Jensen's inequality in finite subdiagonal algebras

TL;DR

This paper generalizes Jensen's inequality within finite subdiagonal algebras of von Neumann algebras, extending previous results and exploring spectral properties of operators under conditional expectations.

Contribution

It proves a more general form of Jensen's inequality involving trace and convex functions, and characterizes equality conditions and spectral inclusion in this context.

Findings

Established a generalized Jensen's inequality for finite subdiagonal algebras.

Characterized conditions for equality involving invertibility and convexity.

Showed the point spectrum of an operator is contained in that of its conditional expectation.

Abstract

Let $\mathscr{M}$ be a finite von Neumann algebra with a faithful normal tracial state $\tau$ and $\mathfrak{A}$ be a finite subdiagonal subalgebra of $\mathscr{M}$ with respect to a $\tau$-preserving faithful normal conditional expectation $\Phi$ on $\mathscr{M}$. Let $\Delta$ denote the Fuglede-Kadison determinant corresponding to $\tau$. For $X \in \mathscr{M}$, define $|X| := (X^*X)^{\frac{1}{2}}$. 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 $\mathscr{M}$ and $f \circ \exp$ is strictly convex, we have $\tau(f( |\Phi(A)|)) = \tau(f( |A|))$ $\Longleftrightarrow \Phi(A) = A$. 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.