Lebesgue and Hardy spaces for symmetric norms I

In this paper, we define and study a class $\mathcal{R}_{c}$ of norms on $L^{\infty}\left( \mathbb{T}\right)$, called $continuous\ rotationally\ symmetric \ norms$, which properly contains the class $\left \{ \left \Vert \cdot \right \Vert _{p}:1\leq p<\infty \right \} .$ For $\alpha \in \mathcal{R}$ we define $L^{\alpha}\left( \mathbb{T}\right)$ and the Hardy space $H^{\alpha}\left( \mathbb{T}\right)$, and we extend many of the classical results, including the dominated convergence theorem, convolution theorems, dual spaces, Beurling-type invariant spaces, inner-outer factorizations, characterizing the multipliers and the closed densely-defined operators commuting with multiplication by $z$. We also prove a duality theorem for a version of $L^{\alpha}$ in the setting of von Neumann algebras.