In 1967, Arveson invented a non-commutative generalization of classical
$H^{\infty},$ known as finite maximal subdiagonal subalgebras, for a finite von
Neumann algebra $\mathcal M$ with a faithful normal tracial state $\tau$. In
2008, Blecher and Labuschagne proved a version of Beurling's theorem on
$H^\infty$-right invariant subspaces in a non-commutative $L^{p}(\mathcal
M,\tau)$ space for $1\le p\le \infty$. In the present paper, we define and
study a class of norms ${\mathcal{N}}_{c}(\mathcal M, \tau)$ on $\mathcal{M},$
called normalized, unitarily invariant, $\Vert \cdot \Vert_{1}$-dominating,
continuous norms, which properly contains the class $\{ \Vert \cdot
\Vert_{p}:1\leq p< \infty \}.$ For $\alpha \in \mathcal{N}_{c}(\mathcal M,
\tau),$ we define a non-commutative $L^{\alpha }({\mathcal{M}},\tau)$ space and
a non-commutative $H^{\alpha}$ space. Then we obtain a version of the
Blecher-Labuschagne-Beurling invariant subspace theorem on $H^\infty$-right
invariant subspaces in a non-commutative $L^{\alpha }({\mathcal{M}},\tau)$
space. Key ingredients in the proof of our main result include a
characterization theorem of $H^\alpha$ and a density theorem for
$L^\alpha(\mathcal M,\tau)$.
