A characterization of absolutely dilatable Schur multipliers

TL;DR

This paper characterizes absolutely dilatable Schur multipliers on $B(L^2(\Omega))$ by linking dilation properties to a trace-form factorization. In the separable setting, a bounded Schur multiplier $\phi$ is absolutely dilatable iff there exists a von Neumann algebra $(N,\tau_N)$ with separable predual and a $w^*$-continuous unitary field $d: \Omega\to N$ such that $\phi(s,t)=\tau_N(d(s)^*d(t))$ for a.e. $(s,t)$. The authors prove the two-way equivalence by deriving from absolute dilation a trace-form representation and, conversely, constructing an explicit dilation from such a representation via infinite tensor products; they also address the discrete case and the non-separable setting. The results provide a comprehensive dilation-theoretic characterization of Schur multipliers, linking non-commutative $L^p$-techniques to operator-valued factorization and transference concepts with potential implications for functional calculus and perturbation theory in non-commutative analysis.

Abstract

Let $M$ be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let $T\colon M\to M$ be a contraction. We say that $T$ is absolutely dilatable if there exist another von Neumann algebra $M'$ equipped with a nsf trace, a $w^*$-continuous trace preserving unital $*$-homomorphim $J\colon M\to M'$ and a trace preserving $*$-automomorphim $U\colon M'\to M'$ such that $T^k=E U^k J$ for all integer $k\geq 0$, where $E\colon M'\to M$ is the conditional expectation associated with $J$. Given a $σ$-finite measure space $(Ω,μ)$, we characterize bounded Schur multipliers $φ\in L^\infty(Ω^2)$ such that the Schur multiplication operator $T_φ\colon B(L^2(Ω))\to B(L^2(Ω))$ is absolutely dilatable. In the separable case, they are characterized by the existence of a von Neumann algebra $N$ with a separable predual, equipped with a normalized normal faithful trace $τ_N$, and of a $w^*$-continuous essentially bounded function $d\colonΩ\to N$ such that $φ(s,t)=τ_N(d(s)^*d(t))$ for almost every $(s,t)\inΩ^2$.

Theorems & Definitions (26)

• Theorem 1.1
• Definition 2.1
• Proposition 2.3
• proof
• Lemma 3.2
• proof
• Lemma 3.4
• proof
• Theorem 4.1
• Lemma 5.1