N-Tuples of weighted noncommutative Orlicz space and some geometrical properties

Abstract

In this article, we present a new concept named the N-tuples weighted noncommutative Orlicz space ⊕j=1nLp,λ(Φj)(ℳ˜,τ){\oplus }_{j=1}^{n}{L}_{p,\lambda }^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ), where L(Φj)(ℳ˜,τ){L}^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ) is the noncommutative Orlicz space. Based on the maximum principle, the Riesz-Thorin interpolation theorem of ⊕j=1nLp,λ(Φj)(ℳ˜,τ){\oplus }_{j=1}^{n}{L}_{p,\lambda }^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ) is given. As applications, we obtain the Clarkson inequality and some other geometrical properties which include the uniform convexity and uniform smoothness of noncommutative Orlicz spaces L(Φs)(ℳ˜,τ),0<s≤1{L}^{\left({\Phi }_{s})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ),0\lt s\le 1.

Keywords

• noncommutative orlicz spaces
• τ-measurable operator
• von neumann algebra
• orlicz function
• riesz-thorin interpolation
• 46l52
• 47l10
• 46a80