Related papers: On $\alpha$-$z$-R\'{e}nyi divergence in the von Ne…

We introduce the notion of the $\alpha$-Haagerup approximation property for $\alpha\in[0,1/2]$ using a one-parameter family of positive cones studied by Araki and show that the $\alpha$-Haagerup approximation property actually does not…

We show that Araki and Masuda's weighted non-commutative vector valued $L_p$-spaces [Araki \& Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched R\'enyi divergences with parameter…

We study some structural aspects of the subspaces of the non-commutative (Haagerup) L_p-spaces associated with a general (non necessarily semi-finite) von Neumann algebra A. If a subspace X of L_p(A) contains uniformly the spaces \ell_p^n,…

We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosaki's variational expression of the relative entropy to an arbitary standard $f$-divergence, from…

We study the $\alpha$-$z$-R\'enyi divergences $D_{\alpha,z}(\psi\|\varphi)$ where $\alpha,z>0$ ($\alpha\ne1$) for normal positive functionals $\psi,\varphi$ on general von Neumann algebras, introduced in [S.~Kato and Y.~Ueda,…

We propose an extension of the sandwiched R\'enyi relative $\alpha$-entropy to normal positive functionals on arbitrary von Neumann algebras, for the values $\alpha>1$. For this, we use Kosaki's definition of noncommutative $L_p$-spaces…

Let $\mathcal{M}$ be a ($\sigma$-finite) von Neumann algebra associated with a normal faithful state $\phi.$ We prove a complex interpolation result for a couple of two (quasi) Haagerup noncommutative $L_p$-spaces $L_{p_0} (\mathcal{M},…

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be a maximal subdiagonal subalgebra of $\mathcal{M}$. We have proved that for $0< p<1$, $H^p(\mathcal{A})$ is…

Around 1967, Arveson invented a striking noncommutative generalization of classical $H^\infty$, known as {\em subdiagonal algebras}, which include a wide array of examples of interest to operator theorists. Their theory extends that of the…

As a continuation of the paper [20] on standard $f$-divergences, we make a systematic study of maximal $f$-divergences in general von Neumann algebras. For maximal $f$-divergences, apart from their definition based on Haagerup's…

On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q…

We study an extension of the sandwiched R\'enyi relative entropies for normal positive functionals on a von Neumann algebra, for parameter values $\alpha\in [1/2,1)$. This work is intended as a continuation of [A. Jen\v{c}ov\'a, Ann. Henri…

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be maximal subdiagonal algebra of $\mathcal{M}$. We prove Stein-Weiss type interpolation theorem of Haagerup…

We revisit Haagerup's enigmatic reduction theorem \cite[Theorems 2.1 \& 3.1]{HJX} showing how that theorem may be extended to general von Neumann algebras $\M$ equipped with an arbitrary faithful normal semifinite weight in a manner which…

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the…

We introduce a natural generalization of the Haagerup property of a finite von Neumann algebra to an arbitrary von Neumann algebra (with a separable predual) equipped with a normal, semi-finite, faithful weight and prove that this property…

Let $\mathfrak A$ be a type 1 subdiagonal algebra in a $\sigma$-finite von Neumann algebra $\mathcal M$ with respect to a faithful normal conditional expectation $\Phi$. We consider a Riesz type factorization theorem in noncommutative $H^p$…

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be maximal subdiagonal subalgebra of $\mathcal{M}$ and $1\le p<\8$. We prove a Beurling-Blecher-Labuschagne…

We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property.

Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's…