Factorization in Haar system Hardy spaces

A Haar system Hardy space is the completion of the linear span of the Haar system $(h_I{)}_I$, either under a rearrangement-invariant norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ or under the associated square function norm \begin{equation*} \Bigl| \sum_Ia_Ih_I \Bigr|_{} = \Bigl| \Bigl( \sum_I a_I^2 h_I^2 \Bigr)^{1/2} \Bigr|. \end{equation} Apart from $L^p$, $1\le p<\mathrm{\infty }$, the class of these spaces includes all separable rearrangement-invariant function spaces on $[0,1]$ and also the dyadic Hardy space $H^1$. Using a unified and systematic approach, we prove that a Haar system Hardy space $Y$ with $Y\mathrm{\ne }C(\mathrm{\Delta })$ ($C(\mathrm{\Delta })$ denotes the continuous functions on the Cantor set) has the following properties, which are closely related to the primariness of $Y$: For every bounded linear operator $T$ on $Y$, the identity $I_Y$ factors either through $T$ or through $I_Y-T$, and if $T$ has large diagonal with respect to the Haar system, then the identity factors through $T$. In particular, we obtain that \begin{equation*} $\mathcal{M}$_Y = { T\in $\mathcal{B}$(Y) : I_Y \ne ATB$\text{ for all }$ A, B\in $\mathcal{B}$(Y) } \end{equation*} is the unique maximal ideal of the algebra $\mathcal{B}(Y)$ of bounded linear operators on $Y$. Moreover, we prove similar factorization results for the spaces ${\mathrm{\ell }}^p(Y)$, $1\le p\le \mathrm{\infty }$, and use them to show that they are primary.