Jensen's inequality in finite subdiagonal algebras

Abstract

Let $\mathscr{M}$ be a finite von Neumann algebra with a faithful normal tracial state $τ$ and $\mathfrak{A}$ be a finite subdiagonal subalgebra of $\mathscr{M}$ with respect to a $τ$-preserving faithful normal conditional expectation $Φ$ on $\mathscr{M}$. Let $Δ$ denote the Fuglede-Kadison determinant corresponding to $τ$. 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. $Δ(Φ(A)) \le Δ(A)$ 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: $τ(f( |Φ(A)|)) \le τ(f( |A|))$ 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 $τ(f( |Φ(A)|)) = τ(f( |A|))$ $\Longleftrightarrow Φ(A) = A$. As an application, we show that for $A \in \mathfrak{A}$ the point spectrum of $A$ is contained in the point spectrum of $Φ(A)$, though such a conclusion does not hold in general for their spectra.