Two tuples of noncommutative Orlicz sequence spaces and some geometry properties
Ma Zhenhua, Jiang Lining
TL;DR
The paper develops a interpolation framework for 2-tuples of noncommutative Orlicz sequence spaces $\bigoplus_{j=1}^{2}S_{\\varphi_{j},p}$ by leveraging the three-lines theorem to prove a noncommutative Riesz–Thorin interpolation theorem. This leads to Clarkson-type inequalities in the noncommutative Orlicz setting and enables explicit bounds for geometry constants, including the von Neumann–Jordan constant and nonsquare constant, for spaces $S_{\\varphi_{s}}$ and Schatten classes $S_{p}$. The results extend interpolation and geometric analysis techniques from classical $L_{p}$ and Schatten spaces to the broader noncommutative Orlicz context. These findings have implications for operator space theory and noncommutative geometry, providing tools to quantify convexity and stability properties of noncommutative Orlicz spaces.
Abstract
The primary contribution of this study lies in proposing a new concept termed $2$-tuples of noncommutative Orlicz sequence spaces $\bigoplus\limits_{j=1}^{2}S_{\varphi_{j},p}$, where $S_{\varphi_{j}}$ denotes a noncommutative Orlicz sequence space. By leveraging the three-line theorem, we establish the Riesz-Thorin interpolation theorem for $\bigoplus\limits_{j=1}^{2}S_{\varphi_{j},p}$. As applications, we derive bound for the nonsquare and von Neumann-Jordan constant of noncommutative Orlicz space $S_{\varphi_{s}} (0<s\leq1)$, where $\varphi_{s}$ is an intermediate function.