Trigonometric bases in noncommutative $L_p(\mathbb{T}^d_θ)$ spaces and associated partial sum operators

Abstract

We develop a harmonic-analytic method for constructing a generalized trigonometric system in noncommutative $L_p(\mathbb{T}^d_θ)$ spaces arising from the strongly continuous representation of $\mathbb{T}^d$ and show that the generalized trigonometric system is a Schauder basis in $L_p(\mathbb{T}^d_θ)$ for $1<p<\infty.$ In particular, we prove that this trigonometric system forms an RUC-basis in $L_p(\mathbb{T}^d_θ)$ for $2<p<\infty.$ Our results provide a noncommutative counterpart of the classical trigonometric basis in $L_p(\mathbb{T}^d)$. Further, we obtain a weak $(1,1)$ type estimate of partial sum operators associated with noncommutative trigonometric systems. This allows us to study uniformly boundedness of partial sum operators between pairs of symmetric spaces that do not necessarily possess nontrivial Boyd indices, extending known results in this direction to the setting of quasi-Banach symmetric spaces.