Abstract:
We give a characterization of the two-weight inequality for a simple vector-valued operator. Special cases of our result have been considered before in the form of the weighted Carleson embedding theorem, the dyadic positive operators of Nazarov, Treil, and Volberg in the square integrable case, and Lacey, Sawyer, Uriarte-Tuero in the L^p case. The main technique of this paper is a Sawyer-style argument and the characterization is for 1 < p < \infty. We are unaware of instances where this operator has been given attention in the two-weight setting before.

Abstract:
In this note we prove Scurry's testing conditions for the boundedness of a sequence-valued averaging positive dyadic operator from a weighted Lp space to a sequence-valued weighted Lp space by using parallel stopping cubes.

Abstract:
Let $M$ be a $B$-probability space. Assume that $B$ itself is a $D$-probability space; then $M$ can be viewed as $D$-probability space as well. Let $X$ be in $M$. We look at the question of relating the properties of $X$ as $B$-valued random variable to its properties as $D$-valued random variable. We characterize freeness of $X$ from $B$ with amalgamation over $D$: (a) in terms of a certain factorization condition linking the $B$-valued and $D$-valued cumulants of $X$, and (b) for $D$ finite-dimensional, in terms of linking the $B$-valued and the $D$-valued Fisher information of $X$. We give an application to random matrices. For the second characterization we derive a new operator-valued description of the conjugate variable and introduce an operator-valued version of the liberation gradient.

Abstract:
Let $B$ be a finite, separable von Neumann algebra. We prove that a $B$-valued distribution $\mu$ that is the weak limit of an infinitesimal array is infinitely divisible. The proof of this theorem utilizes the Steinitz lemma and may be adapted to provide a nonstandard proof of this type of theorem for various other probabilistic categories. We also develop weak topologies for this theory and prove the corresponding compactness and convergence results.

Abstract:
Kullback-Leibler relative-entropy has unique properties in cases involving distributions resulting from relative-entropy minimization. Tsallis relative-entropy is a one parameter generalization of Kullback-Leibler relative-entropy in the nonextensive thermostatistics. In this paper, we present the properties of Tsallis relative-entropy minimization and present some differences with the classical case. In the representation of such a minimum relative-entropy distribution, we highlight the use of the q-product, an operator that has been recently introduced to derive the mathematical structure behind the Tsallis statistics. One of our main results is generalization of triangle equality of relative-entropy minimization to the nonextensive case.

Abstract:
We introduce two kinds of operator-valued norms. One of them is an $L(H)$-valued norm. The other one is an $L(C(K))$-valued norm. We characterize the completeness with respect to a bounded $L(H)$-valued norm. Furthermore, for a given Banach space $\textbf{B}$, we provide an $L(C(K))$-valued norm on $\textbf{B}$. and we introduce an $L(C(K))$-valued norm on a Banach space satisfying special properties.

Abstract:
The note shows that the operator-valued Hardy space $\sH^1$ introduced via Littlewood-Paley $g$-function coincides with the space of $H^1_R(\T, \sL^1)$ of all Bochner integrable operator-valued functions with integrable analytic part. The proof is based on the noncommutative maximal inequality for Poisson group.

Abstract:
We give a new characterization of the two weight inequality for a vector-valued positive operator. Our characterization has a different flavor than the one of Scurry's and H\"{a}nninen's. The proof can be essentially derived from the scalar-valued case.

Abstract:
We present a correspondence between positive operator valued measures (POVMs) and sets of generalized coherent states. Positive operator valued measures describe quantum observables and, similarly to quantum states, also quantum observables can be mixed. We show how the formalism of generalized coherent states leads to a useful characterization of extremal POVMs. We prove that covariant phase space observables related to squeezed states are extremal, while the ones related to number states are not extremal.

Abstract:
For k>6, we determine the minimal area of a compact hyperbolic surface, and an oriented compact hyperbolic surface that can be tiled by embedded regular triangles of angle 2\pi/k. Based on this, all the cases of equality in Laszlo Fejes Toth's triangle bound for hyperbolic surfaces are described.