The continuity of multiplication for two topologies associated with a~Semifinite trace on von Neumann algebra

A. M. Bikchentaev

Lobachevskii journal of mathematics, Tome 14 (2004), pp. 17-24.

Résumé

Let $\mathcal M$ be a semifinite von Neumann algebra in a Hilbert space $\mathcal H$ and $\tau$ be a normal faithful semifinite trace on $\mathcal M$. Let $\mathcal M^{\mathrm{pr}}$ denote the set of all projections in $\mathcal M$, $e$ denote the unit of $\mathcal M$, and ${\|\cdot\|}$ denote the $C^*$-norm on $\mathcal M$. The set of all $\tau$-measurable operators $\widetilde{\mathcal M}$ with sum and product defined as the respective closures of the usual sum and product, is a *-algebra. The sets $$ U(\varepsilon,\delta)=\{x\in\widetilde{\mathcal M}:\|xp\|\le\varepsilon\text{ and }\tau(e-p)\le\delta\text{ for some }p\in\mathcal M^{\mathrm{pr}}\}, \quad \varepsilon>0, \enskip \delta>0, $$ form a base at 0 for a metrizable vector topology $t_\tau$ on $\widetilde{\mathcal M}$, called the measure topology. Equipped with this topology, $\widetilde{\mathcal M}$ is a complete topological *-algebra. We will write $x_i\buildrel{\tau}\over\longrightarrow x$ in case a net $\{x_i\}_{i\in I}\subset\widetilde{\mathcal M}$ converges to $x\in\widetilde{\mathcal M}$ for the measure topology on $\widetilde{\mathcal M}$. By definition, a net $\{x_i\}_{i\in I}\subset\widetilde{\mathcal M}$ converges $\tau$-locally to $x\in\widetilde{\mathcal M}$ (notation: $x_i\buildrel{\tau l}\over\longrightarrow x$) if $x_ip\buildrel{\tau}\over\longrightarrow xp$ for all $p\in\mathcal M^{\mathrm{pr}}$, $\tau(p)\infty$; and a net $\{x_i\}_{i\in I}\subset\widetilde{\mathcal M}$ converges weak $\tau$-locally to $x\in\widetilde{\mathcal M}$ (notation: $x_i\buildrel{w\tau l}\over\longrightarrow x$) if $px_ip\buildrel{\tau}\over\longrightarrow pxp$ for all $p\in\mathcal M^{\mathrm{pr}}$, $\tau(p)\infty$. Theorem 1. {\it Let $x_i,x\in\widetilde{\mathcal M}$. 1. If $x_i\buildrel{\tau l}\over\longrightarrow x $, then $x_iy\buildrel{\tau l}\over\longrightarrow xy$ and $yx_i\buildrel{\tau l}\over\longrightarrow yx$ for every fixed $y\in\widetilde{\mathcal M}$. 2. If $x_i \buildrel{w\tau l}\over\longrightarrow x$, then $x_iy\buildrel{w\tau l}\over\longrightarrow xy$ and $yx_i\buildrel{w\tau l}\over\longrightarrow yx$ for every fixed $y\in\widetilde{\mathcal M}$.} Theorem 2. {\it If $\{x_i\}_{i\in I}\subset\widetilde{\mathcal M}$ is bounded in measure and if $x_i\buildrel{\tau l}\over\longrightarrow x\in\widetilde{\mathcal M}$, then $x_iy\buildrel{\tau}\over\longrightarrow xy$ for all $\tau$-compact $y\in\widetilde{\mathcal M}$.} Theorem 3. {\it Let $x,y,x_i,y_i\in\widetilde{\mathcal M}$ and let a set $\{x_i\}_{i\in I}$ be bounded in measure. If $x_i\buildrel{\tau l}\over\longrightarrow x$ and $y_i\buildrel{\tau l}\over\longrightarrow y$, then $x_iy_i\buildrel{\tau l}\over\longrightarrow xy$.} If $\mathcal M$ is abelian, then the weak $\tau$-local and $\tau$-local convergencies on $\widetilde{\mathcal M}$ coincides with the familiar convergence locally in measure. If $\tau(e)=\infty$, then the boundedness condition cannot be omitted in Theorem 2. If $\mathcal M$ is $\mathcal B(\mathcal H)$ with standard trace, then Theorem 2 for sequences is a “Basic lemma”of the theory of projection methods: If $y$ is compact and $x_n\to x$ strongly, then $x_ny\to xy$ uniformly, i.e. $\|x_ny-xy\|\to 0$ as $n\to\infty$. Theorem 3 means that the mapping $$ (x,y)\mapsto xy\colon(\mathcal B(\mathcal H)_1\times\mathcal B(\mathcal H)\to\mathcal B(\mathcal H)) $$ is strong-operator continuous ($\mathcal B(\mathcal H)_1$ denotes the unit ball of $\mathcal B(\mathcal H)$).

Détail

Keywords: Hilbert space, von Neumann algebra, noncom-mutative integration, measurable operator, semifinite trace, convergence with respect to measure, compact operator, topological algebra.

Bibliographie

[1] Bikchentaev A. M., “On minimality of the topology of convergence in measure on finite von Neumann algebras”, Kolmogorov and contemporary mathematics, Abstracts of international conference in commemoration of the centennial of A. N. Kolmogorov (Moscow, June 16–21, 2003), Moscow. Univ., 2003, 139–140, 915 pp.

[2] Böttcher A. et al., Lectures on operator theory and its applications, Fields Institute monographs, ed. P. Lancaster, Amer. Math. Soc., Providence, Rhode Island, 1996, 340 pp. | MR | Zbl

[3] Ciach L. J., “Some remarks on the convergence in measure and on a dominated sequence of operators measurable with respect to a semifinite von Neumann algebra”, Coll. Math., LV (1988), 109–121 | MR

[4] Dixmier J., Les algebres d'operateurs dans l'espace Hilbertien (algebres de von Neumann), Gauthier-Villars, Paris, 1969, 367 pp. | MR

[5] Dodds P. G., Dodds T. K., and de Pagter B., “Non-commutative Köthe duality”, Trans. Amer. Math. Soc., 339 (1993), 717–750 | DOI | MR | Zbl

[6] Dodds P. G., Dodds T. K., Sukochev F. A. and Tikhonov O. E., “A non-commutative Yosida-Hewitt theorem and applications to convex sets of measurable operators closed locally in measure”, Positivity, 9:3 (2005), 457–484 | DOI | MR | Zbl

[7] Kadison R. V. and Ringrose J. R., Fundamentals of the theory of operator algebras, V. I, Academic Press, New York – London – Paris, 1983, 398 pp. | MR | Zbl

[8] Nelson E., “Notes on non-commutative integration”, J. Funct. Anal., 15 (1974), 103–116 | DOI | MR | Zbl

[9] Segal I., “A non-commutative extension of abstract integration”, Ann. Math., 57 (1953), 401–457 | DOI | MR | Zbl

[10] Skvortsova G. Sh., “Weak sequential completeness of factor spaces of the space of integrable operators”, Russian Mathematics (Iz. VUZ), 45 (2002), 68–71 | MR | Zbl

[11] Terp M., $L^p$ spaces associated with von Neumann algebras, Notes, Copenhagen Univ., 1981, 100 pp.

[12] Yeadon F. J., “Non-commutative $L^p$-spaces”, Math. Proc. Cambridge Phil. Soc., 77 (1975), 91–102 | DOI | MR | Zbl