[Paper Review] On a certain class of operator algebras and their derivations

Sh. A. Ayupov, R. Z. Abdullaev+1 more|ArXiv.org|Aug 9, 2009

Holomorphic and Operator Theory14 references3 citations

TL;DR

This paper introduces finite tracial algebras as the intersection of all noncommutative $L_p$-spaces over a von Neumann algebra $M$ with respect to all faithful normal finite traces. It proves that every derivation on such algebras is inner, generalizing known results on Arens algebras and establishing a key structural property for noncommutative $L_p$-based operator algebras.

ABSTRACT

Given a von Neumann algebra $M$ with a faithful normal finite trace, we introduce the so called finite tracial algebra $M_f$ as the intersection of $L_p$-spaces $L_p(M, μ)$ over all $p \geq 1$ and over all faithful normal finite traces $μ$ on $M.$ Basic algebraic and topological properties of finite tracial algebras are studied. We prove that all derivations on these algebras are inner.

Motivation & Objective

• To define and study finite tracial algebras as intersections of $L_p(M,\mu)$ spaces over all faithful normal finite traces $\mu$ on a von Neumann algebra $M$.
• To investigate topological and algebraic properties of these algebras, particularly their metrizability and reflexivity.
• To establish a complete characterization of derivations on these algebras, showing all are inner.
• To extend known results on derivations in Arens algebras to a broader class of noncommutative $L_p$-based algebras.
• To provide a dual space description for the finite tracial algebra and clarify its topological structure.

Proposed method

• Define the finite tracial algebra $M_f = \bigcap_{\mu \in \mathcal{F}} \bigcap_{p \geq 1} L_p(M,\mu)$, where $\mathcal{F}$ is the set of all faithful normal finite traces on $M$.
• Equip $M_f$ with the topology $t$ generated by the family of $L_p$-norms $\|\cdot\|_p^\mu$ for all $p \geq 1$ and $\mu \in \mathcal{F}$.
• Use the Radon–Nikodym theorem to represent any f.n.f. trace $\mu$ as $\mu(x) = \tau(hx)$ for a positive $h$ in $L^1(M,\tau)$ affiliated with the center of $M$.
• Apply known results on derivations in Arens algebras $L^\omega(M,\tau)$, particularly that derivations on $L^\omega(M,\tau)$ are inner when $\tau$ is finite.
• Leverage the solidity of $M_f$ as a subalgebra of $L^\omega(M,\tau)$ and use Lemma 4.2 to lift the implementing element $a$ of a derivation from $L^\omega(M,\tau)$ into $M_f$ itself.
• Use the Leibniz rule and spectral theory (e.g., invertibility of $1+x$ for $x \geq 0$) to extend the derivation identity from $M$ to the entire algebra $A \supseteq M$.

Experimental results

Research questions

• RQ1 Under what conditions is the finite tracial algebra $M_f$ metrizable or reflexive?
• RQ2 When does $M_f$ coincide as a set with the von Neumann algebra $M$?
• RQ3 What is the structure of the dual space of $M_f$?
• RQ4 Are all derivations on $M_f$ inner, and if so, under what conditions on $M$?
• RQ5 How does the topology on $M_f$ compare to the operator topology and the $L_p$-topologies on $M$?

Key findings

• The finite tracial algebra $M_f$ is a complete locally convex topological $^*$-algebra under the topology generated by all $L_p^\mu$-norms.
• $M_f$ is metrizable or reflexive if and only if the center of $M$ is finite-dimensional, in which case $M_f$ coincides with an Arens algebra $L^\omega(M,\mu)$ for some trace $\mu$.
• The finite tracial algebra $M_f$ coincides with $M$ as a set if and only if $M$ is finite-dimensional or the trace space is trivial, but even then the topology on $M_f$ is strictly finer than the operator topology.
• The dual space of $M_f$ is isometrically isomorphic to the intersection of $L_q(M,\mu)$ spaces over all $q \in (1,\infty)$ and all f.n.f. traces $\mu$, under the duality pairing $\langle x, a \rangle = \tau(ax)$.
• Every derivation on a solid $^*$-subalgebra $A \subseteq L^\omega(M,\tau)$ containing $M$ is inner, with the implementing element $a$ lying in $A$.
• In particular, every derivation on $M_f$ is inner, establishing a noncommutative analog of the classical result for commutative $L_p$-algebras.