Jensen's inequality in finite subdiagonal algebras

Abstract

Let $\mathcal{M}$ be a finite von Neumann algebra with a faithful normal tracial state $\tau$ and $\mathfrak{A}$ be a finite subdiagonal subalgebra of $\mathcal{M}$ with respect to a $\tau$-preserving faithful normal conditional expectation $\Phi$ on $\mathcal{M}$. Let $\Delta$ denote the Fuglede-Kadison determinant corresponding to $\tau$. For $X\in \mathcal{M}$, define $|X|:=(X^{\ast }X{)}^{\frac{1}{2}}$. In 2005, Labuschagne proved the so-called Jensen's inequality for finite subdiagonal algebras i.e. $\Delta (\Phi (A))\le \Delta (A)$ for an operator $A\in \mathfrak{A}$, thus resolving a long-standing open problem posed by Arveson in 1967. In this article, we prove the following more general result: $\tau (f(|\Phi (A)|))\le \tau (f(|A|))$ for $A\in \mathfrak{A}$ and any increasing continuous function $f:[0,\infty )\to \mathbb{R}$ such that $f\circ \exp$ is convex on $\mathbb{R}$. Under the additional hypotheses that $A$ is invertible in $\mathcal{M}$ and $f\circ \exp$ is strictly convex, we have $\tau (f(|\Phi (A)|))=\tau (f(|A|))$ $⟺\Phi (A)=A$. As an application, we show that for $A\in \mathfrak{A}$ the point spectrum of $A$ is contained in the point spectrum of $\Phi (A)$, though such a conclusion does not hold in general for their spectra.

12 pages. Minor changes in notation and presentation. An additional result (Theorem 5.5)

References in corpus (1)

• The Hadamard Determinant Inequality - Extensions to Operators on a Hilbert Space