On geometrical properties of noncommutative modular function spaces (Q887784)

The authors define noncommutative modular function spaces in a way similar to that of noncommutative Banach function spaces. For a semifinite von Neumann algebra \(\mathfrak{M}\) with a faithful normal semifinite trace \(\tau\), and for a convex modular \(\rho\) on \(\mathcal{L}_0([0,\infty))\) with Lebesgue measure, they define \(\tilde{\rho}(x)=\rho(\mu(|x|))\) on the algebra \(\tilde{\mathfrak{M}}\) of \(\tau\)-measurable operators, where \(\mu\) is the generalized singular value function for \(\tau\). Then \(\tilde{\rho}\) is a convex modular functional on \(\tilde{\mathfrak{M}}\), and one can define the noncommutative modular function space \(\mathcal{L}_{\tilde{\rho}}\) as \(\left\{x\in\tilde{\mathfrak{M}}: \tilde{\rho}(\lambda x)\to0 \text{ as } \lambda \to0\right\}.\) These vector spaces can be equipped with (equivalent) Luxemburg and Amemiya norms, turning them into Banach spaces. If \(\rho\) satisfies the \(\Delta_2\) condition, then the spaces are additionally \(\tilde{\rho}\)-complete. It is shown how Orlicz spaces fit into the picture. The rest of the paper is devoted to proving the uniform Opial and the uniform Kadec-Klee properties of the Luxemburg norm with respect to convergence in measure and the \(\tilde{\rho}\)-a.e. convergence, where a sequence \((x_n)\) converges to \(x\) \(\tilde{\rho}\)-a.e. if \(\mu(x_n-x)\to0\) \(\rho\)-a.e.