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 Ufimsk. Mat. Zh., 2019, Volume 11, Issue 3, Pages 3–9 (Mi ufa476)

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

A. M. Bikchentaev

Kazan Federal University, Kremlevskaya str. 18, 420008, Kazan, Russia

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 $\tau =\mathrm{tr}$.

Keywords: Hilbert space, linear operator, von Neumann algebra, normal trace, measurable operators, ideal space, renormalization.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.9773.2017/8.9 The work is supported by subsidy granted to Kazan Federal University for making a state task in the field of scientific activity (1.9773.2017/8.9).

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English version:
Ufa Mathematical Journal, 2019, 11:3, 3–10 (PDF, 380 kB); https://doi.org/10.13108/2019-11-3-3

Bibliographic databases:

UDC: 517.983:517.986
MSC: 46L10, 47C15, 46L51

Citation: A. M. Bikchentaev, “Renormalizations of measurable operator ideal spaces affiliated to semi-finite von Neumann algebra”, Ufimsk. Mat. Zh., 11:3 (2019), 3–9; Ufa Math. J., 11:3 (2019), 3–10

Citation in format AMSBIB
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