James' space $J$ is *subprojective*; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary 11 of [[P.G. Casazza et al. Proc. Amer. Math. Soc. 26 (1977), 294-305]](http://www.ams.org/journals/proc/1977-067-02/S0002-9939-1977-0458129-5/S0002-9939-1977-0458129-5.pdf). 


I would like to know if the dual space $J^\*$ is subprojective. It is known that a non-reflexive subspace of $J^\*$ contains an infinite dimensional subspace complemented in $J^\*$ (Theorem 2 in [A. Andrew. Israel J. Math. 38 (1981), 276-282]), but I do not know what happens for infinite dimensional reflexive subspaces. 



Complemented subspaces in the dual of James' space $J$
**Yes**, $J^\*$ is subprojective. 


Suppose that $M$ is a closed subspace of $J^\*$. By Mazur's theorem, without loss of generality we additionally suppose that $M$ has a normalised basis $(f\_k)\_{k=1}^\infty$. Let $\iota\colon M\to J^\*$ be the inclusion map and let $Q\colon J^{\*\*}\to M^\*$ be its (surjective) adjoint. Denote by $(f^\*\_k)\_{k=1}^\infty$ the coordinate functionals of $(f\_k)\_{k=1}^\infty$. 


Choose a bounded sequence $(z\_n)\_{n=1}^\infty$ in $J^{\*\*}$ such that $Qz\_n = f\_n^\*$ ($n\in \mathbb{N}$). As $J$ is quasi-reflexive, by Rosenthal's $\ell\_1$-theorem, $(z\_n)\_{n=1}^\infty$ contains a weakly Cauchy subsequence, call it $(z^\prime\_n)\_{n=1}^\infty$. As $(z^\prime\_n)\_{n=1}^\infty$ fails to have a convergent subsequence, we may suppose that the sequence $(z^\prime\_{2n+1}-z^\prime\_{2n})\_{n=1}^\infty$ is semi-normalised. However, by a result of Andrew [1, Theorem 2.1], every semi-normalised weakly null sequence in $J$ (so also in $J^{\*\*}$ as $J^{\*\*}\cong J$) contains a sequence spanning a complemented copy of $\ell\_2$, so $(z^\prime\_{2n+1}-z^\prime\_{2n})\_{n=1}^\infty$ contains such subsequence, say $(w\_n)\_{n=1}^\infty$. By reflexivity of $\ell\_2$, $(w\_n)\_{n=1}^\infty$ is shrinking so the adjoint of a projection from $J^{\*\*}$ onto $[w\_n]\_{n=1}^\infty$ is a projection onto a subspace of $M$. As $M\subset J^\* \subset J^{\*\*\*}$, the conclusion follows.


References:


1. A. Andrew, [Spreading basic sequences and subspaces of James' quasi-reflexive space](http://www.mscand.dk/article/viewFile/11904/9920), *Math. Scand.* **48** (1981), 276–282.


