Abstract: Let $E$ be a fully symmetric Banach function space on $[0,1]$ and $\mathcal{M}$ be a finite von Neumann algebra. Let $[x_k]$ be the closed subspace spanned by a sequence $(x_k)$ of freely independent mean zero random variables from $E(\mathcal{M}).$ The subspace $[x_k]$ is complemented in $E(\mathcal{M})$ if and only if the closed subspace spanned by the pairwise orthogonal sequence $(x_k\otimes e_k)$ is complemented in a certain symmetric operator space $Z_E^2(\mathcal{M}\bar{\otimes}\ell_\infty).$ We obtain noncommutative (free) analogues of classical results of Dor and Starbird as well as those of Kadec and Pelczynski. We show that $[x_k]$ is complemented in $L_1(\mathcal{M})$ provided $(x_k)$ is equivalent in $L_1(\mathcal{M})$ to the standard basis of $\ell_2,$ while this never happens in the classical case. We prove that a sequence of freely independent copies of a mean zero random variable $x$ in $L_p(\mathcal{M}),$ $1\leq p<2,$ is equivalent to the standard basis in some Orlicz sequence space $\ell_\Phi$ and give a precise description of the connection between the Orlicz function $\Phi$ and the distribution of the given random variable $x.$ Finally, we prove that $[x_k]$ spanned by a sequence of freely independent copies of a mean zero random variable is complemented in $E(\mathcal{M})$ if and only if $(x_k)$ is equivalent in $E(\mathcal{M})$ to the standard basis of $\ell_2.$ Our method is mainly based on the recent advance on Johnson-Schechtman inequalities in free probability theory.
