In this article, we first study, in the framework of operator theory, Pusz and Woronowiczs functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $[0,infty)^2$. Our constru
ction has special features that functions on $[0,infty)^2$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in $(-infty,infty]$. Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on $(0,infty)$. We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Andos theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.
The Kubo-Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras it is impossible to avoid unbounded operators. In this article we try to extend a notion of connections
to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive $tau$-measurable operators (affiliated with a semi-finite von Neumann algebra equipped with a trace $tau$), (ii) positive elements in Haagerups $L^p$-spaces, (iii) semi-finite normal weights on a von Neumann algebra. Investigation on these generalizations requires some analysis (such as certain upper semi-continuity) on decreasing sequences in various classes. Several results in this direction are proved, which may be of independent interest. Ando studied Lebesgue decomposition for positive bounded operators by making use of parallel sums. Here, such decomposition is obtained in the setting of non-commutative (Hilsum) $L^p$-spaces.
A breakthrough took place in the von Neumann algebra theory when the Tomita-Takesaki theory was established around 1970. Since then, many important issues in the theory were developed through 1970s by Araki, Connes, Haagerup, Takesaki and others, whi
ch are already very classics of the von Neumann algebra theory. Nevertheless, it seems still difficult for beginners to access them, though a few big volumes on the theory are available. These lecture notes are delivered as an intensive course in 2019, April at Department of Mathematical Analysis, Budapest University of Technology and Economics. The course was aimed at giving a fast track study of those main classics of the theory, from which people gain an enough background knowledge so that they can consult suitable volumes when more details are needed.
We improve the existing Ando-Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie-Trotter formula and consider the extension problem of operat
or perspectives to non-invertible positive operators.
Let $mathbb{P}$ be the complete metric space consisting of positive invertible operators on an infinite-dimensional Hilbert space with the Thompson metric. We introduce the notion of operator means of probability measures on $mathbb{P}$, in parallel
with Kubo and Andos definition of two-variable operator means, and show that every operator mean is contractive for the $infty$-Wasserstein distance. By means of a fixed point method we consider deformation of such operator means, and show that the deformation of any operator mean becomes again an operator mean in our sense. Based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures.
We obtain limit theorems for $Phi(A^p)^{1/p}$ and $(A^psigma B)^{1/p}$ as $ptoinfty$ for positive matrices $A,B$, where $Phi$ is a positive linear map between matrix algebras (in particular, $Phi(A)=KAK^*$) and $sigma$ is an operator mean (in particu
lar, the weighted geometric mean), which are considered as certain reciprocal Lie-Trotter formulas and also a generalization of Katos limit to the supremum $Avee B$ with respect to the spectral order.
For $alpha,z>0$ with $alpha e1$, motivated by comparison between different kinds of Renyi divergences in quantum information, we consider log-majorization between the matrix functions begin{align*} P_alpha(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^alpha B^{1
/2}, Q_{alpha,z}(A,B)&:=(B^{1-alphaover2z}A^{alphaover z}B^{1-alphaover2z})^z end{align*} of two positive (semi)definite matrices $A,B$. We precisely determine the parameter $alpha,z$ for which $P_alpha(A,B)prec_{log}Q_{alpha,z}(A,B)$ and $Q_{alpha,z}(A,B)prec_{log}P_alpha(A,B)$ holds, respectively.
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 Haagerups $L^1$-space, we presen
t the general integral expression and the variational expression in terms of reverse tests. From these definition and expressions we prove important properties of maximal $f$-divergences, for instance, the monotonicity inequality, the joint convexity, the lower semicontinuity, and the martingale convergence. The inequality between the standard and the maximal $f$-divergences is also given.
We first develop a theory of conditional expectations for random variables with values in a complete metric space $M$ equipped with a contractive barycentric map $beta$, and then give convergence theorems for martingales of $beta$-conditional expecta
tions. We give the Birkhoff ergodic theorem for $beta$-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the $beta$-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on $M$. Finally, the large derivation property of $beta$-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.
We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosakis variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most o
f the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the Renyi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.
