Renormalizations of measurable operator ideal spaces affiliated to semi-finite von Neumann algebra

Abstract

This work is devoted to non-commutative analogues of classical methods of constructing functional spaces. Let a von Neumann algebra ${\mathcal M}$ of operators act in a Hilbert space $\mathcal{H}$, $\tau$ be a faithful normal semi-finite trace $\mathcal{M}$. Let $ \widetilde{\mathcal{M}}$ be an $\ast$-algebra of $\tau$-measurable operators, $|X|=\sqrt{X^*X}$ for $X \in \widetilde{\mathcal{M}}$. A lineal $\mathcal{E}$ in $\widetilde{\mathcal{M}}$ is called ideal space on $(\mathcal{M}, \tau)$ if 1) $X \in \mathcal{E}$ implies $X^* \in \mathcal{E}$; 2) $X \in \mathcal{E}$, $Y \in \widetilde{\mathcal{M}}$ and $|Y| \leq |X|$ imply $Y \in \mathcal{E}$. Let $\mathcal{E}$, $\mathcal{F}$ be ideal spaces on $(\mathcal{M}, \tau)$. We propose a method of constructing a mapping $\tilde{\rho} \colon \mathcal{E}\to [0, +\infty]$ with nice properties by employing a mapping $\rho$ on a positive cone $\mathcal{E}^+$. At that, if $\mathcal{E}= \mathcal{M}$ and $\rho = \tau$, then $ \tilde{\rho}(X)=\tau (|X|)$ and if the trace $\tau$ is finite, then $ \tilde{\rho}(X)=\|X\|_1$ for all $X\in \mathcal{M}$. We study the case as $\tilde{\rho}(X)$ is equivalent to the original mapping $\rho (|X|)$. Employing mappings on $\mathcal{E}$ and $\mathcal{F}$, we construct a new mapping with nice properties on the sum $\mathcal{E}+\mathcal{F}$. We provide examples of such mappings. The results are new also for $\ast$-algebra $\mathcal{M}=\mathcal{B}(\mathcal{H})$ of all bounded linear operators in $\mathcal{H}$ equipped with a canonical trace $\tau =\mathrm{tr}$.