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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 $α\in \mathcal{R}% _{c}$ we define $L^α\left( \mathbb{T}\right) $ and the Hardy space $H^α\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 $L^α$ in the setting of von Neumann algebras.