses splits.

3 added 1445 characters in body

Here is a more interesting example because the dual is separable, but the range of the projection is not $\ell_p$. Take any separable reflexive space $X$ that fails the approximation property and let $(E_n)$ be an increasing sequence of finite dimensional subspaces of $X$ whose union is dense in $X$. Let $c(E_n)$ be the space of sequences $(x_n)$ with $x_n$ in $E_n$ and $\lim_n x_n$ existing in $X$, normed by $\|(x_n)\|= \sup \|x_n\|$.
Define $c_0(E_n)$ to be $(\sum_{n=1}^\infty E_n)_0$.

Consider the short exact sequence (ses)

$0\to c_0(E_n) \to c(E_n) \to X\to 0$

where the second arrow is the inclusion mapping and the third is the quotient mapping $Q$ defined by $(x_n)\mapsto \lim x_n$. This sequence locally splits (with constant one), hence $Q^*$ maps $X^*$ onto a norm one complemented subspace of $c(E_n)^*$.The ses itself does not split because $c(E_n)$ has the approximation property (even a finite dimensional decomposition).

This example is Proposition 2.4 in

Johnson, William B.; Oikhberg, TimurSeparable lifting property and extensions of local reflexivity, Illinois J. Math. 45 (2001), no. 1, 123–137.

The construction itself is due to W. Lusky

A note on Banach spaces containing $c_0$ or $C_\infty$, J. Funct. Anal. 62 (1985), no. 1, 1–7.

Lusky was interested in the case that $X$ has the bounded approximation property. Then the resulting ses splits.

2 Corrected grammar.

In

Stegall, C. Banach spaces whose duals contain $\ell_1(\Gamma)$ with applications to the study of dual $L_1(\mu)$ spaces. Trans. Amer. Math. Soc. 176 (1973), 463–477

Stegall proved that $\ell_2$ is isometrically isomorphic to a norm one complemented subspace of $X^*$ when $X= (\sum_{n=1}^\infty \ell_2^n)_1$, yet $\ell_2$ does not embed into $X$.

It would be interesting to have an example of this phenomena phenomenon with $X^*$ separable.

1