Jensen's inequality in finite subdiagonal algebras

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 $\mathrm{\Phi }$ on $\mathcal{M}$. Let $\mathrm{\Delta }$ denote the Fuglede-Kadison determinant corresponding to $\tau$. For $X\in \mathcal{M}$, define $|X| := (X^*X)^{\frac{1}{2}}$. In 2005, Labuschagne proved the so-called Jensen’s inequality for finite subdiagonal algebras i.e. $\mathrm{\Delta }(\mathrm{\Phi }(A))\le \mathrm{\Delta }(A)$ 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: $\tau (f(\mathrm{\mid }\mathrm{\Phi }(A)\mathrm{\mid }))\le \tau (f(\mathrm{\mid }A\mathrm{\mid }))$ for $A \in \mathfrak{A}$andanyincreasingcontinuousfunction$f : [0, \infty) \to $\mathbb{R}$$suchthat$f \circ \exp$isconvexon$$\mathbb{R}$$. Under the additional hypotheses that $A$ is invertible in $\mathcal{M}$ and $f \circ \exp$isstrictlyconvex,wehave$\tau(f( |\Phi(A)|)) = \tau(f( |A|))$ $⟺\mathrm{\Phi }(A)=A$. As an application, we show that for $A \in \mathfrak{A}$thepointspectrumof$A$iscontainedinthepointspectrumof$\Phi(A)$, though such a conclusion does not hold in general for their spectra.