Article
Ufa Mathematical Journal
Volume 11, Number 3, pp. 3-10
Renormalizations of measurable operator ideal spaces,
affiliated to a semifinite von Neumann algebra
Bikchentaev A.M.
DOI:10.13108/2019-11-3-3
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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 $\tau =\text{\rm tr}$.