Direct sums of finite dimensional $SL^\infty_n$ spaces

Abstract: $SL^\infty$ denotes the space of functions whose square function is in $L^\infty$, and the subspaces $SL^\infty_n$, $n\in\mathbb{N}$, are the finite dimensional building blocks of $SL^\infty$. We show that the identity operator $I_{SL^\infty_n}$ on $SL^\infty_n$ well factors through operators $T : SL^\infty_N\to SL^\infty_N$ having large diagonal with respect to the standard Haar system. Moreover, we prove that $I_{SL^\infty_n}$ well factors either through any given operator $T : SL^\infty_N\to SL^\infty_N$, or through $I_{SL^\infty_N}-T$. Let $X^{(r)}$ denote the direct sum $\bigl(\sum_{n\in\mathbb{N}0} SL^\infty_n\bigr)_r$, where $1\leq r \leq \infty$. Using Bourgain's localization method, we obtain from the finite dimensional factorization result that for each $1\leq r\leq \infty$, the identity operator $I{X^{(r)}}$ on $X^{(r)}$ factors either through any given operator $T : X^{(r)}\to X^{(r)}$, or through $I_{X^{(r)}} - T$. Consequently, the spaces $\bigl(\sum_{n\in\mathbb{N}_0} SL^\infty_n\bigr)_r$, $1\leq r\leq \infty$, are all primary.